`. Properties defined through bracket notation are excluded from further analysis, as explained in [sect:preconditions]. The *bound properties* *B**i* of module *m**i*, henceforth, refer to all property bindings on the module object *e**i* with the dot notation. In case that the module object is instantiated through object literal notation, all properties appearing in the property-value pairs are, also, appended to the set *B**i*.\nThe module object *e**i* exported by module *m**i* may be constructed within *m**i* or imported from another module. Its instantiation may be declarative, through a function declaration or object literal notation, or may involve the evaluation of an expression (e.g. function invocation). A module object instantiated as (a) an empty object/function with properties attached at runtime, or (b) through object literal notation, is characterized by our method as a *namespace* object. The reason is that such objects serve as containers for data/ function properties that need to be exported by an AMD/CJS module. Namespace objects would be stripped of all their properties after refactoring, and, thus, are eliminated from the set of module features. Figure [fig:namespace-obj] presents a module from `goojs` project that follows the namespace object pattern. The module exports mathematical constants and functions and its original version is depicted in sub-figure (a). Sub-figures (b) and (c) provide syntactic variants that use an empty object or object literal notation.\n[t].5\n```\nfunction MathUtils() {}\n \n MathUtils.DEG_TO_RAD \n = Math.PI/180.0;\n /*... */\n MathUtils.radFromDeg = \n function(degrees){/*... */};\n \n MathUtils.degFromRad = \n function(radians){/*... */};\n /*... */\n module.exports = MathUtils;\n \n```\n[t].5\n```\nvar MathUtils = {};\n \n MathUtils.DEG_TO_RAD \n = Math.PI/180.0;\n /*... */\n MathUtils.radFromDeg = \n function(degrees){/*... */};\n \n MathUtils.degFromRad = \n function(radians){/*... */};\n /*... */\n module.exports = MathUtils;\n \n```\n[t].3\n```\nvar MathUtils = {\n \n DEG_TO_RAD: Math.PI/180.0,\n /*... */,\n radFromDeg: function(degrees){/*... */},\n degFromRad: function(radians){/*... */},\n /*... */\n };\n module.exports = MathUtils;\n \n```\n[fig:namespace-obj]\nLet *e**i* be the module object of a module *m**i* and *F**i* the identified module features. The module object destructuring algorithm computes the set *F**i* for a module *m**i* and can be summarized as following:\n1. Evaluate module object destructuring preconditions (see Section [sect:preconditions]). If any precondition fails then return *F**i* = {*e**i*}, i.e., the module object cannot be destructured.\n2. Identify the set *B**i* of bound properties to *e**i*,\n3. If *e**i* is a *namespace object* return *F**i* = *B**i*. Otherwise, return *F**i* = *B**i* ∪ {*e**i*}, i.e., the module object would be available as a module feature after destructuring\n#### Resolution of global declarations\nGlobal declarations involve explicitly declared and implied global variables, properties bound to platform specific global objects (e.g., `window`, `global`) and top-level declarations in non-modular ES5 source files. We identify the global variables of a module *m**i* through AST traversal and search for (a) top-level `var` statements (non-modular ES5/AMD) and (b) global object property definitions. Implied globals are identified through analysis of AST nodes that represent assignment expressions. Specifically, the variable in the left-hand side of an assignment is sought in the sets of the variables/functions defined within the scope hierarchy formed by the assignment’s scope and its surrounding scopes. The variable comprises an implied global if it is not declared in the hierarchy. Finally, top-level declarations in non-modular ES5 are identified by locating function and variable declaration nodes which do not have function declaration ancestors.\nLet *m**i* be a module and *G**i*, *O**i*, *T**i* refer, respectively to its global variables, global object properties and top-level declarations identified in its AST. The resolution of module features from global declarations involves the following steps:\n1. Evaluate global declaration preconditions (see Section [sect:preconditions]) on *G**i*, *O**i*, *T**i*. In case of any violation return *F**i* = ∅.\n2. For each identified global object property *o**k* ∈ *O**i* search for its presence in the feature set of other modules\n\t1. If *o**k* is also defined in module *m**j*, allocate *o**k* to the module with lower number of imports, so as to normalize coupling among modules,\n\t2. If not found, then *F**i* = *F**i* ∪ {*o**k*}.\n3. Return *F**i* = *F**i* ∪ *G**i* ∪ *T**i*.\nIn case that global declarations’ resolution returns *F**i* = ∅ for any module *m**i*, then the refactoring is abandoned.\n### Resolution of fine-grained module dependencies\nThe resolution of the fine-grained module dependencies involves the identification of the module features that each module needs to import from other modules of the same project. Moreover, it identifies dependencies to external library modules that are treated differently for compatibility reasons.\nLet *d**j* ∈ *D* be a module dependency from module *m**i* to a project module *m**t*. A module dependency represents the import and use in a module *m**i* of a module feature *f**t*, *k* ∈ *F**t*, i.e., exported from module *m**t*. A module dependency is, also, characterized by the type of use *u**j* ∈ {*R*, *W*, *C*, *L*} of *f**t*, *k* within *m**i*. If *f**t*, *k* is only read or called in *m**i*, then *u**j* has a value of *R* or *C*, respectively, If *f**t*, *k* is, also, defined in *m**i* then *u**j* = *W*. Thus, a module dependency is defined as a tuple *d**j* = (*m**i*, *m**t*, *f**t*, *k*, *u**j*). In case that imported module *m**t* corresponds to an external library, the dependency is represented as *d**j* = (*m**i*, *m**t*, ∅, *L*).\nIn CommonJS, we identify imported modules through AST traversal and search for `require` invocations. A typical `require` invocation for a module *m**j* is equivalent to an ES6 namespace import, i.e. the module object exported by *m**j* is returned and assigned to a variable that provides access to the entire feature set *F**j* of *m**j*. A specific property of the imported module *m**j* may also be directly assigned to a variable, through property access on `require` result. This coding idiom emulates an ES6 named import in CommonJS and the receiving variable provides access to a single feature *f**k* ∈ *F**j*. Finally, a module may be imported only for its side effects, i.e., the `require` result is not assigned to any variable.\nIn the case of AMD, we identify imported modules by analyzing the parameters of the `define` module declaration function. Specifically, the first – optional – parameter of the `define` invocation provides a list with the relative paths or identifiers of imported modules. Each module *m**j* is resolved and the module object is provided as parameter to the module definition function (second parameter of the `define` invocation). AMD imports are namespace imports, since parameters of the module definition function provide access to the imported modules’ entire feature set.\nThe identification of module dependencies for each module *m**i* is based on the following algorithm:\n1. For each imported system module *m**j* (AMD/CJS) identify the subset of its module features *A**j* ⊆ *F**j* used in *m**i*\n\t1. Let *u**k* be the type of use of each *a**k* ∈ *A**j*,\n\t2. For each *a**k* ∈ *A**j* add a module dependency, i.e. *D* = *D* ∪ {(*m**i*, *m**j*, *a**k*, *u**k*)}.\n2. For each imported library module *m**j* (AMD/CJS) add a module dependency, i.e. *D* = *D* ∪ {(*m**i*, *m**j*, ∅, *L*)}.\n3. For each access to a global declaration $g\\_k \\notin F\\_i$\n\t1. Let *u**k* be the type of use of *g**k*. Identify the module *m**j* that encapsulates the global declaration, i.e., *g**k* ∈ *F**j*,\n\t2. Add a new module dependency *D* = *D* ∪ {(*m**i*, *m**j*, *g**k*, *u**k*)}.\n### MDG construction algorithm\nOn the basis of the aforementioned definitions and algorithms the construction of the MDG data structure for an ES5 project is summarized below:\n1. Initialize sets *M* = ∅, *D* = ∅\n2. For each source file *f**i* with name *n**i*\n\t1. Evaluate module format preconditions on *f**i* (see Section [sect:preconditions]). In case of any violation, refactoring of *f**i* is abandoned.\n\t2. Resolve the module structure *F**i*. If *F**i* ≠ ∅ then *M* = *M* ∪ {(*n**i*, *F**i*)} else refactoring of *f**i* is abandoned.\n3. For each module *m**i* ∈ *M*\n\t1. Resolve module dependencies *D**i* against modules *M*ʹ = *M* \\ {*m**i*},\n\t2. *D* = *D* ∪ *D**i*\n4. Return (*M*, *D*).\nFigure [fig:mdg-example] presents a fragment of the MDG constructed during this stage for the project `planck.js`. The fragment focuses on module `Math`, its outgoing dependencies and a subset of its incoming dependencies from other modules. The module’s outgoing dependencies correspond to the module features that are imported and used in the module, while incoming dependencies correspond to the module’s features that are exported to client modules. For instance, `Math` imports the feature `create` from the module `create`, while it exports the feature `isFinite` to the client modules `Fixture` and `Body`.\n [fig:mdg-example]\n### Refactoring preconditions\nRefactoring preconditions are evaluated by the refactoring identification algorithm on each module of the analyzed system. They prevent evaluation errors in the refactored code and ensure that the external behaviour of the system is preserved. The identified preconditions, depending on their scope of application, are grouped in three categories: (a) global declarations’ preconditions, (b) module object destructuring preconditions, (c) module format preconditions.\n*Global declarations’ preconditions* involve defining global variables and top-level functions with the same name in different modules:\n1. Each global variable should be introduced in a single location. Conflicting declarations across different files are a potential source of errors.\n2. In non-modular ES5 or AMD, the names of top-level functions should be unique across all JavaScript files. Different top-level functions with the same name in multiple modules lead to function overriding; the binding in the module’s namespace corresponds to the most recently loaded function.\n*Module object destructuring preconditions* involve extracting the properties of the module object that can be independently exported, while preserving the program’s external behaviour:\n1. Object properties should be defined with dot notation. Properties defined with bracket notation may lead to inaccurate conclusions regarding their declaration module, in cases that the object is indexed by a variable.\n2. Module objects should not be fully referenced (e.g. provided in function invocations). Detaching object properties from these objects may lead to errors originating from name aliasing.\n3. Module objects should not be modified. Detaching properties from such objects may lead to errors related to violations of the ES6 specification .\n*Module format preconditions* involve restrictions that are imposed by the language specification. They, also, involve restrictions that are not imposed by ES5 formats, but are introduced in the ES6 specification. The violation of these restrictions may introduce defects in the refactored code or affect the program’s external behaviour:\n1. Function definitions that reference `this` should be declared in files that enforce strict mode. In case that a function is neither an object method nor a constructor, the value of `this` is determined by the module’s code mode. In *non-strict* mode, `this` references the global object. In *strict* mode `this` is `undefined`, as its value is not updated during the function invocation. Since ES6 modules enforce *strict* mode, the transformation of an ES5 source file in *non-strict* mode into an ES6 module would lead to runtime errors originating from the value of `this`.\n2. Invocations of `require` in CJS modules should be located in the module’s top-level scope. Notice that in ES6, the introduction of static imports and exports is allowed only at the module scope. In-place transformation of `require` calls to ES6 static imports would lead to evaluation errors, if they are nested in block statements. This, also, applies to the replacement of return statements with export statements in AMD.\nSource Code Transformation Procedure\n------------------------------------\nThe source code transformation for migration to ES6 modules employs the MDG constructed in the refactoring candidate identification stage. Each MDG node, corresponding to a module *m**i*, is individually refactored in the following steps:\n#### Step 1\nIf *m**i* is an AMD module, clear its dependencies to the AMD API:\n1. Replace the invocation of the `define` function with a list of all statements included in the body of the module definition function.\n2. Replace the return statement included in the statement list with a variable assignment of the form: `var mod_<filename> = <return_expr>`, where `<return_expr>` is the expression returned by the replaced statement.\nThe application of this step to an AMD module is illustrated in Figure [fig:module-migration-amd]. The affected statements are enclosed in a dashed frame and are marked with the step number (1). In the rest of this section, numbers inside parentheses will be used to refer to similarly marked parts of Figures [fig:module-migration-amd], [fig:mathJsRef].\n#### Step 2\nIf *m**i* is a CJS module, clear dependencies to the CommonJS API:\n1. Declare an empty placeholder variable for the module object with the following naming scheme: `var mod_<filename> = null`.\n2. Replace all references to `module.exports` or its alias `exports` with the above variable.\n3. Eliminate all `require` statements.\n[hbt!] [fig:module-migration-amd]\n[hbt!] [fig:mathJsRef]\nFigure [fig:mathJsRef] presents the application of this step to the `Math.js` module of the `planck.js` project. The newly introduced variable `mod_Math`, replaces `module.exports` and holds the reference to the module object in refactored code (2). Variable `math` remains an alias to the module object.\n#### Step 3\nApply module object destructuring by processing each identified feature *f**j* ∈ *F**i* for the module *m**i*. Each feature *f**j* is declared and initialized in ES5 through a property-value pair in object literal notation or through an assignment statement that binds a property to the module object. In order to extract *f**j* from the module object:\n1. Introduce a variable declaration for *f**j* that has the form:\n```\nvar <feature> = <feature_expr>\n```\nwhere `<feature>` corresponds to the name of *f**j*. In case of name conflicts, the feature name is prefixed with the module name, i.e., `<file_name>_`. The term `<feature_expr>` corresponds to the expression used for initialization of *f**j* during its declaration.\n2. Replace references to *f**j* in module *m**i* with the feature name.\n3. Remove the property-value pair or the property binding assignment statement, originally used for *f**j* declaration.\nThe declaration of module features in the ES5 version of `Math.js` (left part of Figure [fig:mathJsRef]), is based on property binding assignments to the `math` alias of the module object. All property bindings can be extracted as accordingly named variable declarations (3), with the exception of `isFinite` that introduces name collision with a global JavaScript function.\n#### Step 4\nDeclare the exported module features of *m**i* through evaluation of all its incoming MDG dependencies.\n1. Initialize exported features *E**i* = ∅\n2. For each incoming MDG edge (*m**k*, *m**i*, *f**j*, *u**j*), originating from a client module *m**k*, add *f**j* to the set of exported features *E**i*. In case that the client module modifies the value of *f**j*, i.e., *u**j* = *W*, declare a mutator function for *f**j* and add it to exported features *E**i*. The following code listing provides a template for mutator function declaration.\n```\nfunction set_<feature>(value) { <feature> = value; }\n```\n3. Declare an export statement listing all exported features in *E**i*. For name consistency, provide an alias and restore the original name of all features that were prefixed for handling name conflicts.\nThe AMD module of Figure [fig:module-migration-amd] provides a single export (4), since the module object could not be further destructured. On the other hand, the CJS module of Figure [fig:mathJsRef], exports a list of features in a single statement (4). In both cases, renamed features are restored to their original version.\nNotice that Step 4 may be adapted in case that the analyzed project is not a standalone application, e.g., it implements a reusable library. The reason is that the part of a library’s public API without incoming MDG dependencies in the codebase, will be accidentally encapsulated and become unavailable to its clients. Step 4, in this case, exports all features *f**j* ∈ *F**i* for each module *m**i*. However, if the public API is invoked in test code, Step 4 can be applied without modifications, as test code may provide the required incoming dependencies.\n#### Step 5\nEstablish module dependencies for *m**i* through evaluation of all its outgoing MDG dependencies.\n1. For each outgoing MDG edge (*m**i*, *m**k*, *f**j*, *u**j*) to a module feature *f**j* of a system module *m**k*, introduce an appropriate named import. The name of *f**j* (`<feature>`) is maintained, or in case of name conflicts it is prefixed with a module identifier (`<prefix>`). The identifier corresponds to the file name of *m**k*, or if the conflict persists it is further prefixed with the name of the parent folder of the module declaration and so on. In case that the imported feature is defined in *m**i*, i.e., *u**j* = *W*, the corresponding mutator function is, also, imported. The introduced named imports follow the pattern:\n```\nimport {<feature> [as <prefix>_<feature>]} from <file>;\n```\n2. For each imported feature *f**j*, replace all references to it with the feature name. In case of assignment statements that define the value of *f**j*, they are replaced with an invocation of the feature’s mutator function.\n3. For each outgoing MDG edge (*m**i*, *m**k*, ∅, *L*) to an external library module, introduce a default import, i.e.,\n```\nimport <lib> from <file>;\n```\nwhere `<lib>` corresponds to the variable name used to reference the library module in ES5 code.\nFigure [fig:module-migration-amd] presents the import of two library modules through *default* import statements (5). In Figure [fig:mathJsRef], the `require` statements are replaced by two named imports that introduce the minimum required module features to the namespace of `Math.js` (5).\nEmpirical evaluation\n====================\nWe have designed and performed an experimental study on the proposed method for automated refactoring of ES5 software projects towards ES6 fine-grained modularity. The *goal* of this study is to analyze the proposed method for the *purpose* of evaluation with respect to effectiveness and practicality from the *perspective* of a developer, that considers the implications of the refactoring and its potential for automation, in the *context* of a set of open source projects.\nResearch Questions\n------------------\nThe experimental study aims at answering the following research questions:\n*RQ1. How prevalent is the presence of ES5 code, modular or not, in real-world applications?* At first, we investigate to which extent ES5 code, that is either non modular or supports the AMD/CJS module formats, is still prevalent in web applications. The results of this study will provide empirical evidence on the scope and practicality of the proposed method.\n*RQ2. Is there a developer intent for independently reusable features within modules?* Our purpose is to explore at which extent the proposed refactoring to ES6 fine-grained modularity is aligned with the semantics of the analyzed modules and the developer’s intent for independently reusable features within modules. The results will support the practicality of the proposed refactoring for effective usage of ES6 language constructs after migration of legacy code.\n*RQ3. What are the effects of the proposed refactoring to system modularity?* Our aim is to study the level of granularity, in terms of individually exported module features, that can be introduced in project modules and the potential for encapsulation of module features within their declaring modules. The research question, also, evaluates potential reduction to module coupling due to fine-grained dependencies introduced by ES6 named imports. The results of this research question will highlight the effectiveness of the proposed method for increasing the granularity of reusable module features and reducing the internal coupling of system modules.\n*RQ4. Does the proposed source code transformation preserve the external behaviour of the analyzed system?* The research question seeks to empirically investigate the soundness of the proposed refactoring, i.e., whether the application of the refactoring does not introduce syntax errors or alter the external behaviour of refactored code. The results of this study will further support the practicality of the proposed method.\nContext Selection\n-----------------\nThe context of the experimental study comprises 19 JavaScript projects. The selection of these projects is based on the following criteria: (a) they are open source, for study replication reasons, (b) they are implemented in ES5 or below, (c) they are either non-modular or use the AMD/CJS module formats as implemented in RequireJS/Node.js respectively, (d) their test suite, if available, can be successfully executed, (e) to the extent possible, they are used as benchmarks in earlier work in this area, e.g., .\n[htb]\nl l l l l l & & & & & \nUltraTetris & & 326 & - & master@5ad237e & silviolucenajunior/UltraTetris \nHangman & & 346 & - & master@b950842 & aurobindodebnath/Hangman \nTicTacToe & & 919 & - & master@346ebe8 & seanpr/TicTacToe \nuki & & 2360 & - & master@6cd2e47 & crcx/uki \nGluttonousSnake & & 378 & - & master@c6b49cb & yyfer/GluttonousSnake \nastix & & 481 & - & master@f0ecc39 & migace/astix \ngame-of-life & & 887 & - & master@1d83874 & devlysh/game-of-life \ntetrisJS & & 1370 & - & master@27712a0 & marneborn/tetrisJS \ndynablaster-js-port & & 3679 & - & master@5b5052e & gardziej/dynablaster-js-port \nbackbone-tableview & & 279 & & master@0c26357 & mbrevda/backbone-tableview \neasystarjs & & 369 & & v0.4.3 & prettymuchbryce/easystarjs \ngeojsonhint & & 473 & & v.2.0.0 & mapbox/geojsonhint \nexpress-session & & 612 & & v1.15.6 & expressjs/session \nunderscore.string & & 873 & & 3.3.4 & esamattis/underscore.string \nmessy & & 1694 & & 6.11.0 & papandreou/messy \nvirtual-dom & & 1978 & & v2.1.1 & Matt-Esch/virtual-dom \nrecipe-parser & & 4134 & & master@626f124 & catesandrew/recipe-parser \nplanck.js & & 10390 & & v0.2.7 & shakiba/planck.js \ngoojs & & 57373 & & v0.16.8 & GooTechnologies/goojs \n[Tab:projectImplementationDetails]\nTable [Tab:projectImplementationDetails] presents the projects used in our evaluation, along with relevant implementation details. Column 2 provides the module format used in each project. Notice that the first four projects are non-modular. Column 3 (SLOC) includes the total lines of JavaScript production code, computed with CLOC[3](#fn3). In case that the codebase of a project includes external libraries, library code is excluded from further analysis. The reason is that these libraries make use of dynamic features of the language (e.g. on-the-fly creation and deletion of object properties, dynamic code generation). These features undermine the results of static analysis and, thus, their refactoring may introduce errors or influence the program’s external behaviour. Column 4 provides the test coverage of each project, in terms covered statements by its test suite. Test coverage is not reported for projects without a test suite. Column 5 specifies the tag/commit of the project’s revision history that was used in this study and Column 6 provides the projects’ Github repository names.\nThe proposed method is implemented as an open source Node.js tool [4](#fn4), which employs jscodeshift for Abstract Syntax Tree (AST) traversal and transformation, JSHintfor the resolution of implied globals and Ternfor the dataflow analysis required in the module dependency identification step ([sect:resolve-module-deps]). The JavaScript code that is embedded in the HTML files of the analyzed web applications is extracted with the help of the jsoup HTML parserThe implemented prototype requires limited user interaction, since the user only needs to specify the application to be refactored and the format that is used for its implementation. The evaluation of the prototype is performed on a workstation, equipped with an Intel-Core i7-7700 CPU @3.60 Ghz and 16GB RAM.\nEvaluation Results\n------------------\n### RQ1. How prevalent is the presence of ES5 code, modular or not, in real-world applications?\nThe first research question explores how relevant is the proposed method with the needs of current web applications. Since the ES6 (ES2015) specification has been fully implemented in most JavaScript runtimes, a question naturally arises as to what extent web applications still use the legacy module formats and have not yet migrated to ES6. To the best of our knowledge, there is no published study that explores the use of AMD/CJS formats and the current adoption of ES6 modules.\nIn order to answer RQ1, we analyzed the code-base of all Github repositories that use JavaScript as their main language and have received at least a minimum of developer engagement in the recent years. We acquired access to the source code of the main branch and the public activity timeline of Github repositories, through the *Github Activity Data* [5](#fn5) and *GithubArchive* [6](#fn6) public BigQuery datasets. Among approximately 4 million repositories that include JavaScript code, we narrowed down our analysis to 14,167 repositories on the basis of the following criteria:\n* the primary language of the repository, in terms of bytes of code, is JavaScript or HTML. We excluded repositories that include Java, Objective-C or C# code, in order to avoid analyzing cross-platform mobile applications, where migration to ES6 is constrained by application development frameworks.\n* each repository has received at least 10 commits and 5 stars since 01.01.2018. Our purpose is to exclude abandoned projects that would bias our results in favor of legacy module formats.\nIn a next stage, we collected all JavaScript files from the aforementioned repositories and classified them as AMD, CJS or ES6 modules. The classification is based on regular expressions that match the syntax of each module format in the source code of each file. Regular expressions are appropriately formulated so that each file is categorized to a single module format and are executed against a BigQuery table with JS file contents. We excluded from our analysis build tool configuration files (e.g. webpack, rollup), as well as source files located at the installation folder of package dependencies (e.g., node\\_modules).\nThe results of JS content classification to module formats are available in Table [Tab:jsmodules-distr]. The table provides the number of JS files (Column 2), among all 14,167 repositories, that declare AMD, CommonJS, ES6 modules or do not use any standard module format. Column 3 estimates the share of all files that fall under these categories. Classification results show that a large share of JS content () is either non-modular or does not use a standard module format. Moreover, about one-third of all JS files () use the standard ES5 module formats, AMD () and CJS (). Finally, the adoption of the ES6 module format characterizes another one-third of all files ().\n[htb]\nl r r\n**Module format** & & \nAMD & 65,457 & 6.7 \nCommonJS & 281,874 & 28.7 \nES6 & 327,367 & 33.3 \nOther & 307,523 & 31.3 \nTotal & 982,221 & 100.0 \n[Tab:jsmodules-distr]\nLegacy module formats and non-modular ES5 code are still widely used in actively maintained open source projects. This stresses the need for methods and tools for their automated migration to ES6 modules in order to reap the benefits of new language features.\n### RQ2. Is there a developer intent for independently reusable features within modules?\nThe second research question is related to the practicality of the proposed refactoring. It’s purpose is to investigate whether the extracted fine-grained module features correspond to elements with similar semantics in the original module design, i.e., elements that belong to the module scope and may be reused independently from each other. Thus, the language constructs (ES6 named imports/exports) that are automatically applied through refactoring, could have been employed by the programmer if they were supported by ES5.\nIn order to answer RQ2, we conducted an exploratory study on frequently occurring patterns on the design of module objects. The context of this study comprised the subset of benchmark projects that use the AMD/CJS formats. Non-modular ES5 projects were excluded, since their source files do not declare module objects. The study involved, at first, the identification of common patterns through code inspection on a sample of modules from each benchmark project. Then, we estimated the frequency of each pattern, across all modules, through a static analysis tool implemented for that purpose. Besides the *namespace object* pattern, already described in [sect:resolve-mod-features], code inspection revealed several module object patterns that can be grouped in two categories: (a) *factory* objects and (b) *utility* objects. The categorization focuses on the purpose of module objects and abstracts variations in their design.\nA *factory object* is a function that implements a template for object creation and initialization. The function is usually a *function constructor*, a factory method or a generic object configuration function. Its purpose is to initialize a new object with attributes and methods through statements in the function body (property bindings and property assignments on `this` reference). Instance methods are, also, attached through property bindings to the factory object’s `prototype` object. Moreover, the module code, usually, includes statements that apply property bindings directly on the factory object. These bindings are extracted as independent module features by the proposed refactoring. They represent static members of the class declaration that is emulated within the module, and can be reused independently from the instances created by the factory object.\nFigure [fig:factory-object] presents a factory object declared in `Vec2` module of `planck.js` project. The module object acts as a function constructor or as a factory method, depending on being invoked with the `new` operator or not. An instance method of `Vec2` objects is declared in lines 21–23. On the other hand, direct property bindings to `Vec2` object, e.g., `zero`, `neo`, represent static utility methods that can be extracted and exported as individual module features.\n```\n/*... */\nmodule.exports = Vec2;\n/*... */\nfunction Vec2(x, y) {\n if (!(this instanceof Vec2)) {\n return new Vec2(x, y);\n }\n if (typeof x === 'undefined') {\n this.x = 0;\n this.y = 0;\n }\n /*... */\n}\nVec2.zero = function() {/*...*/};\nVec2.neo = function(x, y) {/*...*/};\nVec2.clone = function(v) {/*...*/};\nVec2.prototype.toString = function() {\n return JSON.stringify(this);\n};\nVec2.isValid = function(v) {/*...*/}\nVec2.assert = function(o) {/*...*/};\n/*... */\n```\n[fig:factory-object]\nIn the case of *utility objects*, the module object is a non-empty function that serves a domain-specific functionality and does not include bindings to the `this` reference. Furthermore, its prototype object is not extended within the module scope. Bound properties to this module object are, also, extracted and exported as separate module features after refactoring. However, due to the generic structure of *utility objects*, we cannot generalize on the role of extracted module features in module design. Thus, we assume that the individual reuse of these module features might not express the original intent of the developer.\n[htb]\n@ l r r r @ & \n & & & \nGluttonousSnake & & & \nastix & & & \ngame-of-life & & & \ntetrisJS & & & \ndynablaster-js-port & & & \nbackbone-tableview & & & \neasystarjs & & & \ngeojsonhint & & & \nexpress-session & & & \nunderscore.string & & & \nmessy & & & \nvirtual-dom & & & \nrecipe-parser & & & \nplanck.js & & & \ngoojs & & & \n**Total** & & & \n[Tab:moduleStructure]\nTable [Tab:moduleStructure] presents static analysis results on the classification of all module objects in AMD/CJS projects to module pattern categories. Specifically, Columns 2–4 provide for each project the number of module objects that fall, respectively, into the factory, namespace and utility object categories. The results show that the majority of module objects, i.e.,, are classified as factory objects. Moreover, a smaller but significant share of module objects () are classified as namespace objects, while utility objects correspond to of all module objects. Thus, a total of of module object declarations follow the *namespace* or *factory* object pattern, i.e., their module scope emulates, either a container for features or a class. In each one of these declarations, the module object’s bound properties represent features that reside semantically at the module scope and can be individually reused as ES6 named exports. Bound properties in the rest of module declarations, i.e, in *utility* objects, may also be reusable as individual features, but their relevance must be inspected on a case-by-case basis.\nBound properties in of module object declarations correspond to namespace elements or static features of an emulated class, revealing a developer intent for their individual reuse at module level.\n### RQ3: What are the effects of the proposed refactoring to system modularity?\nIn order to answer RQ3, we evaluate the module structure after refactoring, where module objects are destructured and replaced by individually reusable module features. We investigate the reusability of module features across modules, as well as the impact of fine-grained dependencies to module coupling.\n[htb]\n@l r r r r & & \n & & & & \nUltraTetris & 5 & 15 & & \nHangman & 9 & 44 & & \nTicTacToe & 2 & 12 & & \nuki & 3 & 39 & & \nGluttonousSnake & 4 & 10 & & \nastix & 10 & 49 & & \ngame-of-life & 7 & 23 & & \ntetrisJS & 11 & 41 & & \ndynablaster-js-port & 49 & 259 & & \nbackbone-tableview & 12 & 19 & & \neasystarjs & 3 & 11 & & \ngeojsonhint & 3 & 3 & & \nexpress-session & 5 & 9 & & \nunderscore.string & 97 & 191 & & \nmessy & 16 & 111 & & \nvirtual-dom & 31 & 45 & & \nrecipe-parser & 16 & 65 & & \nplanck.js & 55 & 285 & & \ngoojs & 540 & 2045& & \n**Total** & 878 & 3276 & & \n[Tab:exportedFeatures]\nThe effect of the proposed refactoring on the granularity of independently reusable elements within a project, can be inferred by Columns 2, 3 of Table [Tab:exportedFeatures]. Column 2 (Modules) provides the number of module objects declared within each project, while Column 3 (Total Module Features) sums up all module features, i.e., (a) bound properties that are extracted from each module object and (b) global declarations that are allocated and accessed through modules after refactoring. Initially, the units of reuse were coarse-grained module objects, each one declared in its own source file. The proposed method sets the unit of reuse at the granularity of module features, with multiple feature declarations and exports per source file. The figures in Columns 2 and 3 show that this shift in granularity results to an increase in the number of reusable elements per project by a factor of 4 on average. This increase in fine-grained reusable elements reduces module coupling, as will be demonstrated hereafter, while supporting application bundling tools in effective tree-shaking, i.e., identification and elimination of unused code from deployment artifacts .\nThe module features that are, actually, reusable within each project are provided in Column 4 (Exported Module Features). These module features are required and used by at least one module, besides their declaring module. After refactoring, they are declared as *named exports* and are reused in client modules through corresponding *named import* statements. Moreover, the column provides, inside parentheses, the respective fraction of reusable module features against total module features of the project (Column 2). According to figures in Column 4, of all module features are reusable and should be exported, while, in 13 out of 19 projects, these module features exceed.\nHowever, a significant part of module features, especially in non-CommonJS projects, may not be exported but encapsulated in their declaring modules. Notice that CJS projects implement Node.js library packages and, thus, their encapsulated features are estimated by integrating in the static analysis their unit tests, in order to act as client code of their public API. In case that unit tests do not provide sufficient coverage of the public API, encapsulation of module features can be disabled. As revealed by results in Column 5, 20.6% of all module features and 14.3%–87.2% in non-modular ES5 and AMD projects are not reused outside their declaring module. In any case, the results provide evidence on the poor support for module encapsulation in ES5. The proposed refactoring omits ES6 named export statements for these features, restricting their scope and narrowing the public API of modules. Thus, it improves modularity through better encapsulation.\nAs concerning the effect of refactoring on module coupling, we investigate it by comparing the average module coupling of each project before and after the application of refactoring. Module coupling estimation is based on the incoming dependencies (*Fan-in*) and the outgoing dependencies (*Fan-out*) of a module. Moreover, we estimate *Module Instability* in analogy to the well-established *Instability* metric for components or packages . We draw an analogy between an ES6 module and a Java package [7](#fn7) in order to estimate module *Fan-in* and *Fan-out*. A Java package groups classes (and other types or packages) and makes them accessible to other packages through the `public` class modifier. Its outgoing dependencies comprise classes of other packages that are imported by its own classes, while incoming dependencies comprise classes of other packages that import one or more of its own classes. In a similar vein, an ES6 module groups module features, that are declared in its source file, and become accessible to other modules through export statements. An ES6 module has outgoing dependencies to module features that are imported individually or in bulk (named or default imports) from other modules. Incoming dependencies of a module comprise the module features of other modules that import one or more of its own features.\nLet *m**i* be a module, *F**i* its module features and *D**i* its set of outgoing dependencies in the MDG, according to the notation introduced in Section [refactoringCandidateIdentificationProcedure]. Moreover, let *F**i*, *o**u**t* be the set of module features that *m**i* depends on and imports into its source file (external library modules are excluded). The following equation provides a formal specification for *F**i*, *o**u**t*:\n \n$$\\begin{split}\nF\\_{i, out} = & \\bigcup \\{f\\_{t,k}\\ |\\ \\exists d\\in D\\_i\\ s.t. & d=(m\\_i,m\\_t,f\\_{t,k},\\\\_)\\ \\wedge\\ \\\\\n & & m\\_t\\in M \\wedge f\\_{t,k}\\in F\\_t\\} \\\\\n\\end{split}$$ \nThe set *F**i*, *i**n* of module features that depend on module *m**i* contains all module features whose declaration references at least one module feature from *m**i*. Let *m**s* be a module with outgoing dependencies to *m**i* in the MDG, i.e., it imports one or more features from *m**i*. Moreover, let *A**i*, *s* ⊆ *F**s* be the module features of *m**s* that access at least one imported feature from module *m**i*. We compute *F**i*, *i**n* as the union of all sets *A**i*, *s*, where *m**s* is a module that imports features from *m**i*. A formal specification of *F**i*, *i**n* is provided below:\n \n$$\\begin{split}\nF\\_{i, in} = & \\bigcup A\\_{i,s}, \\text{for each}\\ m\\_s \\in M\\ s.t. \\\\\n & (\\exists d\\in D\\_s\\ |\\ d=(m\\_s,m\\_i,f\\_{i,k},\\\\_)\\ \\wedge\\ f\\_{i,k}\\in F\\_i) \\\\\n\\end{split}$$ \nWe estimate metrics *Fan-out* (FO*i*), *Fan-in* (FI*i*) and *Module Instability* (I*i*) for each module *m**i*, on the basis of *F**i*, *o**u**t*, *F**i*, *i**n*:\n \n$$\\begin{aligned}\n\\text{FO}\\_i & = |F\\_{i, out}|, \\text{FI}\\_i = |F\\_{i, in}\\ | \\\\\n\\text{I}\\_i & = \\frac{|F\\_{i, out}|}{|F\\_{i, out}| + |F\\_{i, in}|}\\end{aligned}$$ \n[tb] [tbl:module-coupling]\n@l r r r r r r@ & & & \n& & & & & & \nUltraTetris & 3.20 (0.45) & 0.80 (1.30) & 0.80 (0.45) & 0.80 (0.11) & 0.35 (0.49) & 60 (55) \nHangman & 21.33 (2.29) & 2.56 (3.57) & 9.11 (13.15) & 0.79 (0.25) & 0.37 (0.42) & 58 (44) \nTicTacToe & 2.50 (3.54) & 1.50 (2.12) & 0.50 (0.71) & 0.50 (0.71) & 0.50 (0.71) & 0 ( 0) \nuki & 10.00 (1.00) & 2.67 (3.79) & 1.00 (1.00) & 0.91 (0.09) & 0.44 (0.51) & 53 (50) \nGluttonousSnake & 1.75 (2.87) & 1.75 (2.87) & 0.75 (0.50) & 0.46 (0.54) & 0.46 (0.54) & 0 ( 0) \nastix & 12.80 (13.61) & 10.40 (10.95) & 2.60 (3.10) & 0.71 (0.33) & 0.64 (0.44) & 25 (46) \ngame-of-life & 0.71 (1.11) & 0.71 (1.11) & 1.43 (1.62) & 0.32 (0.43) & 0.32 (0.43) & 0 ( 0) \ntetrisJS & 10.27 (5.50) & 1.18 (2.04) & 1.91 (2.30) & 0.77 (0.31) & 0.37 (0.39) & 51 (42) \ndynablaster-js-port & 3.27 (3.83) & 2.02 (2.76) & 2.73 (4.30) & 0.52 (0.39) & 0.44 (0.37) & 11 (19) \nbackbone-tableview & 2.08 (2.94) & 1.42 (2.31) & 1.08 (1.00) & 0.39 (0.41) & 0.39 (0.41) & 0 ( 0) \neasystarjs & 0.67 (1.15) & 0.67 (1.15) & 0.67 (0.58) & 0.33 (0.58) & 0.33 (0.58) & 0 ( 0) \ngeojsonhint & 0.50 (0.58) & 0.50 (0.58) & 0.50 (0.58) & 0.50 (0.50) & 0.50 (0.50) & 0 ( 0) \nsession & 1.40 (1.67) & 1.00 (1.22) & 8.80 (6.38) & 0.24 (0.43) & 0.23 (0.43) & 9 (21) \nunderscore.string & 2.63 (9.84) & 1.82 (7.58) & 1.02 (3.61) & 0.71 (0.34) & 0.70 (0.34) & 1 ( 7) \nmessy & 2.94 (4.95) & 2.69 (4.88) & 6.31 (10.68) & 0.40 (0.43) & 0.39 (0.42) & 1 ( 3) \nvirtual-dom & 2.30 (3.88) & 1.73 (3.01) & 1.45 (1.79) & 0.49 (0.43) & 0.48 (0.42) & 2 ( 5) \nrecipe-parser & 35.62 (23.87) & 1.06 (1.61) & 13.88 (29.81) & 0.71 (0.40) & 0.28 (0.38) & 52 (44) \nplanck.js & 70.95 (55.75) & 10.22 (7.97) & 24.42 (55.98) & 0.73 (0.39) & 0.60 (0.40) & 20 (27) \ngoojs & 26.40 (44.47) & 16.20 (39.25) & 10.74 (43.94) & 0.70 (0.37) & 0.63 (0.41) & 19 (34) \n[Tab:moduleCouplingMetrics]\nTable [Tab:moduleCouplingMetrics] presents module coupling and instability results for each project, before and after refactoring. Each value is computed as the average of a given metric over all modules in the production code of the respective project. The standard deviation is provided inside parentheses. Columns 2–3 present the average *Fan-out* ($\\overline{\\text{FO}\\_i}$), for the modules of each project, before (ES5) and after refactoring (ES6). The results show a significant decrease of module *Fan-out* in most projects, marginal decrease in one project and no change in 5 of them. The improvement of module *Fan-out* is due to the fine-grained reuse of module features after refactoring, where each module imports the minimum required features from its dependencies. On the other hand, the ES5 version of modules, usually, import entire modules and depend on all features bound to the respective module objects. The average *Fan-in* ($\\overline{\\text{FI}\\_i}$) of modules for each project is presented in Column 4. The values for *Fan-in* are not affected by the refactoring. The reason is that the incoming dependencies of a module remain unchanged, despite its import in client modules as individual module features rather than as a coarse-grained module object.\nThe effect of the refactoring on *Module Instability* is analyzed in Columns 5–6 that provide the average instability of modules across all projects, before and after refactoring, Its average improvement is provided in Column 7. Following the decrease of module *Fan-out*, the instability of modules improves by 9–60% in 10 projects and 1–2% in 3 projects. However, in projects `TicTacToe` and `backbone-tableview` module instability remains the same despite the improvement in *Fan-out*. These projects contain modules with either (a) FI*i* = 0, i.e. their client modules do not define module features that depend on the exported features, or (b) FO*i* = 0, i.e. they do not define module features that depend on the imported features. These modules are not considered in the calculation of $\\overline{\\text{I}\\_i}$after refactoring.\nThe proposed method increases the number of reusable elements per refactored project by a factor of 4 on average, and enables fine-grained reuse through declaration of multiple reusable elements per module. It enforces module encapsulation by restricting the scope of module features that are not used by other modules and reduces module coupling in terms of decreased module Fan-out and module instability.\n### RQ4: Does the proposed source code transformation preserve the external behaviour of the analyzed system?\nTo evaluate how effectively the proposed method preserves the external behaviour of each analyzed project, we executed the project’s test suite after the proposed refactoring. Its successful execution provides an indicator for the reliability of the applied refactoring. For projects without a test suite, we empirically evaluated the correctness of the refactored code through manual code inspection and system usage.\nEvaluation results provide empirical evidence on the soundness of the code transformation, since the test suites and the manual inspection did not reveal changes to the external behaviour of refactored code. Special attention needs to be attributed to the `goojs` project, where the transformation was applied despite the violation of a precondition. Specifically, `goojs` modules contain module dependencies that are established in inner scopes (e.g. inside conditional statements). Therefore, their code violates the module format precondition with respect to nested module dependencies ([sect:preconditions]). Nevertheless, these dependencies did not exhibit side-effects (e.g. modifications to the global scope) and the establishment of these dependencies at the module scope after refactoring did not alter the external behaviour of the system.\nThe proposed transformation preserves each analyzed system’s external behaviour, even in case that it violates the precondition relevant to module dependencies established in inner scopes.\nThreats to validity\n===================\nIn this section, we discuss threats to the validity of empirical evaluation results. We categorize and prioritize the threats to validity according to : internal, external, construct and conclusion validity.\n*Internal validity* threats refer to factors that affect the results of this study but are ignorred or cannot be controlled. A potential threat relates to the quality of the benchmark projects’ test suites that affects the conclusions of RQ4. Since the statement coverage of test suites in CJS projects ranges between – and in 7 of them exceeds (Table [Tab:projectImplementationDetails]), we expect that they are capable of revealing the majority of flaws of the proposed method.\n*External validity* threats affect the generalization of empirical evaluation results. A potential threat concerns the extent at which the selected sample of 19 projects is representative of the population of legacy ES5 projects. We deem that the sample has adequate size and diversity in order to mitigate this threat, since it includes both standalone applications and library projects, with various sizes and coming from different domains. The general applicability of the proposed method is, also, affected by some limitations on handling specific features of the JavaScript language. Specifically, our method does not analyze code which uses dynamic code generation and name aliasing, e.g. resolving modules with variant file paths generated using `__filename` or `__dirname`, properties defined programmatically through the `Object.defineProperty()` method. Static analysis introduces significant challenges in these cases that may undermine the safety of the proposed refactoring. Finally, the method analyzes to a limited extent language features that are forbidden in strict mode , in order to prevent the introduction of defects due to incompatibilities between the ES5 and ES6 specifications. Among benchmark projects, `GluttonousSnake`, `astix`, `game-of-life`, `UltraTetris`, `tetrisJS` and `goojs` make limited use of such features. In order to include them in this study, we manually replace them with equivalent features, permitted in strict mode.\nThreats to *construct validity* concern the correspondence between theoretical constructs and the observations of the empirical evaluation. A potential threat relates to RQ1 and the regex classifiers that we employed in order to categorize JS files to module formats. In order to mitigate this threat we built a test suite for these regular expressions that is publicly available in Github [8](#fn8). Another threat relates to the correct estimation of metrics that support the conclusions of RQ2 and RQ3. In order to contain this threat with performed manual validation of metric calculations for small projects that are included in this study.\n*Conclusion validity* concerns the statistical significance of the results on cause-effect relations studied in the experiment. Since, we conduct an exploratory study, where experimental results and observations shape the findings of the study rather than confirm the validity of certain hypotheses, we evaluate *reliability* as a counterpart of conclusion validity . Threats to *reliability* concern the reproducibility of this study and we mitigate them through publishing the source code and all required artifacts for its replication.\nResults and Discussion\n======================\nThe proposed method automates a large refactoring that migrates an ES5 codebase to ECMAScript 6. The refactoring focuses on the application and effective use of the ES6 language constructs for modularity. This migration represents an important milestone to the evolution of a legacy JavaScript project, since the use of the ES6 module system is a basic requirement for further adoption of language features (e.g. class syntax, promises API) introduced in ES6 or beyond, that boost developer productivity and improve code maintainability.\nThe scope of the proposed refactoring method is not limited to plain syntax transformation, that is supported to an extent by some module bundlers . It additionally alters the design of refactored modules to enable fine-grained reuse of module contents, through destructuring of module objects to multiple individually reusable module features. Module dependencies are, then, redefined and established on the basis of these module features. This improves maintainability, since coupling between the refactored modules is reduced; each refactored module imports the module features that are used by its features, instead of entire module objects. It, also, enables encapsulation, since unused module features are restricted to the scope of their declaration module.\nOur method achieves fine-grained reuse of module contents through effective use of the *named* imports/exports language constructs, instead of their *default* or *namespace* equivalents that establish dependencies on the basis of coarse-grained module objects. Although the latter are used for backwards compatibility with ES5 module formats, named imports/exports provide significant benefits [9](#fn9). Specifically, they enforce name consistency across modules, which improves code understandability and allows for effective “tree-shaking” during code deployment. Moreover, they facilitate other refactoring operations, such as rename refactorings and move feature or extract module refactorings for remodularization. Finally, the evaluation part of this study contributes to the ongoing debate on *named* vs. *default* imports/exports by highlighting the former’s positive effects on module coupling and reusability.\nConclusions – Future Work\n=========================\nWe have proposed a method for automated refactoring of legacy ES5 code, that is either non-modular or uses the AMD/CJS formats, to ES6 modularity with extensive use of the *named import/export* language constructs. Besides migration of ES5 module formats to ES6 modules, the proposed method improves internal quality attributes of refactored code. Specifically, it enforces fine-grained reuse of module contents, through destructuring of module objects to individually reusable module features. On the basis of these features, it optimizes module dependencies by leveraging the ES6 syntax. The core of our method is a static analysis technique that constructs a global model of the analyzed project, the Module Dependence Graph (MDG). The MDG mirrors inferred or explicitly declared modules and their dependencies. On the basis of MDG, we specify the source code transformation for migration to ES6.\nThe proposed method has been implemented in the Node.js platform for the purpose of evaluating its effectiveness and practicality on a collection of open source projects. Evaluation results reveal a developer intent for fine-grained reuse, since of extracted features have semantics that correspond to reusable elements at module scope (namespace properties, static features of emulated classes). The analysis of refactored code shows an increase in the number of reusable elements per project by a factor of four and an improvement in the coupling of refactored modules in terms of decreased module *Fan-out* and module instability. Finally, the execution of projects’ test-suites on refactored code provides evidence on the soundness of the source code transformation.\nOur future work will investigate the potential for further improvements to the modularity of an ES6 codebase through automated refactorings. Specifically, we intend to extend our research across two directions: (a) improvement of module cohesion through redistribution of fine-grained module features among modules, extraction of module features to new modules or consolidation of the contents of different modules, and (b) modernization of the internal structure of modules through the introduction of the ES6 class syntax.\nAcknowledgements\n================\nThe research work was supported by the Hellenic Foundation for Research and Innovation (HFRI) under the HFRI PhD Fellowship grant (Fellowship Number: 696). The authors would, also, like to thank the anonymous reviewers for their useful comments that improved the quality of this work.\n50 natexlab#1#1[1]`#1` [2]#2 [1]#1 [1] [1] [2]#2,,, ()..,,.,,,.,,.,,.,,.,,.,, ()..,,, ().,, ed.,,,.,, ed.,,,.,,,,,,, in:,, pp..,,,,,,,,, in:,, pp..,,,,,, ().,,,,, ().,,,,,,,, ().,,, ().,,, ().,,,,,,. [arXiv:2007.02194](http://arxiv.org/abs/2007.02194).,,,, ().,,,,,, in:, ESEC/FSE 2020,,, p..,,,, ().,,,, ().,,,,, in:,, pp..,,, ().,,,,, in:,, pp..,,,,,, ().,,,,,, in:, OOPSLA ’11,,, pp..,,, ().,,,, in:, ISSTA 2012,,, pp..,,,, in:,.,,,, in:, AWeS ’15,,.,,,,, in:,, pp..,,,,, in:,, pp..,,,, in:,.,,,,,, ().,,,, in:,, pp..,,,,, in:,,, pp..,, ed.,,.,,,.,,,.,,,.,,.,,.,,, ed.,,,.,,.,,.,,,,.,,,,,,, in:, ECOOP’13,,, pp..,,,,,,,,,.\n---\n1. https://nodejs.org/api/modules.html#modules\\_exports[↩](#fnref1)\n2. https://requirejs.org/docs/api.html#data-main[↩](#fnref2)\n3. https://github.com/AlDanial/cloc[↩](#fnref3)\n4. https://github.com/katerinapal/es52es6[↩](#fnref4)\n5. https://console.cloud.google.com/marketplace/product/github/github-repos[↩](#fnref5)\n6. https://www.gharchive.org/[↩](#fnref6)\n7. Or an equivalent construct of another programming language, e.g. `C#` namespace.[↩](#fnref7)\n8. https://github.com/softeng-aueb/es-module-classifier[↩](#fnref8)\n9. https://esdiscuss.org/topic/moduleimport[↩](#fnref9)\n\nAutomated Refactoring of Legacy JavaScript Code to ES6 Modules\n==============================================================\nThe JavaScript language did not specify, until ECMAScript 6 (ES6), native features for streamlining encapsulation and modularity. Developer community filled the gap with a proliferation of design patterns and module formats, with impact on code reusability, portability and complexity of build configurations. This work studies the automated refactoring of legacy ES5 code to ES6 modules with fine-grained reuse of module contents through the *named import/export* language constructs. The focus is on reducing the coupling of refactored modules through destructuring exported module objects to fine-grained module features and enhancing module dependencies by leveraging the ES6 syntax. We employ static analysis to construct a model of a JavaScript project, the Module Dependence Graph (MDG), that represents modules and their dependencies. On the basis of MDG we specify the refactoring procedure for module migration to ES6. A prototype implementation has been empirically evaluated on 19 open source projects. Results highlight the relevance of the refactoring with a developer intent for fine-grained reuse. The analysis of refactored code shows an increase in the number of reusable elements per project and in the coupling of refactored modules. The soundness of the refactoring is empirically validated through code inspection and execution of projects’ test suites.\nRefactoringCode migrationModule object destructuringES6 modulesAMD/CommonJS\n[2]#1 (#2%)\n[1]#1%\nIntroduction\n============\nJavaScript followed, in the last decade, the wide reach of the web platform and turned into a general-purpose programming language for the mobile and server platform. Despite the growing scope and size of JavaScript applications, the language specification did not include, until recently, native features for streamlining encapsulation and modularity . Syntactic constructs for modules and classes, along with other features for writing complex applications (e.g., promises, arrow functions) were introduced in the ECMAScript 2015 (ES6) release . Since ES6 classes represent syntactic “sugar” over function constructors and prototype-based inheritance, they were implemented in browser engines soon after ES6 release. On the other hand, native support for ES6 modules in popular browsers was introduced after the first quarter of 2017 , while it is still an experimental feature of the latest release of the Node.js runtime .\nES6 modules support the development of complex applications through encapsulation of top-level variable and function declarations in a JavaScript file in a new scope, the *module* scope, that differs from the global scope. Thus, top-level declarations do not “pollute” the global scope, preventing name collisions and enabling variable checking. Such incompatibilities in the execution model of ES6 code files with respect to ES5 or earlier versions, required non-trivial extensions to JavaScript engines that needed time to be tested and released.\nAs native support for modularity lagged with respect to the needs of the industry, the developer community filled the gap with the proliferation of design patterns, module formats and tools. The encapsulation of state and behaviour and the avoidance of global scope “pollution” were handled by code patterns (e.g. Immediately Invoked Function Expressions- IIFEs) and module formats, mainly AMD and CommonJS , which introduced interfaces for declaring modules and their dependencies in the browser and the server respectively.\nSince ES6 modules are currently supported in most JavaScript runtimes, developers are confronted with the need to migrate their legacy codebases to the standard module format. Besides simplification of code and build configurations, the migration provides performance improvements, as module loading and dependency management in ES6 are natively supported by the runtime, rather than external libraries. Moreover, special features of ES6 modules improve code maintainability: (a) named imports/exports enforce name consistency within the codebase, facilitate refactoring operations and allow for optimization of application size (tree-shaking) during deployment, (b) named exports act as live immutable bindings that enhance safety by preventing errors relevant to cyclic dependencies , (c) eager resolution of the reachable dependencies of a module (e.g. the application entrypoint), provides fail-fast program behaviour .\nIn this paper, we propose a fully-automated method for migration of an ES5 codebase to ES6 modules. The method applies to code that is either non-modular or uses the AMD/CommonJS module formats and operates on the Module Dependence Graph (MDG), a global model of the codebase. The MDG provides a representation of the code in terms of modules, exported module features and module dependencies, as inferred and optimized by the our method. The proposed migration represents a large and complex refactoring, since it affects the entire code-base and the transformation of each individual source file requires analysis of all incoming and outgoing dependencies. Besides handling the mapping among different module formats, our method focuses on reducing the coupling of refactored modules through: (a) selective destructuring of existing AMD/CommonJS module objects to fine-grained module features, (b) establishment of the minimum required dependencies among modules by leveraging the “named imports/exports” ES6 language construct and (c) encapsulation of global state and functions as cohesive module features.\nAs part of this work, we explore the relevance of the proposed refactoring and evaluate its impact on system modularity and code correctness through an empirical study that provides answers to the following research questions:\n1. How prevalent is the presence of ES5 code, modular or not, in real-world applications?\n2. Is there a developer intent for independently reusable features within modules?\n3. What are the effects of the proposed refactoring to system modularity?\n4. Does the proposed source code transformation preserve the external behaviour of the analyzed system?\nAmong the main findings of our empirical study is the wide use of legacy module formats in open source projects and the frequent presence of code patterns in legacy code that reveal a need for individually reusable module features within modules. As concerning the effects on system modularity, our method contributes to fine-grained reuse among system modules and a reduction in module coupling. Finally, the successful execution of the benchmark projects’ test-suites on refactored code provides empirical evidence on the soundness of the proposed source code transformation.\nThe rest of the paper is organised as follows. Section [relatedwork] presents related work and Section [moduleSystemReview], provides a brief review on modularity support in ES5 and compares it against the features of ES6. In Section [transformationProcedure], we specify our method for refactoring towards ES6 fine-grained modularity. Section [evaluation] presents the details and results of an empirical evaluation of the method, while Section [threats] discusses the threats to its validity. Finally, Section [results] provides a discussion of results and the paper is concluded in Section [Conclusions].\nRelated Work\n============\nRefactoring involves reorganizing source code while preserving its external behaviour . Fowler introduced a series of basic refactorings for handling common code smells in object-oriented code and specialized them on the JavaScript language in the second edition of his book . Recent studies and systematic literature reviews focus on the relationship between code smells and refactoring, especially refactoring towards mitigating architectural problems , and their impact on software quality . Ongoing research expands the application of refactoring techniques to address diverse requirements, beyond design improvements in code or other software artifacts, e.g. UML diagrams . Specifically, several works focus on techniques for improving security and maintainability metrics , improving performance through removing unused features , reasoning about refactoring activities over long periods of time as well as improving software accessibility and achieving business-oriented goals .\nThis work belongs to the research area of refactoring methods and techniques that target the JavaScript language. Relevant works in this area study the automation of refactorings that handle the following concerns: (a) improvement of the code’s runtime performance, (b) improvement of software quality characteristics through generic refactorings or replacement of insecure language features and (c) migration of legacy code to newer language versions. These works are outlined in the rest of this section.\n leverage static analysis techniques for refactoring AJAX applications to shift from the XML to the JSON data format, in order to improve the runtime performance. Furthermore, static and dynamic analysis techniques are proposed towards automating dead code elimination in JavaScript applications. Feldthaus et al. improve modularity through providing a framework for specifying generic refactorings based on pointer analysis. propose a static analysis technique for replacing `eval` with language features that enable reasoning about the program’s behaviour.\nSeveral approaches focus on refactoring JavaScript legacy code towards using language features provided in more recent versions of ECMAScript. The prevalence of callbacks in JavaScript applications and their equivalence to ES6 promises is studied in . The identification of classes and their dependencies in legacy JavaScript is studied in . Extensions of these works provide migration rules for their refactoring towards using ES6 classes . propose a static analysis technique for refactoring non-modular ES5 web applications towards ES6 modules.\nIn this paper, we propose a method for automated refactoring of a code-base implemented in ES5 or earlier versions, towards fine-grained modularity through ES6 *named imports/exports*. Our work handles concerns related to code migration to a newer language version, as well as to the improvement of its internal quality attributes. As compared to earlier work on refactoring legacy ES5 code to the ES6 class syntax , our work is complementary and a basic requirement for its application. The reason is that migration to ES6 modularity is a prerequisite for further application of migrations that incorporate modern language features to a code-base. Concerning our previous work on this area , this paper contributes major extensions in terms of (a) the scope of the refactoring, since it additionally handles AMD/CommonJS code-bases, (b) the effect of the refactoring, as it changes module structure and enables effective usage of ES6 modularity constructs and (c) the evaluation methodology that highlights the effectiveness and practicality of the approach.\nModularity in JavaScript\n========================\nIn this section, we review the evolution of modularity from ES5 to ES6. We briefly present the CommonJS and AMD module formats that are widely used in ES5 and their mostly adopted implementations , followed by the ECMAScript Modules (ESM) feature standardized in ES6. Details on the projects containing the presented fragments are provided in Section [evaluation]. We focus on disparities concerning the encapsulation of module and global state, the granularity of exported definitions and the evaluation order of module dependencies. Moreover, we refer to the imposed limitations with respect to the scope of applicability of the provided features. In line with the overview of the features provided by ES6, we compare ES5 formats against ES6 and highlight their incompatibilities, which complicate automated migration of ES5 code to ES6.\nModularity in ES5\n-----------------\nUnlike static programming languages JavaScript did not provide prior to ES6, a formal mechanism for preserving modularity and encapsulation. The community addressed this need through adopting design patterns, e.g. the Singleton pattern , that inject data and functionality belonging to the global namespace into a single global variable.\n [fig:es5-js-snippets]\nAlthough each JavaScript file is meant to bundle coherent functionality, representing a conceptual module, the variables/functions declared in the file’s top-level scope are bound to the global object after the file is loaded. Global variables are, also, introduced in terms of using non-declared variables (implied globals) in non-strict mode code and attaching properties to the global object. Figure [fig:es5-js-snippets] depicts fragments from ES5 files of the `Hangman` project that are loaded in page `hangman.html`. Upon loading, `content.js` defines the global variable `hintsleft` and makes it available to `hint.js`.\nThe main problem arising from loading files with non-modular ES5 code is that it complicates program state management due to global variables. The presence of global variables in different program units hampers reasoning about the origin of their values, notably when these units are located in different files. Furthermore, it undermines maintainability, since variables loaded from different files increase the likelihood of name conflicts leading to runtime errors or unexpected behaviour. Finally, the lack of a module loading mechanism increases the application’s complexity, since the module loading order must be manually specified. The developer community handled these challenges with the CommonJS and AMD module formats, that support the declaration of modules on the server and the browser, respectively .\n### CommonJS\nThe CommonJS (CJS) proposal specifies an interface for defining and loading modules on the server. A CJS module comprises a piece of reusable JavaScript code which makes objects available to other modules. Figure [fig:commonJSExample] presents code fragments from `express-session` Node.js package.\n [fig:commonJSExample]\nThe top-level variables/functions of a CJS module are bound to the module scope, since CJS restricts the visibility of top-level declarations . Similarly to non-modular ES5, global variables are introduced in terms of implied globals in non-strict mode code , and global object properties. CJS modules are synchronously loaded through `require` , a function which returns the *module object*, i.e. the object containing the variables/functions that are visible beyond the scope of their declaring module . They behave as singleton objects, since their instances are cached at the time of their first load . A variable/function defined in the module scope becomes available to other modules through binding to the `module.exports` object or its alias `exports` [1](#fn1). CommonJS does not pose limitations regarding the scope of applicability of the require/export features, since they can be used in the module’s inner scopes (e.g. nested in conditional statements).\nAs depicted in Figure [fig:commonJSExample], `memory.js` imports the function `Store` exported from `store.js`. The function is defined at the time it is exported from `store.js` due to variable/function declaration hoisting, i.e. a reference to `Store` is bound to the nearest scope in which it is defined.\nNotice that CJS partially enhances encapsulation, since it does not prevent the introduction of implied global variables . The global namespace “pollution” due to implied globals is mitigated through the `strict mode` directive that prohibits certain language features. Finally, software portability cannot be guaranteed, since the format targets at defining modules for server-side tasks. The use of design patterns (e.g Universal Module Definition - UMD ) is needed in order to load CJS modules in the browser. These patterns produce overhead, in terms of memory space and time, in order to resolve the execution environment of the loaded module.\n### Asynchronous Module Definition (AMD)\nThe AMD proposal specifies an interface for declaring and loading modules on browsers . Similarly to non-modular ES5, AMD supports global variables through either the declaration of top-level variables, the definition of implied globals in non-strict mode code or property bindings to the global object.\nAMD modules are loaded either *eagerly* or upon request with the help of the features provided by the format or script loaders, e.g. RequireJS, respectively. By eager loading we mean that upon request for loading an AMD module, all its dependencies are loaded before the execution of the requested module’s code. They are instantiated through the function `define` provided by the format, which returns the module object. However, AMD does not provide a feature for the actual loading of modules upon request, and, thus, script loaders (e.g. RequireJS ) are used for module loading and dependency management.\n [fig:AMDExample]\nSimilarly to the CJS format, the features provided by the AMD format and the respective script loaders can be used in the module’s inner scopes. Moreover, AMD modules are loaded in an asynchronous manner. The module loading and execution order cannot be precisely assumed, which can lead to the introduction of runtime errors[2](#fn2). Finally, the problems encountered in defining AMD modules correspond to these encountered in defining CJS modules.\nFigure [fig:AMDExample] presents AMD module fragments from the `tetrisJS` application, which are loaded with the RequireJS loader . The `TetrisGame.js` module depends on `Score.js`, which is referenced through the entry `app/Score` in the dependency array provided in `define`. The resolution of `Score.js` imports its module object, i.e. the function `Score`, in the scope of the module’s definition function.\nES6 Modules\n-----------\nThe support for code modularity through the AMD/CJS formats is currently replaced by native language features standardized in ECMAScript 6. These features enable defining platform-independent modules which can be loaded without the need for adopting design patterns (e.g. UMD ).\nThe variables/functions defined in the top-level scope of an ES6 module are restricted to the module scope . Global variables are introduced in terms of global object properties, since ES6 modules enforce *strict* mode which prohibits the introduction of implied global variables .\nECMAScript 6 provides a declarative syntax for exporting and importing declarations through `export` and `import` statements. Unlike AMD/CJS modules, which export a single *module object*, i.e. a compound object with the module’s exported declarations, an ES6 module can apply multiple strategies for managing dependencies. Specifically, it may use *named exports* to export multiple *module features*, i.e. independent variable/function declarations, to client modules. Moreover, an ES6 module may selectively import individual module features from another module through static *named* imports. Importing all module features as a compound object is, also, supported through *namespace* imports. Furthermore, each module may declare a specific feature as a *default export* that is imported in client modules through a *default import* statement. Finally, ES6 enables module loading at module execution time through *dynamic* imports. Figure [fig:commonjs-vs-es6] presents the equivalent ES6 module of `store.js` (Figure [fig:commonJSExample]). The module declares the function `Store`, exported through a *named export*, while it imports `Session` from `session.js` through a static *named import*.\n [fig:commonjs-vs-es6]\nAlthough ES6 standardized features for platform-independent modules, their semantics pose a number of restrictions. Specifically, imported declarations introduce immutable bindings to the respective exported declarations in the module’s scope . Moreover, ES6 exports and static imports are permitted only at the top-level scope of the module. The establishment of these statements in the module’s inner scopes introduces evaluation errors.\nCompared to ES5 module formats, ES6 modules support more effective elimination of unused code through the use of *named exports* that simplify the identification of unused module features. Moreover, ES6 module dependencies are eagerly evaluated , before the actual execution of the module’s code. Specifically, the module features that are imported in an ES6 module are resolved during the module’s evaluation stage, in contrast to ES5 formats where they are resolved at module execution time . This enables the identification of software defects relevant to non-declared imported features at the module’s loading stage instead of its execution stage, thus providing fail-fast behaviour . Finally, the ES6 module format is platform-independent, enabling module reuse across different platforms . This, also, enhances testability through mitigating the need for simulating the module’s execution environment.\nRefactoring towards using ES6 modules\n=====================================\nWe propose a method for fully-automated migration of legacy ES5 code to ES6 modules that applies to ES5 projects that are non-modular or employ the AMD/CJS formats as implemented by RequireJS and Node.js , respectively. The method focuses on reducing the coupling of refactored modules, through destructuring existing module objects to fine-grained module features and optimization of module dependencies on the basis of the minimum required module features. Moreover, our method restricts the scope of global declarations, and confines them to the module scope as appropriate module features.\n [fig:motiv-example]\nFigure [fig:motiv-example] illustrates a motivating example, in order to showcase the application and the merits of the proposed refactoring. It presents code from the `planck.js` 2D physics engine for game development and, specifically, code fragments from the `Solver.js` module and its dependencies. `Solver.js` is a core module of `planck.js` and uses declarations from 9 modules of the same project, e.g. it reads configuration properties from the `Settings.js` module and uses utility functions for vector calculations from the `Vec2.js` module. The code is based on the CJS module format, where the reusable declarations of a module are exported as part of a single module object (top part of Figure [fig:motiv-example]). For instance, the `Settings.js` module exports 26 configuration properties, as part of the `Settings` object, `Vec2.js` exports 27 utility functions, as bound properties to the `Vec2` constructor function, and `Math.js` exports 8 functions and a constant as part of the module object. In CommonJS, each client of these modules, e.g., the `Solver.js` module, imports at once all their reusable features through the respective module object. Thus, the module clients depend on complex, coarse-grained objects instead of the elementary features that they actually use. In the case of `Solver.js` and imported modules `Settings.js`, `Vec2.js` and `Math.js`, the former depends on their entire public API, although it uses 8, 4 and 1 of their exported features, respectively.\nThe bottom part of Figure [fig:motiv-example] presents an ES6 version of the code fragments from `Solver.js` and its dependencies, after the application of the proposed refactoring. Notice that the independently reusable features of each module, e.g. configuration properties in `Settings.js` or utility functions in `Vec2.js`, are refactored to standalone declarations and exported to client modules through ES6 *named export* statements. The module structure is simplified, while empty module objects are eliminated, as is the case for the `Settings` object. Moreover, all declarations that a module requires from other modules are introduced to its scope through static *named import* statements. The `Solver.js` module, now imports the minimum required declarations from its dependencies.\nFine-grained module design, in terms of multiple independently reusable module features, accompanied with fine-grained module dependencies through named imports, boosts refactorings, code optimizations and error tracing. Specifically, it streamlines the renaming of module features and their relocation among modules. It, also, supports module bundling tools in effective tree-shaking, i.e., identification and elimination of unused code from deployment artifacts . Moreover, it enforces a static module structure which, in turn, paves the way for static variable checking and type inference , enabling, thus, early error detection. Named imports, also, enable fail-fast program behaviour, since missing imports are resolved at module evaluation stage (probably at application startup), rather than at a later execution stage .\nBesides *named imports* or *named exports*, the proposed method uses ES6 *default import* statements for the migration of module dependencies on external libraries, since they support backwards compatibility with ES5 module formats. As concerning ES6 *namespace* imports, i.e. import of all module features as a compound object, they are not used by our method, since they introduce implicit dependencies among modules. Specifically, in a namespace import, the client module does not explicitly specify the module features that it depends upon. Instead, it depends on all the features of the imported module, including currently unused features or features added in subsequent revisions.\nIn addition to the benefits provided by fine-grained modularity, the proposed refactoring improves code portability and testability , due to the migration of AMD/CJS modules to platform-independent modules and global variable encapsulation. The automation of the proposed refactoring is a challenging task due to incompatibilities of the ES6 module format against ES5 formats, such as read-only bindings for imported module features, top-level declaration of static imports, hoisting and eager loading of static module dependencies, as opposed to on-demand loading in CommonJS etc.\nIn the rest of this section, we specify our method for automated migration to ES6 modules as two distinct processing stages: (a) identification of refactoring candidates, i.e., code fragments that need to be refactored and (b) application of the source code transformation. The specification of our method, also, defines the refactoring preconditions that preclude transformations leading to erroneous code or influencing the analyzed program’s external behaviour.\nRefactoring candidate identification procedure\n----------------------------------------------\nThe identification of refactoring candidates involves: (a) the resolution of the structure of each module, in terms of module features that can be individually exported and reused by other modules, and (b) the resolution of fine-grained module dependencies, on the basis of the minimum required module features that have to be imported by each module after refactoring.\nThe outcome of this procedure is a directed graph, the Module Dependence Graph (MDG), that models the desired module structure after refactoring and supports the source code transformation procedure. The MDG concept, initially introduced in our previous work , is redefined here to provide a unified representation of the module structure in modular and non-modular JavaScript code. Specifically, it comprises a pair *M**D**G* = (*M*, *D*), where *M* represents the set of graph nodes. MDG nodes correspond to the ES6 modules that will be introduced after refactoring. The set *M* is initially populated with one node for each JavaScript file of the analyzed project. The set *D* represents graph edges. Each edge models a dependency between two modules of the refactored system.\n### Resolution of module structure\nThis step involves the identification of the module features for each module in *M*. Let *m**i* ∈ *M* be a module and *n**i* be its module name used in AMD/CJS or its module path in the case of non-modular ES5. We define *m**i* as a pair *m**i* = (*n**i*, *F**i*), where *F**i* represents the set of module features of *m**i*. Module features comprise (a) properties of the module object – in AMD/CJS – that can be extracted to module scope and get individually exported, (b) the module object per se, in case that it is not stripped of all its content, (c) global declarations that are assigned to modules in order to restrict their scope.\n#### Module object destructuring\nThe identification of object properties that can be extracted as standalone module features and their subsequent extraction from the AMD/CJS module object is termed in this work as *module object destructuring*. The identification procedure involves recovery of the module structure from its AST representation and is based on the analysis of statements that apply property bindings to the module object.\nProperty binding statements comprise assignments whose left-hand side expression is an object property access through dot/bracket notation, for instance `foo.bar = <expression>`. Properties defined through bracket notation are excluded from further analysis, as explained in [sect:preconditions]. The *bound properties* *B**i* of module *m**i*, henceforth, refer to all property bindings on the module object *e**i* with the dot notation. In case that the module object is instantiated through object literal notation, all properties appearing in the property-value pairs are, also, appended to the set *B**i*.\nThe module object *e**i* exported by module *m**i* may be constructed within *m**i* or imported from another module. Its instantiation may be declarative, through a function declaration or object literal notation, or may involve the evaluation of an expression (e.g. function invocation). A module object instantiated as (a) an empty object/function with properties attached at runtime, or (b) through object literal notation, is characterized by our method as a *namespace* object. The reason is that such objects serve as containers for data/ function properties that need to be exported by an AMD/CJS module. Namespace objects would be stripped of all their properties after refactoring, and, thus, are eliminated from the set of module features. Figure [fig:namespace-obj] presents a module from `goojs` project that follows the namespace object pattern. The module exports mathematical constants and functions and its original version is depicted in sub-figure (a). Sub-figures (b) and (c) provide syntactic variants that use an empty object or object literal notation.\n[t].5\n```\nfunction MathUtils() {}\n \n MathUtils.DEG_TO_RAD \n = Math.PI/180.0;\n /*... */\n MathUtils.radFromDeg = \n function(degrees){/*... */};\n \n MathUtils.degFromRad = \n function(radians){/*... */};\n /*... */\n module.exports = MathUtils;\n \n```\n[t].5\n```\nvar MathUtils = {};\n \n MathUtils.DEG_TO_RAD \n = Math.PI/180.0;\n /*... */\n MathUtils.radFromDeg = \n function(degrees){/*... */};\n \n MathUtils.degFromRad = \n function(radians){/*... */};\n /*... */\n module.exports = MathUtils;\n \n```\n[t].3\n```\nvar MathUtils = {\n \n DEG_TO_RAD: Math.PI/180.0,\n /*... */,\n radFromDeg: function(degrees){/*... */},\n degFromRad: function(radians){/*... */},\n /*... */\n };\n module.exports = MathUtils;\n \n```\n[fig:namespace-obj]\nLet *e**i* be the module object of a module *m**i* and *F**i* the identified module features. The module object destructuring algorithm computes the set *F**i* for a module *m**i* and can be summarized as following:\n1. Evaluate module object destructuring preconditions (see Section [sect:preconditions]). If any precondition fails then return *F**i* = {*e**i*}, i.e., the module object cannot be destructured.\n2. Identify the set *B**i* of bound properties to *e**i*,\n3. If *e**i* is a *namespace object* return *F**i* = *B**i*. Otherwise, return *F**i* = *B**i* ∪ {*e**i*}, i.e., the module object would be available as a module feature after destructuring\n#### Resolution of global declarations\nGlobal declarations involve explicitly declared and implied global variables, properties bound to platform specific global objects (e.g., `window`, `global`) and top-level declarations in non-modular ES5 source files. We identify the global variables of a module *m**i* through AST traversal and search for (a) top-level `var` statements (non-modular ES5/AMD) and (b) global object property definitions. Implied globals are identified through analysis of AST nodes that represent assignment expressions. Specifically, the variable in the left-hand side of an assignment is sought in the sets of the variables/functions defined within the scope hierarchy formed by the assignment’s scope and its surrounding scopes. The variable comprises an implied global if it is not declared in the hierarchy. Finally, top-level declarations in non-modular ES5 are identified by locating function and variable declaration nodes which do not have function declaration ancestors.\nLet *m**i* be a module and *G**i*, *O**i*, *T**i* refer, respectively to its global variables, global object properties and top-level declarations identified in its AST. The resolution of module features from global declarations involves the following steps:\n1. Evaluate global declaration preconditions (see Section [sect:preconditions]) on *G**i*, *O**i*, *T**i*. In case of any violation return *F**i* = ∅.\n2. For each identified global object property *o**k* ∈ *O**i* search for its presence in the feature set of other modules\n\t1. If *o**k* is also defined in module *m**j*, allocate *o**k* to the module with lower number of imports, so as to normalize coupling among modules,\n\t2. If not found, then *F**i* = *F**i* ∪ {*o**k*}.\n3. Return *F**i* = *F**i* ∪ *G**i* ∪ *T**i*.\nIn case that global declarations’ resolution returns *F**i* = ∅ for any module *m**i*, then the refactoring is abandoned.\n### Resolution of fine-grained module dependencies\nThe resolution of the fine-grained module dependencies involves the identification of the module features that each module needs to import from other modules of the same project. Moreover, it identifies dependencies to external library modules that are treated differently for compatibility reasons.\nLet *d**j* ∈ *D* be a module dependency from module *m**i* to a project module *m**t*. A module dependency represents the import and use in a module *m**i* of a module feature *f**t*, *k* ∈ *F**t*, i.e., exported from module *m**t*. A module dependency is, also, characterized by the type of use *u**j* ∈ {*R*, *W*, *C*, *L*} of *f**t*, *k* within *m**i*. If *f**t*, *k* is only read or called in *m**i*, then *u**j* has a value of *R* or *C*, respectively, If *f**t*, *k* is, also, defined in *m**i* then *u**j* = *W*. Thus, a module dependency is defined as a tuple *d**j* = (*m**i*, *m**t*, *f**t*, *k*, *u**j*). In case that imported module *m**t* corresponds to an external library, the dependency is represented as *d**j* = (*m**i*, *m**t*, ∅, *L*).\nIn CommonJS, we identify imported modules through AST traversal and search for `require` invocations. A typical `require` invocation for a module *m**j* is equivalent to an ES6 namespace import, i.e. the module object exported by *m**j* is returned and assigned to a variable that provides access to the entire feature set *F**j* of *m**j*. A specific property of the imported module *m**j* may also be directly assigned to a variable, through property access on `require` result. This coding idiom emulates an ES6 named import in CommonJS and the receiving variable provides access to a single feature *f**k* ∈ *F**j*. Finally, a module may be imported only for its side effects, i.e., the `require` result is not assigned to any variable.\nIn the case of AMD, we identify imported modules by analyzing the parameters of the `define` module declaration function. Specifically, the first – optional – parameter of the `define` invocation provides a list with the relative paths or identifiers of imported modules. Each module *m**j* is resolved and the module object is provided as parameter to the module definition function (second parameter of the `define` invocation). AMD imports are namespace imports, since parameters of the module definition function provide access to the imported modules’ entire feature set.\nThe identification of module dependencies for each module *m**i* is based on the following algorithm:\n1. For each imported system module *m**j* (AMD/CJS) identify the subset of its module features *A**j* ⊆ *F**j* used in *m**i*\n\t1. Let *u**k* be the type of use of each *a**k* ∈ *A**j*,\n\t2. For each *a**k* ∈ *A**j* add a module dependency, i.e. *D* = *D* ∪ {(*m**i*, *m**j*, *a**k*, *u**k*)}.\n2. For each imported library module *m**j* (AMD/CJS) add a module dependency, i.e. *D* = *D* ∪ {(*m**i*, *m**j*, ∅, *L*)}.\n3. For each access to a global declaration $g\\_k \\notin F\\_i$\n\t1. Let *u**k* be the type of use of *g**k*. Identify the module *m**j* that encapsulates the global declaration, i.e., *g**k* ∈ *F**j*,\n\t2. Add a new module dependency *D* = *D* ∪ {(*m**i*, *m**j*, *g**k*, *u**k*)}.\n### MDG construction algorithm\nOn the basis of the aforementioned definitions and algorithms the construction of the MDG data structure for an ES5 project is summarized below:\n1. Initialize sets *M* = ∅, *D* = ∅\n2. For each source file *f**i* with name *n**i*\n\t1. Evaluate module format preconditions on *f**i* (see Section [sect:preconditions]). In case of any violation, refactoring of *f**i* is abandoned.\n\t2. Resolve the module structure *F**i*. If *F**i* ≠ ∅ then *M* = *M* ∪ {(*n**i*, *F**i*)} else refactoring of *f**i* is abandoned.\n3. For each module *m**i* ∈ *M*\n\t1. Resolve module dependencies *D**i* against modules *M*ʹ = *M* \\ {*m**i*},\n\t2. *D* = *D* ∪ *D**i*\n4. Return (*M*, *D*).\nFigure [fig:mdg-example] presents a fragment of the MDG constructed during this stage for the project `planck.js`. The fragment focuses on module `Math`, its outgoing dependencies and a subset of its incoming dependencies from other modules. The module’s outgoing dependencies correspond to the module features that are imported and used in the module, while incoming dependencies correspond to the module’s features that are exported to client modules. For instance, `Math` imports the feature `create` from the module `create`, while it exports the feature `isFinite` to the client modules `Fixture` and `Body`.\n [fig:mdg-example]\n### Refactoring preconditions\nRefactoring preconditions are evaluated by the refactoring identification algorithm on each module of the analyzed system. They prevent evaluation errors in the refactored code and ensure that the external behaviour of the system is preserved. The identified preconditions, depending on their scope of application, are grouped in three categories: (a) global declarations’ preconditions, (b) module object destructuring preconditions, (c) module format preconditions.\n*Global declarations’ preconditions* involve defining global variables and top-level functions with the same name in different modules:\n1. Each global variable should be introduced in a single location. Conflicting declarations across different files are a potential source of errors.\n2. In non-modular ES5 or AMD, the names of top-level functions should be unique across all JavaScript files. Different top-level functions with the same name in multiple modules lead to function overriding; the binding in the module’s namespace corresponds to the most recently loaded function.\n*Module object destructuring preconditions* involve extracting the properties of the module object that can be independently exported, while preserving the program’s external behaviour:\n1. Object properties should be defined with dot notation. Properties defined with bracket notation may lead to inaccurate conclusions regarding their declaration module, in cases that the object is indexed by a variable.\n2. Module objects should not be fully referenced (e.g. provided in function invocations). Detaching object properties from these objects may lead to errors originating from name aliasing.\n3. Module objects should not be modified. Detaching properties from such objects may lead to errors related to violations of the ES6 specification .\n*Module format preconditions* involve restrictions that are imposed by the language specification. They, also, involve restrictions that are not imposed by ES5 formats, but are introduced in the ES6 specification. The violation of these restrictions may introduce defects in the refactored code or affect the program’s external behaviour:\n1. Function definitions that reference `this` should be declared in files that enforce strict mode. In case that a function is neither an object method nor a constructor, the value of `this` is determined by the module’s code mode. In *non-strict* mode, `this` references the global object. In *strict* mode `this` is `undefined`, as its value is not updated during the function invocation. Since ES6 modules enforce *strict* mode, the transformation of an ES5 source file in *non-strict* mode into an ES6 module would lead to runtime errors originating from the value of `this`.\n2. Invocations of `require` in CJS modules should be located in the module’s top-level scope. Notice that in ES6, the introduction of static imports and exports is allowed only at the module scope. In-place transformation of `require` calls to ES6 static imports would lead to evaluation errors, if they are nested in block statements. This, also, applies to the replacement of return statements with export statements in AMD.\nSource Code Transformation Procedure\n------------------------------------\nThe source code transformation for migration to ES6 modules employs the MDG constructed in the refactoring candidate identification stage. Each MDG node, corresponding to a module *m**i*, is individually refactored in the following steps:\n#### Step 1\nIf *m**i* is an AMD module, clear its dependencies to the AMD API:\n1. Replace the invocation of the `define` function with a list of all statements included in the body of the module definition function.\n2. Replace the return statement included in the statement list with a variable assignment of the form: `var mod_<filename> = <return_expr>`, where `<return_expr>` is the expression returned by the replaced statement.\nThe application of this step to an AMD module is illustrated in Figure [fig:module-migration-amd]. The affected statements are enclosed in a dashed frame and are marked with the step number (1). In the rest of this section, numbers inside parentheses will be used to refer to similarly marked parts of Figures [fig:module-migration-amd], [fig:mathJsRef].\n#### Step 2\nIf *m**i* is a CJS module, clear dependencies to the CommonJS API:\n1. Declare an empty placeholder variable for the module object with the following naming scheme: `var mod_<filename> = null`.\n2. Replace all references to `module.exports` or its alias `exports` with the above variable.\n3. Eliminate all `require` statements.\n[hbt!] [fig:module-migration-amd]\n[hbt!] [fig:mathJsRef]\nFigure [fig:mathJsRef] presents the application of this step to the `Math.js` module of the `planck.js` project. The newly introduced variable `mod_Math`, replaces `module.exports` and holds the reference to the module object in refactored code (2). Variable `math` remains an alias to the module object.\n#### Step 3\nApply module object destructuring by processing each identified feature *f**j* ∈ *F**i* for the module *m**i*. Each feature *f**j* is declared and initialized in ES5 through a property-value pair in object literal notation or through an assignment statement that binds a property to the module object. In order to extract *f**j* from the module object:\n1. Introduce a variable declaration for *f**j* that has the form:\n```\nvar <feature> = <feature_expr>\n```\nwhere `<feature>` corresponds to the name of *f**j*. In case of name conflicts, the feature name is prefixed with the module name, i.e., `<file_name>_`. The term `<feature_expr>` corresponds to the expression used for initialization of *f**j* during its declaration.\n2. Replace references to *f**j* in module *m**i* with the feature name.\n3. Remove the property-value pair or the property binding assignment statement, originally used for *f**j* declaration.\nThe declaration of module features in the ES5 version of `Math.js` (left part of Figure [fig:mathJsRef]), is based on property binding assignments to the `math` alias of the module object. All property bindings can be extracted as accordingly named variable declarations (3), with the exception of `isFinite` that introduces name collision with a global JavaScript function.\n#### Step 4\nDeclare the exported module features of *m**i* through evaluation of all its incoming MDG dependencies.\n1. Initialize exported features *E**i* = ∅\n2. For each incoming MDG edge (*m**k*, *m**i*, *f**j*, *u**j*), originating from a client module *m**k*, add *f**j* to the set of exported features *E**i*. In case that the client module modifies the value of *f**j*, i.e., *u**j* = *W*, declare a mutator function for *f**j* and add it to exported features *E**i*. The following code listing provides a template for mutator function declaration.\n```\nfunction set_<feature>(value) { <feature> = value; }\n```\n3. Declare an export statement listing all exported features in *E**i*. For name consistency, provide an alias and restore the original name of all features that were prefixed for handling name conflicts.\nThe AMD module of Figure [fig:module-migration-amd] provides a single export (4), since the module object could not be further destructured. On the other hand, the CJS module of Figure [fig:mathJsRef], exports a list of features in a single statement (4). In both cases, renamed features are restored to their original version.\nNotice that Step 4 may be adapted in case that the analyzed project is not a standalone application, e.g., it implements a reusable library. The reason is that the part of a library’s public API without incoming MDG dependencies in the codebase, will be accidentally encapsulated and become unavailable to its clients. Step 4, in this case, exports all features *f**j* ∈ *F**i* for each module *m**i*. However, if the public API is invoked in test code, Step 4 can be applied without modifications, as test code may provide the required incoming dependencies.\n#### Step 5\nEstablish module dependencies for *m**i* through evaluation of all its outgoing MDG dependencies.\n1. For each outgoing MDG edge (*m**i*, *m**k*, *f**j*, *u**j*) to a module feature *f**j* of a system module *m**k*, introduce an appropriate named import. The name of *f**j* (`<feature>`) is maintained, or in case of name conflicts it is prefixed with a module identifier (`<prefix>`). The identifier corresponds to the file name of *m**k*, or if the conflict persists it is further prefixed with the name of the parent folder of the module declaration and so on. In case that the imported feature is defined in *m**i*, i.e., *u**j* = *W*, the corresponding mutator function is, also, imported. The introduced named imports follow the pattern:\n```\nimport {<feature> [as <prefix>_<feature>]} from <file>;\n```\n2. For each imported feature *f**j*, replace all references to it with the feature name. In case of assignment statements that define the value of *f**j*, they are replaced with an invocation of the feature’s mutator function.\n3. For each outgoing MDG edge (*m**i*, *m**k*, ∅, *L*) to an external library module, introduce a default import, i.e.,\n```\nimport <lib> from <file>;\n```\nwhere `<lib>` corresponds to the variable name used to reference the library module in ES5 code.\nFigure [fig:module-migration-amd] presents the import of two library modules through *default* import statements (5). In Figure [fig:mathJsRef], the `require` statements are replaced by two named imports that introduce the minimum required module features to the namespace of `Math.js` (5).\nEmpirical evaluation\n====================\nWe have designed and performed an experimental study on the proposed method for automated refactoring of ES5 software projects towards ES6 fine-grained modularity. The *goal* of this study is to analyze the proposed method for the *purpose* of evaluation with respect to effectiveness and practicality from the *perspective* of a developer, that considers the implications of the refactoring and its potential for automation, in the *context* of a set of open source projects.\nResearch Questions\n------------------\nThe experimental study aims at answering the following research questions:\n*RQ1. How prevalent is the presence of ES5 code, modular or not, in real-world applications?* At first, we investigate to which extent ES5 code, that is either non modular or supports the AMD/CJS module formats, is still prevalent in web applications. The results of this study will provide empirical evidence on the scope and practicality of the proposed method.\n*RQ2. Is there a developer intent for independently reusable features within modules?* Our purpose is to explore at which extent the proposed refactoring to ES6 fine-grained modularity is aligned with the semantics of the analyzed modules and the developer’s intent for independently reusable features within modules. The results will support the practicality of the proposed refactoring for effective usage of ES6 language constructs after migration of legacy code.\n*RQ3. What are the effects of the proposed refactoring to system modularity?* Our aim is to study the level of granularity, in terms of individually exported module features, that can be introduced in project modules and the potential for encapsulation of module features within their declaring modules. The research question, also, evaluates potential reduction to module coupling due to fine-grained dependencies introduced by ES6 named imports. The results of this research question will highlight the effectiveness of the proposed method for increasing the granularity of reusable module features and reducing the internal coupling of system modules.\n*RQ4. Does the proposed source code transformation preserve the external behaviour of the analyzed system?* The research question seeks to empirically investigate the soundness of the proposed refactoring, i.e., whether the application of the refactoring does not introduce syntax errors or alter the external behaviour of refactored code. The results of this study will further support the practicality of the proposed method.\nContext Selection\n-----------------\nThe context of the experimental study comprises 19 JavaScript projects. The selection of these projects is based on the following criteria: (a) they are open source, for study replication reasons, (b) they are implemented in ES5 or below, (c) they are either non-modular or use the AMD/CJS module formats as implemented in RequireJS/Node.js respectively, (d) their test suite, if available, can be successfully executed, (e) to the extent possible, they are used as benchmarks in earlier work in this area, e.g., .\n[htb]\nl l l l l l & & & & & \nUltraTetris & & 326 & - & master@5ad237e & silviolucenajunior/UltraTetris \nHangman & & 346 & - & master@b950842 & aurobindodebnath/Hangman \nTicTacToe & & 919 & - & master@346ebe8 & seanpr/TicTacToe \nuki & & 2360 & - & master@6cd2e47 & crcx/uki \nGluttonousSnake & & 378 & - & master@c6b49cb & yyfer/GluttonousSnake \nastix & & 481 & - & master@f0ecc39 & migace/astix \ngame-of-life & & 887 & - & master@1d83874 & devlysh/game-of-life \ntetrisJS & & 1370 & - & master@27712a0 & marneborn/tetrisJS \ndynablaster-js-port & & 3679 & - & master@5b5052e & gardziej/dynablaster-js-port \nbackbone-tableview & & 279 & & master@0c26357 & mbrevda/backbone-tableview \neasystarjs & & 369 & & v0.4.3 & prettymuchbryce/easystarjs \ngeojsonhint & & 473 & & v.2.0.0 & mapbox/geojsonhint \nexpress-session & & 612 & & v1.15.6 & expressjs/session \nunderscore.string & & 873 & & 3.3.4 & esamattis/underscore.string \nmessy & & 1694 & & 6.11.0 & papandreou/messy \nvirtual-dom & & 1978 & & v2.1.1 & Matt-Esch/virtual-dom \nrecipe-parser & & 4134 & & master@626f124 & catesandrew/recipe-parser \nplanck.js & & 10390 & & v0.2.7 & shakiba/planck.js \ngoojs & & 57373 & & v0.16.8 & GooTechnologies/goojs \n[Tab:projectImplementationDetails]\nTable [Tab:projectImplementationDetails] presents the projects used in our evaluation, along with relevant implementation details. Column 2 provides the module format used in each project. Notice that the first four projects are non-modular. Column 3 (SLOC) includes the total lines of JavaScript production code, computed with CLOC[3](#fn3). In case that the codebase of a project includes external libraries, library code is excluded from further analysis. The reason is that these libraries make use of dynamic features of the language (e.g. on-the-fly creation and deletion of object properties, dynamic code generation). These features undermine the results of static analysis and, thus, their refactoring may introduce errors or influence the program’s external behaviour. Column 4 provides the test coverage of each project, in terms covered statements by its test suite. Test coverage is not reported for projects without a test suite. Column 5 specifies the tag/commit of the project’s revision history that was used in this study and Column 6 provides the projects’ Github repository names.\nThe proposed method is implemented as an open source Node.js tool [4](#fn4), which employs jscodeshift for Abstract Syntax Tree (AST) traversal and transformation, JSHintfor the resolution of implied globals and Ternfor the dataflow analysis required in the module dependency identification step ([sect:resolve-module-deps]). The JavaScript code that is embedded in the HTML files of the analyzed web applications is extracted with the help of the jsoup HTML parserThe implemented prototype requires limited user interaction, since the user only needs to specify the application to be refactored and the format that is used for its implementation. The evaluation of the prototype is performed on a workstation, equipped with an Intel-Core i7-7700 CPU @3.60 Ghz and 16GB RAM.\nEvaluation Results\n------------------\n### RQ1. How prevalent is the presence of ES5 code, modular or not, in real-world applications?\nThe first research question explores how relevant is the proposed method with the needs of current web applications. Since the ES6 (ES2015) specification has been fully implemented in most JavaScript runtimes, a question naturally arises as to what extent web applications still use the legacy module formats and have not yet migrated to ES6. To the best of our knowledge, there is no published study that explores the use of AMD/CJS formats and the current adoption of ES6 modules.\nIn order to answer RQ1, we analyzed the code-base of all Github repositories that use JavaScript as their main language and have received at least a minimum of developer engagement in the recent years. We acquired access to the source code of the main branch and the public activity timeline of Github repositories, through the *Github Activity Data* [5](#fn5) and *GithubArchive* [6](#fn6) public BigQuery datasets. Among approximately 4 million repositories that include JavaScript code, we narrowed down our analysis to 14,167 repositories on the basis of the following criteria:\n* the primary language of the repository, in terms of bytes of code, is JavaScript or HTML. We excluded repositories that include Java, Objective-C or C# code, in order to avoid analyzing cross-platform mobile applications, where migration to ES6 is constrained by application development frameworks.\n* each repository has received at least 10 commits and 5 stars since 01.01.2018. Our purpose is to exclude abandoned projects that would bias our results in favor of legacy module formats.\nIn a next stage, we collected all JavaScript files from the aforementioned repositories and classified them as AMD, CJS or ES6 modules. The classification is based on regular expressions that match the syntax of each module format in the source code of each file. Regular expressions are appropriately formulated so that each file is categorized to a single module format and are executed against a BigQuery table with JS file contents. We excluded from our analysis build tool configuration files (e.g. webpack, rollup), as well as source files located at the installation folder of package dependencies (e.g., node\\_modules).\nThe results of JS content classification to module formats are available in Table [Tab:jsmodules-distr]. The table provides the number of JS files (Column 2), among all 14,167 repositories, that declare AMD, CommonJS, ES6 modules or do not use any standard module format. Column 3 estimates the share of all files that fall under these categories. Classification results show that a large share of JS content () is either non-modular or does not use a standard module format. Moreover, about one-third of all JS files () use the standard ES5 module formats, AMD () and CJS (). Finally, the adoption of the ES6 module format characterizes another one-third of all files ().\n[htb]\nl r r\n**Module format** & & \nAMD & 65,457 & 6.7 \nCommonJS & 281,874 & 28.7 \nES6 & 327,367 & 33.3 \nOther & 307,523 & 31.3 \nTotal & 982,221 & 100.0 \n[Tab:jsmodules-distr]\nLegacy module formats and non-modular ES5 code are still widely used in actively maintained open source projects. This stresses the need for methods and tools for their automated migration to ES6 modules in order to reap the benefits of new language features.\n### RQ2. Is there a developer intent for independently reusable features within modules?\nThe second research question is related to the practicality of the proposed refactoring. It’s purpose is to investigate whether the extracted fine-grained module features correspond to elements with similar semantics in the original module design, i.e., elements that belong to the module scope and may be reused independently from each other. Thus, the language constructs (ES6 named imports/exports) that are automatically applied through refactoring, could have been employed by the programmer if they were supported by ES5.\nIn order to answer RQ2, we conducted an exploratory study on frequently occurring patterns on the design of module objects. The context of this study comprised the subset of benchmark projects that use the AMD/CJS formats. Non-modular ES5 projects were excluded, since their source files do not declare module objects. The study involved, at first, the identification of common patterns through code inspection on a sample of modules from each benchmark project. Then, we estimated the frequency of each pattern, across all modules,"},"data_id":{"kind":"string","value":"arxiv_0184743"}}},{"rowIdx":46190,"cells":{"content":{"kind":"string","value":"m*\\*(1 − max*u* ∈ U*φ**u*)− 1, which can provide significant computational benefits.\nThus, we obtain a three-strep approach to compute an approximate solution to the network monitoring problem and evaluate its optimality gap. Next, we further study our solution for problem instances with particular characteristics.\nSpecial Cases\n-------------\n### Non-overlapping monitoring sets\nWe consider problem instances for which monitoring sets are pairwise disjoint, i.e., *U**x* ∩ *U**x*′ = ∅ for every *x* ≠ *x*′ ∈ X. Such instances occur when it is desirable to reduce sensor interference or the energetic cost of the network. In other contexts such as in security games, disjoint monitoring is naturally satisfied. For such instances, we show that our approach in Section [sec:Approach] optimally solves the network monitoring problem:\n[theorem:analyticalsolutionspecialcase] If the monitoring sets are pairwise disjoint, then $z^\\*\\_{({\\overline{\\mathcal{M}}})} = z^\\*\\_{({\\mathcal{M}})}$. Furthermore, any monitoring strategy satisfying for *C* = X is an optimal solution of.\nFor instances when monitoring sets do not overlap, we find that problem can be optimally solved by first selecting the *k*X\\* most critical sensor locations and monitoring them with probabilities given by ; and then by utilizing the coordination algorithm to construct a monitoring strategy that randomizes the placement of *r* sensors and is consistent with the marginal probabilities in.\n### Homogeneous security levels\nWe now consider instances where the security levels are identical across the components in U. In such cases, we refine the bounds on the optimal value of using the minimum set cover and maximum set packing sizes (*n*\\* and *m*\\*):\n[prop:identical] If the security levels are identical and equal to *φ*, then:\n \n$$\\begin{aligned}\nz^\\*\\_{({\\overline{\\mathcal{M}}})} \\hspace{-0.05cm}=\\hspace{-0.05cm} \\varphi + (1-\\varphi) \\frac{r}{n^\\*} \\hspace{-0.05cm}\\leq\\hspace{-0.03cm} z^\\*\\_{({\\mathcal{M}})} \\hspace{-0.05cm} \\leq \\hspace{-0.03cm} \\min\\bigg\\{1,\\varphi + (1-\\varphi) \\frac{r}{m^\\*}\\bigg\\}.\\end{aligned}$$ \nIn such instances, we find that and can be simplified and the bounds on *z*(M)\\* can be computed by solving a minimum set cover and maximum set packing problem, as in. If *C*\\* denotes a minimum set cover, then a monitoring strategy that optimizes $z^\\*\\_{({\\overline{\\mathcal{M}}})}$ monitors each location in *C*\\* with identical probability (given by ), and can be constructed by cycling the *r* sensors over *C*\\* (see for reference).\nAlternative Approaches\n======================\nIn order to compare our solution approach derived in Section [sec:Approach], we next consider two alternative approaches for solving the network monitoring problem. In particular, we show how the column generation algorithm and multiplicative weights update algorithm can be applied to solve.\nColumn Generation Algorithm\n---------------------------\nProblem can be formulated as a linear program with a large number (${|{\\mathcal{X}}| \\choose r} + 1$) of variables and a small number (∣U∣ + 1) of constraints. Such problems can potentially be solved using the column generation algorithm, as long as the pricing problem can be solved efficiently.\nSpecifically, at each iteration of the column generation algorithm, the following restricted master problem is solved:\n \n$$\\begin{aligned}\n{3}\n\\max \\ & z \\tag{${\\mathcal{M}}\\_\\mathcal{I}$}\\label{eq:P\\_I}\\\\\n\\text{s.t.} \\ & z - \\sum\\_{X\\in \\mathcal{I}}\\sigma^1\\_X f(X,u) \\leq 0, \\ && {\\ForAll\\,}u \\in {\\mathcal{U}}\\label{dual1}\\\\\n& \\sum\\_{X \\in \\mathcal{I}}\\sigma^1\\_X = 1,\\label{dual2}\\\\\n& \\sigma^1\\_X \\geq 0, && {\\ForAll\\,}X \\in \\mathcal{I},\\nonumber\\end{aligned}$$ \nwhere I ⊆ A is a subset of feasible sensor positionings. To determine if the optimal solution of is optimal for, the algorithm then determines the variable in with highest reduced cost. Let (*α*\\*, *β*\\*) denote the optimal dual variables of, associated with constraints -. Then, the variable with highest reduced cost can be determined by solving the following pricing problem:\n \n$$\\begin{aligned}\n\\max\\_{X\\in{\\mathcal{A}}} - \\beta^\\* +\\sum\\_{u\\in{\\mathcal{U}}} \\alpha^\\*\\_u f(X,u).\\end{aligned}$$ \nIn fact, the pricing problem for is a maximum weighted covering problem, and can be solved using the following IP:\n \n$$\\begin{aligned}\n{3}\n\\max\\_{y,z} \\ & - \\beta^\\* + \\sum\\_{u \\in {\\mathcal{U}}} \\alpha^\\*\\_u (\\varphi\\_u + (1-\\varphi\\_u)y\\_u)\\tag{$\\mathcal{W}$}\\label{eq:O}\\\\\n\\text{s.t.} \\ & y\\_u \\leq \\sum\\_{\\{x \\in {\\mathcal{X}}\\, | \\, u \\in U\\_x\\}} z\\_x,\\ && {\\ForAll\\,}u \\in {\\mathcal{U}}\\label{covering\\_pb\\_1}\\\\\n& \\sum\\_{x \\in {\\mathcal{X}}} z\\_x = r,\\\\\n&y\\_u \\in \\{0,1\\}, && {\\ForAll\\,}u \\in {\\mathcal{U}}\\\\\n& z\\_x \\in \\{0,1\\}, && {\\ForAll\\,}x \\in {\\mathcal{X}}\\label{covering\\_pb\\_4}.\\end{aligned}$$ \nIn this IP, the binary variables *z**x*, for *x* ∈ X, represent the nodes that receive a sensor, and the binary variables *y**u*, for *u* ∈ U, represent the components that are monitored by at least one sensor.\nIf *z*(W)\\* = 0, then the optimal solution of is optimal for, and the algorithm terminates. If instead *z*(W)\\* > 0, then given an optimal solution (*y*\\*, *z*\\*) of, the algorithm adds *X*\\* = supp(*z*\\*) to the set I of variable indices, and solves the new restricted master problem.\nMultiplicative Weights Update Algorithm\n---------------------------------------\nProblem is also equivalent to a simultaneous two-person zero-sum game, where the first player is the network operator who places *r* sensors across the networked system and the second player is an attacker who targets a component. The payoff function is given by *f*, which the operator (resp. attacker) wants to maximize (resp. minimize).\nIn fact, devised a multiplicative weights update algorithm for computing approximate mixed Nash equilibria of such simultaneous games, as long as one player has a small number of actions and the other player can efficiently compute best responses. The algorithm runs for *N* iterations, and at each iteration *t* ∈ ⟦1, *N*⟧, generates a probability distribution *σ*2, *t* over U for the attacker as well a feasible sensor positioning *X**t* ∈ A for the network operator. The algorithm starts by setting *σ*2, 1 to the uniform distribution over U. Then, at each iteration *t* ∈ ⟦1, *N*⟧, the algorithm determines the operator’s best response to *σ*2, *t* by solving the following problem: \n*X**t* ∈ argmax*X* ∈ AE*u* ∼ *σ*2, *t*[*f*(*X*, *u*)]. \n In our setting, this problem can also be formulated as a maximum weighted covering problem and can be solved using the following IP:\n \n$$\\begin{aligned}\n{3}\n\\max\\_{y,z} \\ & \\sum\\_{u \\in {\\mathcal{U}}} \\sigma^{2,t}\\_u (\\varphi\\_u + (1-\\varphi\\_u)y\\_u)\\tag{$\\mathcal{W}^\\prime$}\\label{eq:W'}\\\\\n\\text{s.t.} \\ & \\eqref{covering\\_pb\\_1}-\\eqref{covering\\_pb\\_4}.\\nonumber\\end{aligned}$$ \nThen, the algorithm updates the attacker’s strategy using the following multiplicative weights update: \n$${\\ForAll\\,}u \\in {\\mathcal{U}}, \\ \\sigma^{2,t+1}\\_u = \\sigma^{2,t}\\_u \\frac{\\eta^{f(X^t,u)}}{\\sum\\_{u^\\prime \\in {\\mathcal{U}}} \\sigma^{2,t}\\_{u^\\prime} \\eta^{f(X^t,{u^\\prime})}},$$ \n where $\\eta = \\left(1 + \\sqrt{2 \\frac{\\ln |{\\mathcal{U}}|}{N}}\\right)^{-1}$. The authors in showed that the monitoring strategy *σ̄*1 ∈ Δ1 that uniformly randomizes over the *N* best responses *X*1, …, *X**N* satisfies:\n \n$$\\begin{aligned}\n\\min\\_{u \\in {\\mathcal{U}}} \\mathbb{E}\\_{X\\sim \\bar{\\sigma}^1}[f(X,u)] \\geq z^\\*\\_{(\\mathcal{M})} - \\sqrt{\\frac{2 \\ln |{\\mathcal{U}}|}{N}} - \\frac{\\ln|{\\mathcal{U}}|}{N}.\\end{aligned}$$ \nThus, if $N = 4\\lceil \\frac{\\ln |{\\mathcal{U}}|}{\\epsilon^2}\\rceil$ for *ε* > 0, then the (absolute) optimality gap associated with the monitoring strategy *σ̄*1 generated by this algorithm is upper bounded by *ε*.\nIn fact, showed that each iteration of this algorithm can be conducted in polynomial time by approximately solving using a greedy algorithm. However, the resulting guarantee on the monitoring strategy *σ̄*1 becomes\n \n$$\\begin{aligned}\n\\min\\_{u \\in {\\mathcal{U}}} \\mathbb{E}\\_{X\\sim \\bar{\\sigma}^1}[f(X,u)] \\geq (1-1/e) z^\\*\\_{(\\mathcal{M})} - \\epsilon,\\end{aligned}$$ \nfor $T = 4\\lceil \\frac{\\ln |{\\mathcal{U}}|}{\\epsilon^2}\\rceil$. Since commercial solvers are now highly efficient at solving classical IPs, we instead optimally solve at each iteration of the algorithm.\nWe now move to computational experiments, where we evaluate the three approaches presented in this article and compare their running times, as well as the performance and implementability of the monitoring strategies they generate.\nComputational Study\n===================\nIn this section, we describe how our strategic network monitoring problem can be applied for contamination detection in water distribution systems and actuator protection in networked control systems. Our experiments are coded in Julia and all optimization problems are solved using the Gurobi solver v9.0 on a computer with 2.3-GHz, 8-Core Intel Core i9 processor and 32 GB of RAM.\nApplication I: Contamination Detection in Water Distribution Systems\n--------------------------------------------------------------------\nWe first consider the security problem where the operator of a water distribution system aims to allocate sensors to monitor their network against adversarially-induced contamination events. We model the water distribution system as a directed graph, with the vertices representing pumps, junctions, and water tanks, while the edges represent pipes. The direction of every edge is adopted to be in the direction of the water flow. We assume that the operator has access to contaminant detection sensors that can be deployed on valve access points or fire hydrants and measure water quality indicators such as electrical conductivity, free and total chlorine, turbidity, and oxygen reduction potential in the water.\nIn this application, the set of vulnerable components U (resp. set of sensor locations X) is given by the set of edges (resp. nodes) of the graph modeling the water network. Monitoring sets are constructed through simulations using a hydraulic network solver that tracks the advection and reaction dynamics from a contaminant intrusion event. In practice, the security levels of components can be assessed based on previously deployed security measures and their accessibility to an adversary. One can then use different security scales to determine quantitative values for the security levels based on that assessment. In this application, we adopt the scale {0.2, 0.4, 0.6, 0.8} for security levels, and randomly assign them to each component.\nNext, we implement the three solution approaches described in Sections [sec:Method] and [sec:Alt] on the water distribution system ky5 . This anonymized real-world water network from Kentucky comprises 496 pipes and 420 nodes (junctions, water tanks, pumps), satisfies a demand of 2.28 million gallons of water per day, and spans 52.3 miles. In Fig. [fig:bounds], we illustrate the bounds derived in Theorem [prop:bounds] and computed by solving the generalized covering set problem and the nonlinear set packing problem, respectively.\n[scale=1]\n[ grid=major, grid style=dashed,gray!30, xlabel=Number of sensors *r*, ylabel=Lowest post-security level $\\min\\limits\\_{u \\in {\\mathcal{U}}}\\mathbb{E}\\_{X\\sim \\sigma^1}[f(X,u)]$, font=, xmin=0, xmax=19, ymin=0.2, ymax=1, ytick=0.2, 0.4,0.6,0.8,1, xtick=0, 2,4,6,8,10,12,14,16,18, legend pos=south east, legend cell align=left, legend style=draw=none, font =, ]\ntable[x=rs, y=Upper Bound, col sep=comma, comment chars=%] ky5resultsobj.csv;\ntable[x=rs, y=Optimal Value, col sep=comma, comment chars=%] ky5resultsobj.csv;\ntable[x=rs, y=Lower Bound Marg, col sep=comma, comment chars=%] ky5resultsobj.csv;\n[fig:bounds]\nFrom Fig. [fig:bounds], we observe that the bounds obtained from and are close to the optimal value of. On average, the optimality gap given by the bounds does not exceed 4.7 %, while the actual optimality gap is equal to 3.6 % on average.\nIn general, we observe that *z*(M)\\* increases nonlinearly with respect to the number of sensors. When *r* is small, the optimal monitoring strategy focuses on protecting the most vulnerable components (with a security level of 0.2). As the operator has access to more sensors, they randomize their placement to increase the post-security level of less critical components. With 19 sensors, the operator can monitor all components and ensure a post-security level of 1 for the entire network.\nNext, we compare the running times of the various approaches. Specifically, we illustrate the time to (a) solve and get an upper bound on *z*(M)\\*, (b) solve and construct a monitoring strategy using the coordination algorithm (CA), (c) solve using column generation (CG), and (d) solve using multiplicative weights updates (MWU) with *N* = 2, 480 iterations to ensure a theoretical (absolute) optimality gap of *ε* = 0.1. Their running times are illustrated in Fig. [fig:times].\n[scale=1]\n[ grid=major, grid style=dashed,gray!30, xlabel=Number of sensors *r*, ylabel=Running time [s], font=, xmin=1, xmax=17, ymin=0.01, ymax=1000, xtick=1,3,5,7,9,11,13,15,17, legend columns=4, legend style=/tikz/column 2/.style= column sep=0pt,, font =,legend cell align=left,at=(0.5,-0.18),anchor=north, ]\ntable[x=rs, y=Time Lower Bound Marg, col sep=comma, comment chars=%] ky5resultstimes.csv;\ntable[x=rs, y=Time Upper Bound, col sep=comma, comment chars=%] ky5resultstimes.csv;\ntable[x=rs, y=Time Optimal Value, col sep=comma, comment chars=%] ky5resultstimes.csv;\ntable[x=rs, y=Time Multiplicative Weight Update, col sep=comma, comment chars=%] ky5resultstimes.csv;\n[fig:times]\nFig. [fig:times] shows that the upper bound on *z*(M)\\* derived in Theorem [prop:bounds] can be efficiently computed; by solving, we obtain the upper bound in less than 0.2 seconds. We also observe that the number of sensors has limited impact on the running time.\nFurthermore, we find that CG is approximately 100 times faster than MWU. This is in part due to the number of iterations that are required for MWU to theoretically guarantee a small optimality gap. Interestingly, we find that CG can efficiently compute an optimal solution of when the number of sensors is small. However, as *r* increases, the running time of CG increases exponentially. Finally, we find that our approximate monitoring strategy, which achieves the lower bound in Theorem [prop:bounds] and is obtained by solving + CA, can be efficiently computed.\nFinally, we compare the implementability of the solutions generated by the three approaches by illustrating their node basis sizes in Fig. [fig:nodebasis].\n[scale=1]\n[ grid=major, grid style=dashed,gray!30, xlabel=Number of sensors *r*, ylabel=Node basis size ∣X(*σ*1)∣, font=, xmin=1, xmax=17, ymin=0, ymax=200, xtick=1,3,5,7,9,11,13,15,17, legend style=font =,draw=none,legend cell align=left,at=(0.8,0.37),anchor=north, ]\ntable[x=rs, y=Node Basis Size Marg, col sep=comma, comment chars=%] ky5results.csv;\ntable[x=rs, y=Node Basis Size at Optimality, col sep=comma, comment chars=%] ky5results.csv;\ntable[x=rs, y=Node Basis Size Multiplicative Weight Update, col sep=comma, comment chars=%] ky5results.csv;\n[fig:nodebasis]\nFig. [fig:nodebasis] clearly shows the implementability advantage of our monitoring strategy. In particular, we find that our solution requires monitoring at most 20 different locations and achieves an excellent worst-case post-security level (from Fig. [fig:bounds]). In contrast, the optimal post-security level achieved using CG or MWU requires monitoring up to 180 different locations. From an implementation perspective, our monitoring strategy is considerably easier to translate into a daily/weekly scheduling of monitoring operations.\nApplication II: Protecting Actuators from Extended Replay Attacks\n-----------------------------------------------------------------\nWe now show how our findings can be used to protect actuators in a networked control system from extended replay attacks. The networked system consists of a set of physical states X and actuators U, as well as fixed unprotected sensors. We consider an attacker who aims to conduct an undetectable attack on an actuator using an extended replay strategy. More precisely an extended replay attack on an actuator is undetectable if the sensor measurements remain in the steady state during the attack. To this end, the attacker must also compromise a subset of components (i.e., unprotected sensors and other actuators) to ensure that the attack is undetected. Such subset of components typically depends on the dynamics of the networked system, and can be efficiently determined in large-scale systems,.\nImportantly, the vulnerability of an actuator *u* with respect to extended replay attacks can be determined based on its security index *δ**u*, which represents the minimum number of sensors and actuators, including *u* itself, that the attacker must compromise to conduct an undetectable extended replay attack against *u*, in the absence of protected sensors. If no undetectable extended replay attack against *u* is possible, then *δ**u* = + ∞. We can then scale these security indices and define security levels for this application, following our modeling framework. Specifically, we adopt an identical scale as in the previous application, and define the security level of each actuator *u* ∈ U as follows:\n \n$$\\begin{aligned}\n\\varphi\\_u =&\\textstyle 0.2 \\cdot \\mathds{1}\\_{[\\delta\\_u \\in (0,5] ]} \n+ 0.4 \\cdot \\mathds{1}\\_{\\{\\delta\\_u \\in (5,15] \\}}+ 0.6 \\cdot \\mathds{1}\\_{\\{\\delta\\_u \\in (15,20] \\}} \\\\\n&\\textstyle\n+ 0.8 \\cdot \\mathds{1}\\_{\\{\\delta\\_u \\in (20,+\\infty)\\}}+ 1.0 \\cdot \\mathds{1}\\_{\\{\\delta\\_u= +\\infty\\}}.\\end{aligned}$$ \n[!htbp] [tab:ieee]\nccccccccccccc & & & & & & \n *r* & & + CA & CG & MWU & & + CA & CG & MWU & & + CA & CG & MWU \n50 & &0.0 & 0.0 & 0.3 & & 1.6 & 71.3 & 227.7 & & 223 & 373 & 430 \n100 & &0.0 & 0.4 & 0.4 & & 1.9 & 600 & 245.1 & & 318 & 868 & 737 \n150 & &0.0 & 1.1 & 0.4 & & 13.2 & 600 & 302.6 & & 328 & 981 & 824 \n200 & &0.0 & 8.0 & 0.3 & & 5.1 & 600 & 412.2 & & 328 & 1009 & 827 \n250 & &0.0 & 15.0 & 0.3 & & 174.2 & 600 & 581.2 & & 344 & 1024 & 1115 \n300 & &0.0 & 24.0 & 0.3 & & 39.2 & 600 & 591.0 & & 344 & 1086 & 612 \nTo defend the networked system, we assume the operator possesses protected sensors that cannot be compromised by the attacker and that can be positioned within the network. We assume that each positioned sensor can only measure one physical state, which is a commonly adopted model of large–scale systems . In particular, a protected sensor can detect if the state it measures deviates from its steady state value during an extended replay attack. Thus, actuator *u* ∈ U belongs to the monitoring set of state *x* ∈ X if placing a protected sensor at *x* prevents the attacker from conducting an undetectable extended replay attack on *u*.\nWe now consider the IEEE 2383 bus power system. We model the system using linearized swing equations where the generators are represented by two states (rotor angle and frequency) and load buses with one state (voltage angle) . We assume that all ∣X∣ = 3037 states are measurable, and that the attacker can conduct an attack using ∣U∣ = 1042 loads (we randomly selected 30% of the loads to be attackable). Table [tab:ieee] compares the performance of the different solution approaches on this networked system. Note that we set a time limit of 10 minutes for each approach.\nWe observe that for this network, our solution approach optimally solves the network monitoring problem. Indeed, the upper bound obtained by solving guarantees that the monitoring strategy computed from + CA is optimal for for every number of sensors. Although we observe some variance in the running time of our approach, the average running time for this network is 29 seconds. In general, we find that the hardest instances to solve are for which the number of sensors to position is high but not too close to the minimum set cover size. On the other hand, MWU achieves a very high performance, but requires 393 seconds on average. This illustrates the fact that the empirical performance of MWU is better than the theoretical guarantees. In contrast, we find that CG scales poorly with the number of sensors, which is interesting given that MWU and CG solve the very similar IPs and. Finally, we similarly note that the node basis (i.e., the locations that are monitored with positive probability) of our monitoring strategy is up to 3 times smaller than when utilizing CG or MWU, thus providing a solution that is more easily implementable in practice.\nSummary and Extension of Solution Methodology\n---------------------------------------------\nIn summary, we find that by extracting the structure of the network monitoring problem, we could derive an efficient solution approach. Our approach solves two IPs and runs the coordination algorithm to directly obtain a monitoring strategy with strong performance guarantees and that is easier to implement in practice. This contrasts with the classical CG and MWU algorithms that iteratively refine their solutions and generally run more slowly.\nInterestingly, more flexibility can be embedded in our solution approach. For instance, if the operator is interested in an even simpler monitoring strategy, we can add the cardinality constraint ∑*x* ∈ X*y**x* ≤ *s* in to find the best monitoring strategy of that monitors at most *s* locations. This permits the operator to achieve the desired tradeoff between solution implementability and guaranteed post-security level.\nFinally, we note that the three solution approaches can be combined for an even better performance. Indeed, for the instances for which our approach does not provide an optimal solution to the network monitoring problem, we can utilize our solution as a warm start for CG and/or MWU, which will then iteratively improve it.\nConclusion\n==========\nIn this article, we studied the problem of strategically monitoring large-scale networked systems with heterogeneous component criticalities using multiple sensors. Specifically, we formulated a large-scale maximin problem where the network operator selects a randomized placement of their protected sensors to improve the post-security level of the network’s most critical components against an attacker.\nWe developed a three-step solution approach that leverages the structure of the network monitoring problem. First, we solved a generalized covering set problem to determine the marginal probabilities to monitor each location. Then, we utilized a combinatorial algorithm to randomize the positioning of the multiple sensors and obtain a monitoring strategy that is consistent with the marginal probabilities. Finally, we solved a nonlinear set packing problem to evaluate our solution’s optimality gap. We also investigated problem instances with special structures, for which we refined our solution approach and optimality gaps.\nWe then adapted two classical solution methods for solving our monitoring problem based on column generation and multiplicative weights updates, and compared them on real-world benchmark water distribution and power systems. Our computational study revealed that our solution approach outperforms the classical methods as the size of the problem increases. In particular, we find that our approach can efficiently compute near optimal solutions for large instances and are significantly simpler to implement.\nAs part of future work, it would be interesting to investigate how the solution method developed in this article (combining smaller IPs and combinatorial algorithms) can be applied to solve problems where an attacker targets multiple components simultaneously, or where sensors are faulty.\nWe consider an optimal solution *σ*1\\* of and we set *C* = X(*σ*1\\*). Then, *z*(R*C*)\\* ≥ min*x* ∈ X(*σ*1\\*)E*X* ∼ *σ*1\\*[*f̄*(*X*, *x*)]. Furthermore, $\\min\\_{u \\notin U\\_C} \\varphi\\_u = \\min\\_{u \\notin U\\_{{\\mathcal{X}}(\\sigma^{1^\\*})}} \\varphi\\_u$. Therefore, $z^\\*\\_{({\\overline{\\mathcal{M}}})} \\leq \\max\\_{C \\subseteq {\\mathcal{X}}} \\min\\{z^\\*\\_{(\\mathcal{R}\\_C)},\\min\\_{u \\notin U\\_C} \\varphi\\_u\\}$.\nTo show the reverse inequality, let $C^\\* \\in \\arg\\max\\_{C \\subseteq {\\mathcal{X}}} \\min\\{z^\\*\\_{(\\mathcal{R}\\_C)},\\min\\_{u \\notin U\\_C} \\varphi\\_u\\}$. We also consider an optimal solution *σ*1\\* of (R*C*\\*). Without loss of generality, we can assume that X(*σ*1\\*) ⊆ *C*\\* (i.e., *σ*1\\* only places sensors within *C*\\*). Then, min*x* ∈ X(*σ*1\\*)E*X* ∼ *σ*1\\*[*f̄*(*X*, *x*)] ≥ *z*(R*C*\\*)\\*. Furthermore, let $u^\\prime \\notin U\\_{{\\mathcal{X}}(\\sigma^{1^\\*})}$. If *u*′ ∈ *U**C*\\* \\ *U*X(*σ*1\\*), let *x*′ ∈ *C*\\* \\ X(*σ*1\\*) such that *u*′ ∈ *U**x*′. We then obtain:\n \n$$\\begin{aligned}\n\\varphi\\_{u^\\prime} \\geq \\overline{\\varphi}\\_{x^\\prime} = \\mathbb{E}\\_{X \\sim \\sigma^{1^\\*}}[\\bar{f}(X,x^\\prime)] \\geq z^\\*\\_{(\\mathcal{R}\\_{C^\\*})}.\\end{aligned}$$ \nIf instead $u^\\prime \\notin U\\_{C^\\*}$, then $\\varphi\\_{u^\\prime} \\geq \\min\\_{u \\notin U\\_{C^\\*}} \\varphi\\_u$. In conclusion, $z^\\*\\_{({\\overline{\\mathcal{M}}})} = \\max\\_{C \\subseteq {\\mathcal{X}}} \\min\\{z^\\*\\_{(\\mathcal{R}\\_C)},\\min\\_{u \\notin U\\_C} \\varphi\\_u\\}$.\nWe first show that $z\\_{({\\mathcal{M}})}^\\* \\geq z\\_{({\\overline{\\mathcal{M}}})}^\\*$: Let *σ*1\\* ∈ Δ1 be an optimal solution of and let *u*′ ∈ U. If *u*′ ∈ *U*X(*σ*1\\*), then let *x*′ ∈ X(*σ*1\\*) such that *u*′ ∈ *U**x*′. Since $\\varphi\\_{u^\\prime} \\geq \\overline{\\varphi}\\_{x^\\prime}$ and for every *X* ∈ A, $ \\mathds{1}\\_{\\{x^\\prime \\notin X\\}} \\geq \\mathds{1}\\_{\\{u^\\prime \\notin U\\_X\\}}$, then\n \n$$\\begin{aligned}\n \\mathbb{E}\\_{X\\sim\\sigma^{1^\\*}}[f(X,u^\\prime)] &\\geq \\mathbb{E}\\_{X\\sim\\sigma^{1^\\*}}[1 - (1 - \\overline{\\varphi}\\_{x^\\prime}) \\mathds{1}\\_{\\{x^\\prime \\notin X\\}}]\\\\\n& \\geq \\min\\_{\\quad\\mathclap{x \\in {\\mathcal{X}}(\\sigma^{1^\\*})}\\quad} \\mathbb{E}\\_{X\\sim\\sigma^{1^\\*}}[1 - (1 - \\overline{\\varphi}\\_{x}) \\mathds{1}\\_{\\{x \\notin X\\}}].\\end{aligned}$$ \nIf instead $u^\\prime \\notin U\\_{{\\mathcal{X}}(\\sigma^{1^\\*})}$, then:\n \n$$\\begin{aligned}\n&\\mathbb{E}\\_{X\\sim\\sigma^{1^\\*}}[f(X,u^\\prime)] = \\varphi\\_{u^\\prime} \\geq \\min\\_{u \\notin U\\_{{\\mathcal{X}}(\\sigma^{1^\\*})} } \\varphi\\_u.\\end{aligned}$$ \nThus, $z^\\*\\_{(\\mathcal{M})} \\geq z^\\*\\_{({\\overline{\\mathcal{M}}})}$. We next show that for every *C* ⊆ X of size greater than *r*, $z^\\*\\_{(\\mathcal{R}\\_C)} = 1 - \\frac{k^\\*\\_C-{r}}{S^{k^\\*}\\_C}$.\nWe rewrite *C* = {*x*1, …, *x**n*} with *n* > *r* and indices satisfying $\\overline{\\varphi}\\_{x\\_1} \\leq \\cdots \\leq \\overline{\\varphi}\\_{x\\_n}$. Then necessarily, *k**C*\\* ≥ *r* + 1 since\n \n$$\\begin{aligned}\n1 - \\frac{r + 1 - r}{\\sum\\_{l=1}^{r+1} (1 - \\overline{\\varphi}\\_{x\\_l})^{-1}} \\geq 1 - (1 - \\overline{\\varphi}\\_{x\\_{r+1}}) = \\overline{\\varphi}\\_{x\\_{r+1}}.\\end{aligned}$$ \nLet *σ*1′ ∈ Δ1 be a monitoring strategy satisfying. The existence of such a monitoring strategy follows from Farkas’ lemma (for instance, see ), since\n \n$$\\begin{aligned}\n \\sum\\_{l=1}^{k^\\*\\_C} 1 - \\frac{k^\\*\\_C-{r}}{(1-\\overline{\\varphi}\\_{x})S^{k^\\*}\\_C} = r,\n \\end{aligned}$$ \nand for every *l* ∈ ⟦1, *k**C*\\*⟧,\n \n$$\\begin{aligned}\n1 \\geq 1 - \\frac{k^\\*\\_C-{r}}{(1-\\overline{\\varphi}\\_{x\\_l})S^{k^\\*}\\_C} \\geq 1 - \\frac{k^\\*\\_C-{r}}{(1-\\overline{\\varphi}\\_{x\\_{k^\\*\\_C}})S^{k^\\*}\\_C} \\geq 0.\n \\end{aligned}$$ \nWe next consider *x* ∈ *C*. If *x* ∈ {*x*1, …, *x**k**C*\\*}, then:\n \n$$\\begin{aligned}\n\\mathbb{E}\\_{X \\sim \\sigma^{1^\\prime}}[\\bar{f}(X,x)] &= \\overline{\\varphi}\\_x + (1-\\overline{\\varphi}\\_x)\\mathbb{P}\\_{X \\sim \\sigma^{1^\\prime}}(x \\in X) \\\\\n&= 1 - \\frac{k^\\*\\_C-{r}}{S^{k^\\*}\\_C}.\\end{aligned}$$ \nIf instead *x* ∈ *C* \\ {*x*1, …, *x**k**C*\\*}, then:\n \n$$\\begin{aligned}\n\\mathbb{E}\\_{X \\sim \\sigma^{1^\\prime}}[\\bar{f}(X,x)] &= \\overline{\\varphi}\\_x \\geq \\overline{\\varphi}\\_{x\\_{k^\\*\\_C+1}} \\\\\n&> 1 - \\frac{k^\\*\\_C + 1 - r}{S^{k^\\*}\\_C + (1 - \\overline{\\varphi}\\_{x\\_{k^\\*\\_C+1}})^{-1}},\\end{aligned}$$ \nwhich is equivalent to\n \n$$\\begin{aligned}\n\\mathbb{E}\\_{X \\sim \\sigma^{1^\\prime}}[\\bar{f}(X,x)] > 1 - \\frac{k^\\*\\_C - r}{S^{k^\\*}\\_C}.\\end{aligned}$$ \nThus, $z^\\*\\_{(\\mathcal{R}\\_C)} \\geq 1 - \\frac{k^\\*\\_C-{r}}{S^{k^\\*}\\_C}$.\nNow, let us assume by contradiction that $z^\\*\\_{(\\mathcal{R}\\_C)}> 1 - \\frac{k^\\*\\_C-{r}}{S^{k^\\*}\\_C}$. Let *σ*1\\* be an optimal solution of *z*(R*C*)\\*. Then, we deduce that:\n \n$$\\begin{aligned}\n{\\ForAll\\,}x \\in C, \\ \\mathbb{P}\\_{X\\sim \\sigma^{1^\\*}}(x \\in X) > 1 - \\frac{k^\\*\\_C-{r}}{(1-\\overline{\\varphi}\\_x)S^{k^\\*}\\_C},\\end{aligned}$$ \nwhich induces the following contradiction:\n \n$$\\begin{aligned}\nr &= \\sum\\_{X \\in {\\mathcal{A}}} \\sigma^{1^\\*}\\_X |X| = \\sum\\_{x \\in {\\mathcal{X}}} \\sum\\_{X \\in {\\mathcal{A}}} \\sigma^{1^\\*}\\_X \\mathds{1}\\_{\\{x \\in X\\}} \\\\\n&=\\sum\\_{x \\in {\\mathcal{X}}} \\mathbb{P}\\_{X\\sim \\sigma^{1^\\*}}(x \\in X) > \\sum\\_{l=1}^{k^\\*\\_C} 1 - \\frac{k^\\*\\_C-{r}}{(1-\\overline{\\varphi}\\_{x\\_l})S^{k^\\*}\\_C} = r.\\end{aligned}$$ \nIn conclusion, $z^\\*\\_{(\\mathcal{R}\\_C)}= 1 - \\frac{k^\\*\\_C-{r}}{S^{k^\\*}\\_C}$, and an optimal solution is given by *σ*1′ satisfying.\nWe note that for every *C* ⊆ X of size no more than *r*, *z*(R*C*)\\* = 1 (which is obtained by placing a sensor on each location in *C*). Also note that in such cases, U \\ *U**C* ≠ ∅ and $1 - \\frac{k^\\*\\_C-{r}}{S^{k^\\*}\\_C}\\geq 1 \\geq \\min\\_{u \\notin U\\_C} \\varphi\\_u$. Thus, from Lemma [lem:Angle], we deduce that \n$$z^\\*\\_{(\\overline{\\mathcal{M}})} = \\max\\_{C \\subseteq {\\mathcal{X}}} \\min\\bigg\\{1 - \\frac{k^\\*\\_C-{r}}{S^{k^\\*}\\_C},\\min\\_{\\mathclap{u \\notin U\\_C}}\\varphi\\_u\\bigg\\}.$$ \nFinally, we prove the upper bound on *z*(M)\\*. Since *f*(*X*, *u*) ≤ 1 for every *X* ∈ A and every *u* ∈ U, then *z*(M)\\* ≤ 1. Furthermore, by strong duality, we know that *z*(M)\\* = min*σ*2 ∈ Δ2max*X* ∈ AE*u* ∼ *σ*2[*f*(*X*, *u*)], where $\\Delta^2 \\coloneqq \\{\\sigma^2 \\in [0,1]^{|{\\mathcal{U}}|} \\ | \\ \\sum\\_{u \\in {\\mathcal{U}}}\\sigma^2\\_u = 1\\}.$ Then, given a set packing *T* ∈ T, we define *σ*2′ ∈ Δ2 as follows:\n \n$$\\begin{aligned}\n\\sigma^{2^\\prime}\\_u = \\begin{cases} \\frac{1}{(1-\\varphi\\_u)S\\_T}, & \\text{ if } u \\in T\\\\\n0, & \\text{ otherwise.}\\end{cases}\n \\end{aligned}$$ \nWe can then derive the following upper bound.\n \n$$\\begin{aligned}\n z^\\*\\_{({\\mathcal{M}})} &\\leq \\max\\_{X\\in {\\mathcal{A}}}\\mathbb{E}\\_{u\\sim \\sigma^{2^\\prime}}[f(X,u)] \\\\\n &= \\max\\_{X\\in {\\mathcal{A}}}\\sum\\_{u \\in T} \\sigma^{2^\\prime}\\_u (1 - (1-\\varphi\\_u)\\mathds{1}\\_{\\{u \\notin U\\_X\\}})\\\\\n & = 1 - \\frac{1}{S\\_T}\\min\\_{X\\in {\\mathcal{A}}}\\sum\\_{u \\in T}\\mathds{1}\\_{\\{u \\notin U\\_X\\}} \\leq 1 - \\frac{|T|-r}{S\\_T}\n \\end{aligned}$$ \nIn conclusion, $z\\_{({\\mathcal{M}})}^\\*\\leq \\min\\{1,\\min\\_{T \\in {\\mathcal{T}}} 1 - \\frac{|T|-{r}}{S\\_T}\\}$.\nLet (*y*\\*, *z*\\*, *ρ*\\*) be an optimal solution of, and we set *C* = supp(*y*\\*). Then, can be rewritten as follows:\n \n$$\\begin{aligned}\n{3}\n\\max\\_y &\\min\\bigg\\{\\max\\_\\rho \\min\\_{x \\in {\\operatorname{supp}}(y)} && \\overline{\\varphi}\\_x + (1- \\overline{\\varphi}\\_x) \\rho\\_x, \\min\\_{u \\notin U\\_{{\\operatorname{supp}}(y)}} \\varphi\\_u\\bigg\\}.\\nonumber\\\\\n \\text{s.t.} \\ & 0 \\leq \\rho\\_x \\leq y\\_x, \\ &&{\\ForAll\\,}x \\in {\\mathcal{X}}\\label{cons\\_1}\\\\\n & \\sum\\_{x \\in {\\mathcal{X}}}\\rho\\_x = r,\\label{cons\\_2}\\\\\n & y\\_x \\in \\{0,1\\}, \\ &&{\\ForAll\\,}x \\in {\\mathcal{X}}.\\nonumber\\end{aligned}$$ \nThe equivalence between and then follows from Lemma [lem:Angle], equality, and the existence of monitoring strategies satisfying constraints -.\nWe argued in the main text that Problem is indeed a MIP formulation of the following problem $\\min\\_{T \\in {\\mathcal{T}}} \\frac{S\\_T}{|T|-r}$. In particular, at optimality, constraints and *M*2 ensure that at optimality all *t**l* variables are equal to 0 except for the index corresponding to the set packing size, thus ensuring the validity of the formulation. Then, $\\min\\{1,\\min\\_{T \\in {\\mathcal{T}}} 1 - \\frac{|T|-{r}}{S\\_T}\\} = 1 - \\frac{1}{z^\\*\\_{(\\mathcal{P})}}.$ Note that we used the convention that if a is infeasible (which occurs if there is no set packing of size greater than *r*), then *z*(P)\\* = + ∞.\nWe consider an instance where *U**x* ∩ *U**x*′ = ∅ for every *x* ≠ *x*′ ∈ X. We index the locations in X = {*x*1, …, *x**n*} in nondecreasing order of their criticalities. Then, we consider the set packing *T*′ = {*u*1, …, *u**k*X\\*} ∈ T, where for every *l* ∈ ⟦1, *k*X\\*⟧, $\\varphi\\_{u\\_l} = \\overline{\\varphi}\\_{x\\_l}$. In other words, *T*′ selects the most critical component within each of the first *k*X\\* monitoring sets. As a consequence, ∣*T*′∣ = *k*X\\* and *S**T*′ = *S*X*k*\\*. Since *U*X = U, then from Theorem [prop:bounds], we obtain:\n \n$$\\begin{aligned}\nz^\\*\\_{(\\mathcal{R}\\_{\\mathcal{X}})} &= 1 - \\frac{k^\\*\\_{\\mathcal{X}}- r}{S^{k^\\*}\\_{\\mathcal{X}}} \\leq z\\_{({\\overline{\\mathcal{M}}})}^\\*\\leq z^\\*\\_{({\\mathcal{M}})} \\\\\n&\\leq 1 - \\frac{|T^\\prime| - r}{S\\_{T^\\prime}} = 1 - \\frac{k^\\*\\_{\\mathcal{X}}- r}{S^{k^\\*}\\_{\\mathcal{X}}}.\\end{aligned}$$ \nThus, $z^\\*\\_{({\\mathcal{M}})} = z^\\*\\_{({\\overline{\\mathcal{M}}})}$. Furthermore, any monitoring strategy *σ*1′ ∈ Δ1 satisfying for *C* = X, which is optimal for (RX), is then optimal for.\nWe suppose that for every component, the security levels are identical and equal to *φ*. Let *C* ⊆ X. Then, we find that *k**C*\\* = ∣*C*∣ since for every *k* ∈ ⟦1, ∣*C*∣⟧, $\\frac{k-r}{k}(1-\\varphi) \\leq 1- \\varphi$. Therefore, $z^\\*\\_{({\\overline{\\mathcal{M}}})} = \\max\\_{C \\subseteq {\\mathcal{X}}} \\min\\{1 - \\frac{|C| - r}{|C|}(1-\\varphi), \\min\\_{u \\notin U\\_C} \\varphi \\}$.\nNote that for any *C* ⊆ X, $1 - \\frac{|C| - r}{|C|}(1-\\varphi) = \\varphi + \\frac{r}{|C|} (1-\\varphi) \\geq \\varphi$. Therefore,\n \n$$\\begin{aligned}\nz^\\*\\_{({\\overline{\\mathcal{M}}})} = \\max\\_{C\\in \\mathcal{S}} \\varphi + \\frac{r}{|C|} (1-\\varphi) = \\varphi + \\frac{r}{n^\\*}(1-\\varphi),\\end{aligned}$$ \nwhere S is the collection of set covers. Similarly,\n \n$$\\begin{aligned}\n\\min\\_{T\\in{\\mathcal{T}}} 1 - \\frac{|T| - r}{S\\_T} = \\min\\_{T\\in{\\mathcal{T}}} 1 + \\frac{r}{|T|}(1-\\varphi) = 1 + \\frac{r}{m^\\*}(1-\\varphi).\\end{aligned}$$ \nJezdimir Milošević received the M.Sc. degree in electrical engineering and computer science from the University of Belgrade, Belgrade, Serbia, in 2015 and the Ph.D. in electrical engineering and computer science from the KTH Royal Institute of Technology, Stockholm, Sweden, in 2020.\nHe is currently a research and development engineer in Scania’s Autonomous Transport Solutions, Södertälje, Sweden. His research interests are within cyber-security of industrial control systems.\nMathieu Dahan received the M.S. and Ph.D. degrees in computational science and engineering from the Massachusetts Institute of Technology, Cambridge, MA, USA, in 2016 and 2019, respectively.\nHe is currently an Assistant Professor in the School of Industrial and Systems Engineering at the Georgia Institute of Technology, Atlanta, GA, USA. His research interests are in combinatorial optimization, game theory, and predictive analytics, with applications to service and healthcare operations, humanitarian systems, and logistics and supply chain management.\nSaurabh Amin received the Ph.D. degree in systems engineering from the University of California, Berkeley, Berkeley, CA, USA, in 2011.\nHe is currently a Professor with the Department of Civil and Environmental Engineering. He is a member of the Laboratory for Information and Decision Systems with the Massachusetts Institute of Technology, Cambridge, MA, USA. His fields of expertise include control and optimization, applied game theory, and networks. His research focuses on the design and implementation of resilient monitoring and control algorithms for networked infrastructure systems.\nHenrik Sandberg (F’23) received the M.Sc. degree in engineering physics and the Ph.D. degree in automatic control from Lund University, Lund, Sweden, in 1999 and 2004, respectively.\nHe is currently a Professor with the Division of Decision and Control Systems, KTH Royal Institute of Technology, Stockholm, Sweden. His current research interests include security of cyber-physical systems, power systems, model reduction, and fundamental limitations in control.\nDr. Sandberg was a recipient of the Best Student Paper Award from the IEEE Conference on Decision and Control in 2004, an Ingvar Carlsson Award from the Swedish Foundation for Strategic Research in 2007, and a Consolidator Grant from the Swedish Research Council in 2016. He has served on the editorial boards of IEEE Transactions on Automatic Control and the IFAC Journal Automatica. and the IFAC Journal Automatica.\n---\n1. Manuscript received April 6, 2023. The work of JM and HS was supported in part by the Swedish Civil Contingencies Agency (project CERCES2) and Digital Futures (project DEMOCRITUS). The work of MD was supported by the Georgia Tech new faculty start up grant. The work of SA was supported in part by AFOSR grant “Building Attack Resilience into Complex Networks” and NSF grant # 2039771.[↩](#fnref1)\n2. Jezdimir Milošević was with the School of Electrical Engineering and Computer Science, KTH Royal Institute of Technology, Stockholm, Sweden. He is now with Scania Autonomous Transport Solutions (e-mail: jezdimirmilosevic@gmail.com).[↩](#fnref2)\n3. Mathieu Dahan is with the School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332 USA (e-mail: mathieu.dahan@isye.gatech.edu).[↩](#fnref3)\n4. Saurabh Amin is with the Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139 USA (e-mail: amins@mit.edu).[↩](#fnref4)\n5. Henrik Sandberg is with the School of Electrical Engineering and Computer Science, KTH Royal Institute of Technology, Stockholm, Sweden (e-mail: hsan@kth.se).[↩](#fnref5)\n\nStrategic Monitoring of Networked Systems with Heterogeneous Security Levels\n============================================================================\n**We consider a strategic network monitoring problem involving the operator of a networked system and an attacker. The operator aims to randomize the placement of multiple protected sensors to monitor and protect components that are vulnerable to attacks. We account for the heterogeneity in the components’ security levels and formulate a large-scale maximin optimization problem. After analyzing its structure, we propose a three-step approach to approximately solve the problem. First, we solve a generalized covering set problem and run a combinatorial algorithm to compute an approximate solution. Then, we compute approximation bounds by solving a nonlinear set packing problem. To evaluate our solution approach, we implement two classical solution methods based on column generation and multiplicative weights updates, and test them on real-world water distribution and power systems. Our numerical analysis shows that our solution method outperforms the classical methods on large-scale networks, as it efficiently generates solutions that achieve a close to optimal performance and that are simple to implement in practice.**\nStrategic Network Monitoring, Optimization, Game Theory, Networked Control Systems, Other Applications\nIntroduction\n============\nnetworked systems, such as water distribution or power systems, are lucrative targets for malicious attackers. Indeed, several attacks have already occurred , demonstrating the need for rapid development of effective defense strategies for these systems. An essential component of every defense strategy is attack detection , which can be accomplished by allocating multiple sensors for system monitoring. Yet, for large-scale systems, the number of available sensors is likely to be insufficient to ensure complete coverage. A strategic adversary can exploit this deficiency to target the most critical components of the system that are unmonitored. While the literature on sensor placement has studied fixed and randomized sensing strategies, none of these works has accounted for the heterogeneous criticality of the network components, and its impact on monitoring decisions. Hence, the question that arises is how to strategically allocate multiple sensors to monitor a large-scale system of critical networked components.\nSpecifically, we focus on determining a randomized scheduling of monitoring operations to maximize the protection of a networked system against a strategic adversary. In this network monitoring problem, the network operator seeks to randomize the placement of multiple protected sensors to monitor the components and protect them against an attacker who aims to sabotage a system component while remaining undetected. To better account for the attacker’s behavior and preference, we consider that the components have possibly heterogeneous security levels, which represent the corresponding levels of effort required by an adversary to target them. This model is motivated by risk management, where one first identifies critical system components in a risk assessment, and then allocates protection resources accordingly. By monitoring a component with a protected sensor, the operator maximizes the security level of that component. Then, to protect the system against a strategic adversary, we assume that the operator aims to maximize the lowest expected security level across all components.\n### Our contributions\nIn our first set of contributions, we analyze the structure of the network monitoring problem and gather insights on the impact of the problem characteristics on the operator’s optimal strategy. This leads us to a three-step approach for efficiently computing an approximate monitoring strategy and evaluating its optimality gap. First, we formulate and solve a mixed-integer program (MIP) to compute generalized covering sets and obtain marginal probabilities of placing sensors at each location. Second, we employ the combinatorial algorithm of to construct a randomized monitoring strategy that is consistent with the marginal probabilities computed from the MIP. Finally, by leveraging our structural insights, we formulate a MIP to compute nonlinear set packings and obtain an optimality gap on our approximate solution to the network monitoring problem.\nIn our second set of contributions, we refine our solution approach and analysis for problem instances with special structures. In particular, we find that our solution approach optimally solves the network monitoring problem when the sets of components that are monitored from each sensor location are mutually disjoint. Furthermore, when security levels are identical, our solution approach can be simplified by solving the minimum set cover and maximum set packing problems.\nWe also describe and implement two classical solution methods for solving the network monitoring problem based on column generation and multiplicative weights updates. In particular, we show that these algorithms can be utilized, as the pricing problem for the column generation algorithm and the best-response problem for the multiplicative weights update algorithm, can both be formulated as maximum weighted covering problems.\nIn our last set of contributions, we evaluate our solution approach based on generalized covering sets and nonlinear set packings and compare it with the two classical solution methods using two security applications. In the first application, we consider the problem of detecting contaminants injected in a water distribution system. In the second application, we consider the problem of protecting actuators in a large-scale power system against extended replay attacks, during which the attacker uses actuators in addition to sensors to lead an undetectable attack . Our computational results show that our solution approach provides excellent solutions to the network monitoring problem and outperforms the two classical methods in different aspects: It generates solutions with low optimality gaps, and its running time is marginally impacted by the number of available sensors. In contrast, the running time of the other methods exponentially increases with respect to the number of sensors. Finally, the monitoring strategies generated by our solution approach are significantly simpler to implement in practice compared to the ones generated by the classical methods, thus reinforcing the value of our solution approach derived from structural insights.\n### Related work\nOptimization and game theoretic models have been developed for studying various security related problems, including the development of defense strategies , the design of anomaly detectors , the allocation of security budget , and the placement of sensors . For instance, consider bilevel and trilevel optimization problems to model defender-attacker interactions where each player selects a pure strategy. In contrast, we focus on randomized strategies, which are recognized to be more effective when the number of sensors to deploy is limited .\nOther models that have been studied include inspection problems and search games, in which a searcher is concerned with the optimal way of looking for a hidden adversary in a search space; see for example. In particular, the recent literature on hide-and-seek games investigates the problem of coordinating multiple inspection resources for detecting hidden objects in boxes. However, such problems do not account for the impact of sensing range and the network topology on the monitoring strategies. Still, we leverage the algorithm of as part of our solution approach for computing randomized monitoring strategies with small supports.\nFinally, the existing literature on sensor placement considers developing both fixed and randomized monitoring strategies. The placement problems considered previously aimed to minimize estimation error , achieve optimal coverage , detect faults , or improve the system’s security . The study is related to ours, and proposes two static sensor placement strategies to improve the security of actuators in large-scale systems. Our focus is instead on randomized strategies, which can be translated into random scheduling of inspections that can be performed on a day-to-day basis by utility or security personnel.\nOur problem is also related to the simultaneous zero-sum game in , where the operator places multiple sensors to maximize the number of detected (homogeneous) attacks. Recent work extended this model by considering the imperfect detection by sensors, and the different types of sensing resources, for which they derive heuristic approaches. In addition, formulated a game in which sensors are positioned in the nodes of a networked control system to detect attacks on them, and considered imperfect detection through a linear filter that processes the sensors’ measurements to detect attacks. The authors derived equilibrium results using tools from structured systems and graph theory. However, to the best of our knowledge, no article investigates the coordination of *multiple* sensing resources to *strategically* monitor a *networked* system with components of *heterogeneous* criticalities.\nPreliminary results (without proofs) have been presented in. Specifically, investigate a simpler solution approach based on minimal set covers, maximal set packings, and column generation. They also provide approximation bounds under restrictive assumptions. By further leveraging the problem structure, we generalize their results and obtain a more efficient solution approach with stronger approximation bounds. We also provide a more comprehensive evaluation of our solution approach through an extended computational study that involves two other solution algorithms and two security applications.\n### Organization\nSection [section:game] introduces and formulates the strategic network monitoring problem. In Section [sec:Method], we derive and analyze our solution approach based on generalized covering sets and nonlinear set packings. We then present two alternative solution methods based on column generation and multiplicative weights updates in Section [sec:Alt]. Section [section:computations] evaluates and compares these solution approaches through a computational study. Finally, Section [section:ch8conclusion] summarizes our findings and provides avenues for future work.\nProblem Formulation\n===================\nWe consider a networked system consisting of a set of components U. Each component *u* ∈ U faces a risk of attack by an adversary who wishes to disrupt or take control of the network. For every *u* ∈ U, let *φ**u* ∈ [0, 1] denote its *security level*, i.e., the effort that the attacker needs to spend in order to successfully target *u* and compromise its operation. Security levels can for example be estimated based on previously deployed security measures, configuration of the system, or monetary cost for conducting an attack against the components. A component *u* for which *φ**u* = 0 (resp. *φ**u* = 1) is unsafe (resp. highly secure).\nThe network operator can increase the security levels of some components by positioning *r* sensors within a set of locations X. Specifically, a sensor positioned at location *x* ∈ X can monitor and secure a subset of components *U**x* ⊆ U. For every *x* ∈ X, we refer to *U**x* as the monitoring set of *x*. Typically, monitoring sets are computed from the characteristics of the sensors and the network. If multiple sensors are positioned at locations *X* ⊆ X, then the monitored and secured components are given by $U\\_X \\coloneqq \\cup\\_{x \\in X}U\\_x$.\nWe let ${\\mathcal{A}}\\coloneqq \\{X \\subseteq {\\mathcal{X}}\\ | \\ |X| \\leq {r}\\}$ denote the set of actions of the operator. Thus, for every sensor positioning *X* ∈ A and every component *u* ∈ U, the resulting *post-security level* of *u* is given by $f(X,u) \\coloneqq \\varphi\\_u \\mathds{1}\\_{\\{u \\notin U\\_X\\}} + \\mathds{1}\\_{\\{u \\in U\\_X\\}}$. Specifically, the post-security levels of all monitored components are maximal and equal to one. Without loss of generality, we assume the following: 1) *U**x* ≠ ∅ for every *x* ∈ X; 2) every component can be monitored from at least one sensor location; and 3) *φ**u* < 1 for every *u* ∈ U.\nNext, we illustrate the monitoring model with an example.\nConsider the networked system and monitoring model illustrated in Fig. [fig:example].\n[scale=1.25,main node/.style=circle,draw,inner sep = 0.15cm,main node2/.style=circle,draw,inner sep = 0.10cm,] (-0.5, 0) ellipse [x radius = 15mm, y radius = 6mm]; (1.5, 0) ellipse [x radius = 15mm, y radius = 6mm]; (1.5, -1.3) ellipse [x radius = 6mm, y radius = 15mm];\n(-0.5, 0) ellipse [x radius = 15mm, y radius = 6mm]; (1.5, 0) ellipse [x radius = 15mm, y radius = 6mm]; (1.5, -1.3) ellipse [x radius = 6mm, y radius = 15mm];\n(1) at (-0.5, 0.4) *x*1; (2) at (2.5, 0.15) *x*2; (3) at (1.8, -1) *x*3;\n(e1) at (-1.5, 0) *u*1; (e2) at (-0.5, -0.3) *u*2; (e3) at (0.5, 0) *u*3; (e4) at (1.5, -0.3) *u*4; (e5) at (2.0, 0.25) *u*5; (e6) at (1.2, -1.3) *u*6; (e7) at (1.5, -2.3) *u*7;\n(green sensor) at (1.8, -1.3); (yellow sensor) at (2.5, -0.15);\n[fig:example]\nIn this example, the set of sensor locations is X = {*x*1, *x*2, *x*3}, the set of components is U = {*u*1, …, *u*7}, and the monitoring sets are *U**x*1 = {*u*1, *u*2, *u*3}, *U**x*2 = {*u*3, *u*4, *u*5}, and *U**x*3 = {*u*4, *u*6, *u*7}. The security levels are given by *φ**u*3 = *φ**u*4 = 0.2 (red in the figure), *φ**u*1 = *φ**u*5 = *φ**u*7 = 0.5 (blue in the figure), and *φ**u*2 = *φ**u*6 = 0.8 (green in the figure). We assume that the operator has two sensors that they place on *X* = {*x*2, *x*3}. Then, each component *u* in *U**X* = {*u*3, *u*4, *u*5, *u*6, *u*7} is monitored and has a post-security level equal to *f*(*X*, *u*) = 1. On the other hand, the remaining components *u*1 and *u*2 are not monitored, and their post-security levels are equal to their original security levels: *f*(*X*, *u*1) = 0.5 and *f*(*X*, *u*2) = 0.8. Note that *u*1 has the lowest resulting post-security level and is the most vulnerable to attacks.\nIn such combinatorial security settings, the network operator can significantly benefit from randomizing the positioning of their sensors. Thus, we consider the problem where the operator is interested in positioning their sensors in a randomized manner to protect the network against an attacker who would target the most vulnerable component. In other words, the operator aims to maximize the lowest expected post-security level across all network’s components. Let $\\Delta^1\\coloneqq\\{\\sigma^{1} \\in [0,1]^{|\\mathcal{A}|} \n\\ | \\\n \\sum\\_{X \\in \\mathcal{A}} \\sigma^1\\_{X} =1 \\}$ denote the set of mixed strategies of the operator, i.e., the set of probability distributions over A; here, *σ**X*1 denotes the probability that the operator positions sensors on locations *X* ∈ A. Then, the problem of strategic network monitoring can be formulated as\n \n$$\\begin{aligned}\n \\label{eq:P}\n\\max\\_{\\sigma^1 \\in \\Delta^1} \\min\\_{u \\in {\\mathcal{U}}} \\mathbb{E}\\_{X\\sim\\sigma^1}[f(X,u)]. \\tag{$\\mathcal{M}$}\n \\end{aligned}$$ \nProblem can be formulated as a linear program with ∣A∣ + 1 variables and ∣U∣ + 1 constraints. However, the cardinality of A grows combinatorially with respect to the number of available sensors. Thus, standard methods for solving linear programs cannot be used for large-scale systems, and different approaches are needed to compute an optimal or approximate monitoring strategy. In Section [sec:Method], we develop an approach based on mixed-integer programs (MIPs) to compute an approximate solution of, while in Section [sec:Alt] we discuss alternative approaches based on column generation and multiplicative weights updates.\nMain Results\n============\nIn this section, we analyze the structure of the network monitoring problem and derive an approximate solution approach. Our approach first solves a generalized covering set problem and runs a combinatorial algorithm to construct a monitoring strategy with small support. Then, it computes an optimality gap by solving a nonlinear set packing problem. We further investigate problem instances with special structures, for which we refine our approach and optimality gaps.\nPreliminaries\n-------------\nTo carry out our analysis, we first introduce additional quantities and notations: For any optimization problem (O), we denote by *z*(O)\\* its optimal value. Then, for every location *x* ∈ X, let $\\overline{\\varphi}\\_x \\coloneqq \\min\\_{u \\in U\\_x} \\varphi\\_u$ denote the lowest security level of a component in the monitoring set of *x*. This quantity will impact the operator’s monitoring strategy, as the objective is to maximize the worst-case component post-security level. We quantify the *criticality* of a location *x* ∈ X (resp. component *u* ∈ U) from the value of $\\overline{\\varphi}\\_x$ (resp. *φ**u*): If $\\overline{\\varphi}\\_x$ (resp. *φ**u*) is low, we say that *x* ∈ X (resp. *u* ∈ U) is more critical. We additionally define, for every sensor positioning *X* ∈ A and every location *x* ∈ X, $\\bar{f}(X,x) \\coloneqq \\overline{\\varphi}\\_x \\mathds{1}\\_{\\{x \\notin X\\}} + \\mathds{1}\\_{\\{x \\in X\\}}$ to be the lowest post-security level within the monitoring set *U**x*.\nFor any subset of locations *C* ⊆ X (resp. any subset of components *T* ⊆ U), we denote as $\\overline{\\varphi}\\_C$ (resp. *φ**T*) the following vector $(\\overline{\\varphi}\\_x)\\_{x \\in C}$ (resp. (*φ**u*)*u* ∈ *T*), sorted in nondecreasing order. For every subset of components *T* ⊆ U, we also define the following quantity $S\\_T \\coloneqq \\sum\\_{u \\in T}(1-\\varphi\\_u)^{-1}$.\nFor any vector *y* ∈ R*n*, we denote its support as ${\\operatorname{supp}}(y) \\coloneqq \\{i \\in \\llbracket1,n\\rrbracket \\ | \\ y\\_i \\neq 0\\}$, where ⟦1, *n*⟧ = {1, …, *n*}. Then, given a monitoring strategy *σ*1 ∈ Δ1, we define its *node basis* as ${\\mathcal{X}}(\\sigma^1) \\coloneqq \\cup\\_{X \\in {\\operatorname{supp}}(\\sigma^1)} X$, that is, the set of nodes that are monitored with positive probability under the monitoring strategy *σ*1. Note that node bases provide a good indicator of the practical implementability of a monitoring strategy: A strategy that monitors a small number of locations can easily be implemented in practice via a randomized scheduling of operations.\nWe consider the collection of set covers S of our networked system. Specifically, a subset of locations *C* ⊆ X is a *set cover* if *U**C* = U, i.e., every component is monitored from at least one location in *C*. Let *n*\\* denote the minimum set cover size. We note that if *r* ≥ *n*\\*, then an optimal solution of the network monitoring problem consists of placing sensors on a set cover and ensuring a post-security level of 1 for every component. Henceforth, we assume that *r* < *n*\\*.\nFinally, we consider the collection of set packings T of our networked system. Specifically, a subset of components *T* ⊆ U is a *set packing* if ∣*T* ∩ *U**x*∣ ≤ 1 holds for every *x* ∈ X. Set packings indicate the *spread* of the network, which directly impacts the monitoring strategy of the operator: Since components in a set packing must be monitored from distinct sensor locations, a network with large set packings is more challenging to protect. We denote the maximum set packing size as *m*\\*.\nStructural Analysis\n-------------------\nWe first analyze the structure of the network monitoring problem and derive bounds on its optimal value. To obtain a lower bound on *z*(M)\\*, we simplify the problem by supposing that when the operator places a sensor at location *x* ∈ X, they protect the component *u* within the monitoring set *U**x* that is the most critical. The premise is that the post-security level achieved for any component in *U**x* will be at least that of component *u*. Formally, we relax the objective function of and consider the following problem:\n \n$$\\begin{aligned}\n\\label{eq:Q}\n \\max\\_{\\sigma^1\\in \\Delta^1} \\min\\bigg\\{\\ \\min\\_{\\quad \\mathclap{x \\in {\\mathcal{X}}(\\sigma^1)}\\quad } \\mathbb{E}\\_{X\\sim\\sigma^1}[\\bar{f}(X,x)], \\ \\min\\_{\\quad \\mathclap{u \\notin U\\_{{\\mathcal{X}}(\\sigma^1)}}\\quad }\\varphi\\_u\\bigg\\}.\n\\tag{$\\overline{\\mathcal{M}}$}\\end{aligned}$$ \nProblem selects a monitoring strategy *σ*1 that partitions the set of components U into two subsets: the subset of components U \\ *U*X(*σ*1) that are never monitored by *σ*1 and for which the post-security levels are given by the corresponding original security levels; and the subset of components *U*X(*σ*1) that are monitored with positive probability. In this latter set, *σ*1 guarantees a post-security level for every component in *U*X(*σ*1) by improving the lowest security level within each monitoring set *U**x* for *x* ∈ X(*σ*1).\nIn fact, we provide in the next lemma a different perspective on problem to guide our analysis:\n[lem:Angle] Problem satisfies the following property:\n \n$$\\begin{aligned}\nz^\\*\\_{({\\overline{\\mathcal{M}}})} = \\max\\_{C \\subseteq {\\mathcal{X}}} \\min\\Big\\{z^\\*\\_{(\\mathcal{R}\\_C)},\\min\\_{u \\notin U\\_C} \\varphi\\_u\\Big\\},\\end{aligned}$$ \nwhere for every *C* ⊆ X, *z*(R*C*)\\* is the optimal value of\n \n$$\\begin{aligned}\n\\label{eq:R}\n\\max\\_{\\sigma^1 \\in \\Delta^1} \\min\\_{x \\in C} \\mathbb{E}\\_{X \\sim \\sigma^1}[\\bar{f}(X,x)].\\tag{$\\mathcal{R}\\_C$}\\end{aligned}$$ \nThus, can be solved by first determining the set of locations *C* over which to randomize the placement of sensors, and then determining the monitoring strategy that optimizes. Given a preselected subset of *n* locations *C* = {*x*1, …, *x**n*} ⊆ X, indexed such that $\\overline{\\varphi}\\_{x\\_1} \\leq \\cdots \\leq \\overline{\\varphi}\\_{x\\_n}$, we provide some intuition on the monitoring strategy *σ*1 that maximizes min*x* ∈ *C*E*X* ∼ *σ*1[*f̄*(*X*, *x*)]. Initially, the goal is to randomize the placement of the sensors among the locations in *C* so as to equalize the post-security level of the most critical component in each of their monitoring sets. However, because of the heterogeneity of the security levels, the operator should instead focus on the most critical locations and not monitor the most secure locations. The number of most critical locations in *C* to monitor is given by\n \n$$\\begin{aligned}\nk^\\*\\_C\\coloneqq \\max\\bigg\\{k \\in \\llbracket 1,n\\rrbracket \\ \\bigg| \\ \\overline{\\varphi}\\_{x\\_k} \\leq 1 - \\frac{k - {r}}{\\sum\\_{l=1}^k (1-\\overline{\\varphi}\\_{x\\_l})^{-1}}\\bigg\\}.\\end{aligned}$$ \nFor simplicity, we also denote $S^{k^\\*}\\_C \\coloneqq \\sum\\_{l=1}^{k^\\*\\_C}(1-\\overline{\\varphi}\\_C)^{-1}$.\nUsing this property and the collection of set packings T, we can then derive the following main theorem:\n[prop:bounds] The optimal value of the network monitoring problem is bounded as follows:\n \n$$\\begin{aligned}\n z\\_{({\\overline{\\mathcal{M}}})}^\\* &= \\max\\_{C \\subseteq {\\mathcal{X}}} \\min\\bigg\\{1 - \\frac{k^\\*\\_C-{r}}{S^{k^\\*}\\_C},\\min\\_{\\mathclap{u \\notin U\\_C}}\\varphi\\_u\\bigg\\} \\\\\n& \\leq z\\_{({\\mathcal{M}})}^\\*\\leq \\min\\bigg\\{1,\\min\\_{T \\in {\\mathcal{T}}} 1 - \\frac{|T|-{r}}{S\\_T}\\bigg\\}.\n \\end{aligned}$$ \nFurthermore, for any set of *n* > *r* locations *C* = {*x*1, …, *x**n*} ⊆ X indexed in nondecreasing order of their criticalities, any monitoring strategy *σ*1′ satisfying the following conditions is an optimal solution of :\n \n$$\\begin{aligned}\n\\mathbb{P}\\_{\\hspace{-0.05cm}X\\sim\\sigma^{1^\\prime}}(x \\hspace{-0.02cm}\\in\\hspace{-0.02cm} X) \\hspace{-0.02cm}= \\hspace{-0.02cm} \\begin{cases} 1 \\hspace{-0.02cm}-\\hspace{-0.02cm} \\dfrac{k^\\*\\_C-{r}}{(1-\\overline{\\varphi}\\_{x})S^{k^\\*}\\_C}, & \\hspace{-0.15cm}\\text{if } \\hspace{-0.05cm} x \\hspace{-0.02cm}\\in\\hspace{-0.02cm} \\{x\\_1,\\dots,x\\_{k^\\*\\_C}\\}\\\\ 0, & \\hspace{-0.15cm}\\text{otherwise.}\\end{cases} \\label{sigma\\_marg}\\end{aligned}$$ \nInterestingly, from this theorem, we find that solving problem indeed provides a lower bound on the optimal value of. Furthermore, given a subset of sensor locations *C* ⊆ X of size greater than *r*, the post-security level that can be guaranteed for any component in *U**C* is given by $1 - \\frac{k^\\*\\_C-{r}}{S^{k^\\*}\\_C}$. To achieve this post-security level, the network operator must ensure that the probability that each location in *C* is monitored follows. Importantly, we find that the *k**C*\\* most critical locations must be monitored, with a probability that is decreasing with the associated lowest security level. In contrast, the remaining locations should not be monitored, as their security levels are already larger than the post-security level achieved for the *k**C*\\* most critical locations (by definition of *k**C*\\*). We note that a monitoring strategy satisfying is guaranteed to exist by Farkas’ lemma.\nFrom Theorem [prop:bounds], we also find that upper bounds on the optimal value of the network monitoring problem can be computed from set packings. We will leverage this finding to efficiently compute optimality gaps for our solutions.\nSolution Approach\n-----------------\nFrom our structural analysis and Theorem [prop:bounds], we next derive a three-step approach to approximately solve the network monitoring problem. Steps 1 and 2 compute a monitoring strategy that optimally solves the relaxed problem, and step 3 computes a set packing that achieves the upper bound from Theorem [prop:bounds].\nIn the first step, we formulate a MIP to compute $z^\\*\\_{({\\overline{\\mathcal{M}}})}$ as well as the marginal probabilities of monitoring each location at optimality of. In particular, we leverage the fact that the objective function in can be expressed using these marginal probabilities. Indeed, for every *σ*1 ∈ Δ1 and *x* ∈ X,\n \n$$\\begin{aligned}\n\\mathbb{E}\\_{X\\sim \\sigma^1}[\\bar{f}(X,x)] = \\overline{\\varphi}\\_x + (1- \\overline{\\varphi}\\_x)\\mathbb{P}\\_{X\\sim\\sigma^1}(x \\in X).\\label{marg\\_f}\\end{aligned}$$ \nUsing Lemma [lem:Angle], we then formulate the following MIP, which we refer to as a *generalized covering set problem*:\n \n$$\\begin{aligned}\n{3}\n\\max\\_{y,z,\\rho} \\ & z \\tag{$\\mathcal{C}$}\\label{eq:M}\\\\\n\\text{s.t.} \\ &z \\leq \\overline{\\varphi}\\_{x} + (1 - \\overline{\\varphi}\\_{x})\\rho\\_x + M\\_1(1-y\\_x), \\ &&{\\ForAll\\,}x \\in {\\mathcal{X}}\\label{term1}\\\\\n& z \\leq \\varphi\\_u + M\\_1\\sum\\_{\\quad \\mathclap{\\{x\\in {\\mathcal{X}}\\, | \\, u \\in U\\_x\\}}\\quad } y\\_x, &&{\\ForAll\\,}u \\in {\\mathcal{U}}\\label{term2}\\\\\n& 0 \\leq \\rho\\_x \\leq y\\_x, && {\\ForAll\\,}x \\in {\\mathcal{X}}\\\\\n& \\sum\\_{x \\in {\\mathcal{X}}} \\rho\\_x = r,\\label{resource\\_constraint}\\\\\n& y\\_x \\in \\{0,1\\}, && {\\ForAll\\,}x \\in {\\mathcal{X}}.\\nonumber\\end{aligned}$$ \nHere, for every *x* ∈ X, the binary variable *y**x* determines if *x* can receive a sensor, while the continuous variable *ρ**x* determines the marginal probability that *x* receives a sensor. *M*1 is a “big M”, which we can set to 1 − min*u* ∈ U*φ**u*. Given *C* = {*x* ∈ X ∣ *y**x* = 1}, we then observe that constraints (combined with ) model min*x* ∈ *C*E*X* ∼ *σ*1[*f̄*(*X*, *x*)], while constraints model $\\min\\_{u \\notin U\\_C} \\varphi\\_u$. In fact, we show the equivalence between and :\n[prop:LB] Problems and have identical optimal values: $z^\\*\\_{({\\overline{\\mathcal{M}}})} = z^\\*\\_{(\\mathcal{C})}$. Furthermore, a monitoring strategy *σ*1\\* ∈ Δ1 is an optimal solution of if and only if there exists an optimal solution (*y*\\*, *z*\\*, *ρ*\\*) of satisfying *ρ**x*\\* = P*X* ∼ *σ*1\\*(*x* ∈ *X*) for every *x* ∈ X.\nThus, the optimal value of can be computed by solving. In addition, given an optimal solution (*y*\\*, *z*\\*, *ρ*\\*) of, *ρ*\\* provides the marginal probabilities that each node must be monitored at optimality of.\nThe second step of our approach consists of reconstructing an optimal solution of from *ρ*\\*. To this end, we utilize the combinatorial algorithm derived in that computes a probability distribution *σ*1\\* ∈ Δ1 satisfying P*X* ∼ *σ*1\\*(*x* ∈ *X*) = *ρ**x*\\* for every *x* ∈ X. From Proposition [prop:LB], this guarantees the optimality of *σ*1\\* for.\nSpecifically, the algorithm iteratively expresses a vector *ρ* ∈ [0, 1]∣X∣ satisfying as a convex combination of the incidence vector of a sensor positioning *X* ∈ A, and another vector *ρ*′ ∈ [0, 1]∣X∣ satisfying and with at least one additional integral component compared to *ρ*\\*. We refer the reader to for the detailed algorithm, which we henceforth refer to as the *coordination algorithm*.\nImportantly, given *n* = ∣supp(*ρ*\\*)∣ the number of locations that must receive a sensor with positive probability, this algorithm runs in time *O*(*n*2) and constructs a monitoring strategy of support size *n* + 1. We recall that a monitoring strategy with small support and/or small node basis can more easily be implemented in practice. We will numerically evaluate the node basis sizes of our solutions in our computational study.\nThe final step of our approach consists of evaluating the optimality gap of our solution for the network monitoring problem. To this end, we derive a MIP to efficiently compute the upper bound on *z*(M)\\* from Theorem [prop:bounds]. Specifically, we aim to compute a set packing *T* ∈ T that minimizes $1 - \\frac{|T| - r}{S\\_T}$, which is equivalent to finding a set packing of size greater than *r* that minimizes $\\frac{S\\_T}{|T| - {r}}$. Although this is a *nonlinear set packing problem*, we can reformulate it as the following MIP:\n \n$$\\begin{aligned}\n{3}\n\\min\\_{t,y,z} & \\displaystyle\\sum\\_{l={r}+1}^{|{\\mathcal{U}}|} \\frac{1}{l - {r}} t\\_l\\tag{$\\mathcal{P}$}\\label{eq:N}\\\\\n\\text{s.t.} & \\sum\\_{u \\in U\\_x} y\\_u \\leq 1, && {\\ForAll\\,}x \\in {\\mathcal{X}}\\label{cons\\_sp}\\\\\n& \\sum\\_{u \\in {\\mathcal{U}}} y\\_u = \\sum\\_{l={r}+ 1}^{|{\\mathcal{U}}|} l z\\_l, \\label{cons\\_nl\\_1}\\\\\n& \\sum\\_{l={r}+ 1}^{|{\\mathcal{U}}|} z\\_l = 1,\\label{cons\\_nl\\_2}\\\\\n&t\\_l \\hspace{-0.05cm}\\geq \\hspace{-0.12cm}\\sum\\_{u \\in {\\mathcal{U}}} (1\\hspace{-0.05cm}-\\hspace{-0.05cm}\\varphi\\_u)^{-1} y\\_u \\hspace{-0.05cm}-\\hspace{-0.05cm} M\\_2 (1\\hspace{-0.03cm}-\\hspace{-0.03cm}z\\_l), && {\\ForAll\\,}l \\hspace{-0.05cm}\\in\\hspace{-0.05cm} \\llbracket {r}\\hspace{-0.05cm}+\\hspace{-0.05cm} 1,|{\\mathcal{U}}|\\rrbracket\\label{cons\\_product\\_1}\\\\\n& t\\_l \\geq 0, && {\\ForAll\\,}l \\hspace{-0.05cm}\\in\\hspace{-0.05cm} \\llbracket {r}\\hspace{-0.05cm}+\\hspace{-0.05cm} 1,|{\\mathcal{U}}|\\rrbracket\\label{cons\\_product\\_2}\\\\\n& z\\_l \\in \\{0,1\\}, && {\\ForAll\\,}l \\hspace{-0.05cm}\\in\\hspace{-0.05cm} \\llbracket {r}\\hspace{-0.05cm}+\\hspace{-0.05cm} 1,|{\\mathcal{U}}|\\rrbracket\\nonumber\\\\\n& y\\_u \\in \\{0,1\\}, && {\\ForAll\\,}u \\in {\\mathcal{U}}.\\nonumber\\end{aligned}$$ \nSpecifically, the binary variables *y**u*, for *u* ∈ U select the components to be part of the set packing. To parametrize the nonlinearity of the problem, we introduced for every *l* ∈ ⟦*r* + 1, ∣U∣⟧ the binary variable *z**l*, which is equal to 1 if the set packing is of size *l*. Finally, for every *l* ∈ ⟦*r* + 1, ∣U∣⟧, the continuous variable *t**l* represents the product *z**l*∑*u* ∈ U(1 − *φ**u*)− 1*y**u*. Constraints ensure that supp(*y*) is a set packing. Constraints - select the set packing size. Then, constraints - ensure that at optimality, the objective value is equal to ∑*u* ∈ supp(*y*)(1 − *φ**u*)− 1/(∣supp(*y*)∣ − *r*). *M*2 is also a “big M”, which we can set to ∣U∣(1 − max*u* ∈ U*φ**u*)− 1.\nAs a consequence, the upper bound on the optimal value of from Theorem [prop:bounds] can computed by solving :\n[prop:UB] The following equality holds: \n$$\\min\\bigg\\{1,\\min\\_{T \\in {\\mathcal{T}}} 1 - \\frac{|T|-{r}}{S\\_T}\\bigg\\} = 1 - \\frac{1}{z^\\*\\_{(\\mathcal{P})}}.$$ \nWe note that the number of variables in can be reduced in practice by computing an upper bound on the maximum set packing size in the networked system. Given the maximum set packing size *m*\\* (or an upper bound), then we can reduce the variables in from ∣U∣ + 2(∣U∣ − *r*) to ∣U∣ + 2(*m*\\* − *r*), and set *M*2 to *m*\\*(1 − max*u* ∈ U*φ**u*)− 1, which can provide significant computational benefits.\nThus, we obtain a three-strep approach to compute an approximate solution to the network monitoring problem and evaluate its optimality gap. Next, we further study our solution for problem instances with particular characteristics.\nSpecial Cases\n-------------\n### Non-overlapping monitoring sets\nWe consider problem instances for which monitoring sets are pairwise disjoint, i.e., *U**x* ∩ *U**x*′ = ∅ for every *x* ≠ *x*′ ∈ X. Such instances occur when it is desirable to reduce sensor interference or the energetic cost of the network. In other contexts such as in security games, disjoint monitoring is naturally satisfied. For such instances, we show that our approach in Section [sec:Approach] optimally solves the network monitoring problem:\n[theorem:analyticalsolutionspecialcase] If the monitoring sets are pairwise disjoint, then $z^\\*\\_{({\\overline{\\mathcal{M}}})} = z^\\*\\_{({\\mathcal{M}})}$. Furthermore, any monitoring strategy satisfying for *C* = X is an optimal solution of.\nFor instances when monitoring sets do not overlap, we find that problem can be optimally solved by first selecting the *k*X\\* most critical sensor locations and monitoring them with probabilities given by ; and then by utilizing the coordination algorithm to construct a monitoring strategy that randomizes the placement of *r* sensors and is consistent with the marginal probabilities in.\n### Homogeneous security levels\nWe now consider instances where the security levels are identical across the components in U. In such cases, we refine the bounds on the optimal value of using the minimum set cover and maximum set packing sizes (*n*\\* and *m*\\*):\n[prop:identical] If the security levels are identical and equal to *φ*, then:\n \n$$\\begin{aligned}\nz^\\*\\_{({\\overline{\\mathcal{M}}})} \\hspace{-0.05cm}=\\hspace{-0.05cm} \\varphi + (1-\\varphi) \\frac{r}{n^\\*} \\hspace{-0.05cm}\\leq\\hspace{-0.03cm} z^\\*\\_{({\\mathcal{M}})} \\hspace{-0.05cm} \\leq \\hspace{-0.03cm} \\min\\bigg\\{1,\\varphi + (1-\\varphi) \\frac{r}{m^\\*}\\bigg\\}.\\end{aligned}$$ \nIn such instances, we find that and can be simplified and the bounds on *z*(M)\\* can be computed by solving a minimum set cover and maximum set packing problem, as in. If *C*\\* denotes a minimum set cover, then a monitoring strategy that optimizes $z^\\*\\_{({\\overline{\\mathcal{M}}})}$ monitors each location in *C*\\* with identical probability (given by ), and can be constructed by cycling the *r* sensors over *C*\\* (see for reference).\nAlternative Approaches\n======================\nIn order to compare our solution approach derived in Section [sec:Approach], we next consider two alternative approaches for solving the network monitoring problem. In particular, we show how the column generation algorithm and multiplicative weights update algorithm can be applied to solve.\nColumn Generation Algorithm\n---------------------------\nProblem can be formulated as a linear program with a large number (${|{\\mathcal{X}}| \\choose r} + 1$) of variables and a small number (∣U∣ + 1) of constraints. Such problems can potentially be solved using the column generation algorithm, as long as the pricing problem can be solved efficiently.\nSpecifically, at each iteration of the column generation algorithm, the following restricted master problem is solved:\n \n$$\\begin{aligned}\n{3}\n\\max \\ & z \\tag{${\\mathcal{M}}\\_\\mathcal{I}$}\\label{eq:P\\_I}\\\\\n\\text{s.t.} \\ & z - \\sum\\_{X\\in \\mathcal{I}}\\sigma^1\\_X f(X,u) \\leq 0, \\ && {\\ForAll\\,}u \\in {\\mathcal{U}}\\label{dual1}\\\\\n& \\sum\\_{X \\in \\mathcal{I}}\\sigma^1\\_X = 1,\\label{dual2}\\\\\n& \\sigma^1\\_X \\geq 0, && {\\ForAll\\,}X \\in \\mathcal{I},\\nonumber\\end{aligned}$$ \nwhere I ⊆ A is a subset of feasible sensor positionings. To determine if the optimal solution of is optimal for, the algorithm then determines the variable in with highest reduced cost. Let (*α*\\*, *β*\\*) denote the optimal dual variables of, associated with constraints -. Then, the variable with highest reduced cost can be determined by solving the following pricing problem:\n \n$$\\begin{aligned}\n\\max\\_{X\\in{\\mathcal{A}}} - \\beta^\\* +\\sum\\_{u\\in{\\mathcal{U}}} \\alpha^\\*\\_u f(X,u).\\end{aligned}$$ \nIn fact, the pricing problem for is a maximum weighted covering problem, and can be solved using the following IP:\n \n$$\\begin{aligned}\n{3}\n\\max\\_{y,z} \\ & - \\beta^\\* + \\sum\\_{u \\in {\\mathcal{U}}} \\alpha^\\*\\_u (\\varphi\\_u + (1-\\varphi\\_u)y\\_u)\\tag{$\\mathcal{W}$}\\label{eq:O}\\\\\n\\text{s.t.} \\ & y\\_u \\leq \\sum\\_{\\{x \\in {\\mathcal{X}}\\, | \\, u \\in U\\_x\\}} z\\_x,\\ && {\\ForAll\\,}u \\in {\\mathcal{U}}\\label{covering\\_pb\\_1}\\\\\n& \\sum\\_{x \\in {\\mathcal{X}}} z\\_x = r,\\\\\n&y\\_u \\in \\{0,1\\}, && {\\ForAll\\,}u \\in {\\mathcal{U}}\\\\\n& z\\_x \\in \\{0,1\\}, && {\\ForAll\\,}x \\in {\\mathcal{X}}\\label{covering\\_pb\\_4}.\\end{aligned}$$ \nIn this IP, the binary variables *z**x*, for *x* ∈ X, represent the nodes that receive a sensor, and the binary variables *y**u*, for *u* ∈ U, represent the components that are monitored by at least one sensor.\nIf *z*(W)\\* = 0, then the optimal solution of is optimal for, and the algorithm terminates. If instead *z*(W)\\* > 0, then given an optimal solution (*y*\\*, *z*\\*) of, the algorithm adds *X*\\* = supp(*z*\\*) to the set I of variable indices, and solves the new restricted master problem.\nMultiplicative Weights Update Algorithm\n---------------------------------------\nProblem is also equivalent to a simultaneous two-person zero-sum game, where the first player is the network operator who places *r* sensors across the networked system and the second player is an attacker who targets a component. The payoff function is given by *f*, which the operator (resp. attacker) wants to maximize (resp. minimize).\nIn fact, devised a multiplicative weights update algorithm for computing approximate mixed Nash equilibria of such simultaneous games, as long as one player has a small number of actions and the other player can efficiently compute best responses. The algorithm runs for *N* iterations, and at each iteration *t* ∈ ⟦1, *N*⟧, generates a probability distribution *σ*2, *t* over U for the attacker as well a feasible sensor positioning *X**t* ∈ A for the network operator. The algorithm starts by setting *σ*2, 1 to the uniform distribution over U. Then, at each iteration *t* ∈ ⟦1, *N*⟧, the algorithm determines the operator’s best response to *σ*2, *t* by solving the following problem: \n*X**t* ∈ argmax*X* ∈ AE*u* ∼ *σ*2, *t*[*f*(*X*, *u*)]. \n In our setting, this problem can also be formulated as a maximum weighted covering problem and can be solved using the following IP:\n \n$$\\begin{aligned}\n{3}\n\\max\\_{y,z} \\ & \\sum\\_{u \\in {\\mathcal{U}}} \\sigma^{2,t}\\_u (\\varphi\\_u + (1-\\varphi\\_u)y\\_u)\\tag{$\\mathcal{W}^\\prime$}\\label{eq:W'}\\\\\n\\text{s.t.} \\ & \\eqref{covering\\_pb\\_1}-\\eqref{covering\\_pb\\_4}.\\nonumber\\end{aligned}$$ \nThen, the algorithm updates the attacker’s strategy using the following multiplicative weights update: \n$${\\ForAll\\,}u \\in {\\mathcal{U}}, \\ \\sigma^{2,t+1}\\_u = \\sigma^{2,t}\\_u \\frac{\\eta^{f(X^t,u)}}{\\sum\\_{u^\\prime \\in {\\mathcal{U}}} \\sigma^{2,t}\\_{u^\\prime} \\eta^{f(X^t,{u^\\prime})}},$$ \n where $\\eta = \\left(1 + \\sqrt{2 \\frac{\\ln |{\\mathcal{U}}|}{N}}\\right)^{-1}$. The authors in showed that the monitoring strategy *σ̄*1 ∈ Δ1 that uniformly randomizes over the *N* best responses *X*1, …, *X**N* satisfies:\n \n$$\\begin{aligned}\n\\min\\_{u \\in {\\mathcal{U}}} \\mathbb{E}\\_{X\\sim \\bar{\\sigma}^1}[f(X,u)] \\geq z^\\*\\_{(\\mathcal{M})} - \\sqrt{\\frac{2 \\ln |{\\mathcal{U}}|}{N}} - \\frac{\\ln|{\\mathcal{U}}|}{N}.\\end{aligned}$$ \nThus, if $N = 4\\lceil \\frac{\\ln |{\\mathcal{U}}|}{\\epsilon^2}\\rceil$ for *ε* > 0, then the (absolute) optimality gap associated with the monitoring strategy *σ̄*1 generated by this algorithm is upper bounded by *ε*.\nIn fact, showed that each iteration of this algorithm can be conducted in polynomial time by approximately solving using a greedy algorithm. However, the resulting guarantee on the monitoring strategy *σ̄*1 becomes\n \n$$\\begin{aligned}\n\\min\\_{u \\in {\\mathcal{U}}} \\mathbb{E}\\_{X\\sim \\bar{\\sigma}^1}[f(X,u)] \\geq (1-1/e) z^\\*\\_{(\\mathcal{M})} - \\epsilon,\\end{aligned}$$ \nfor $T = 4\\lceil \\frac{\\ln |{\\mathcal{U}}|}{\\epsilon^2}\\rceil$. Since commercial solvers are now highly efficient at solving classical IPs, we instead optimally solve at each iteration of the algorithm.\nWe now move to computational experiments, where we evaluate the three approaches presented in this article and compare their running times, as well as the performance and implementability of the monitoring strategies they generate.\nComputational Study\n===================\nIn this section, we describe how our strategic network monitoring problem can be applied for contamination detection in water distribution systems and actuator protection in networked control systems. Our experiments are coded in Julia and all optimization problems are solved using the Gurobi solver v9.0 on a computer with 2.3-GHz, 8-Core Intel Core i9 processor and 32 GB of RAM.\nApplication I: Contamination Detection in Water Distribution Systems\n--------------------------------------------------------------------\nWe first consider the security problem where the operator of a water distribution system aims to allocate sensors to monitor their network against adversarially-induced contamination events. We model the water distribution system as a directed graph, with the vertices representing pumps, junctions, and water tanks, while the edges represent pipes. The direction of every edge is adopted to be in the direction of the water flow. We assume that the operator has access to contaminant detection sensors that can be deployed on valve access points or fire hydrants and measure water quality indicators such as electrical conductivity, free and total chlorine, turbidity, and oxygen reduction potential in the water.\nIn this application, the set of vulnerable components U (resp. set of sensor locations X) is given by the set of edges (resp. nodes) of the graph modeling the water network. Monitoring sets are constructed through simulations using a hydraulic network solver that tracks the advection and reaction dynamics from a contaminant intrusion event. In practice, the security levels of components can be assessed based on previously deployed security measures and their accessibility to an adversary. One can then use different security scales to determine quantitative values for the security levels based on that assessment. In this application, we adopt the scale {0.2, 0.4, 0.6, 0.8} for security levels, and randomly assign them to each component.\nNext, we implement the three solution approaches described in Sections [sec:Method] and [sec:Alt] on the water distribution system ky5 . This anonymized real-world water network from Kentucky comprises 496 pipes and 420 nodes (junctions, water tanks, pumps), satisfies a demand of 2.28 million gallons of water per day, and spans 52.3 miles. In Fig. [fig:bounds], we illustrate the bounds derived in Theorem [prop:bounds] and computed by solving the generalized covering set problem and the nonlinear set packing problem, respectively.\n[scale=1]\n[ grid=major, grid style=dashed,gray!30, xlabel=Number of sensors *r*, ylabel=Lowest post-security level $\\min\\limits\\_{u \\in {\\mathcal{U}}}\\mathbb{E}\\_{X\\sim \\sigma^1}[f(X,u)]$, font=, xmin=0, xmax=19, ymin=0.2, ymax=1, ytick=0.2, 0.4,0.6,0.8,1, xtick=0, 2,4,6,8,10,12,14,16,18, legend pos=south east, legend cell align=left, legend style=draw=none, font =, ]\ntable[x=rs, y=Upper Bound, col sep=comma, comment chars=%] ky5resultsobj.csv;\ntable[x=rs, y=Optimal Value, col sep=comma, comment chars=%] ky5resultsobj.csv;\ntable[x=rs, y=Lower Bound Marg, col sep=comma, comment chars=%] ky5resultsobj.csv;\n[fig:bounds]\nFrom Fig. [fig:bounds], we observe that the bounds obtained from and are close to the optimal value of. On average, the optimality gap given by the bounds does not exceed 4.7 %, while the actual optimality gap is equal to 3.6 % on average.\nIn general, we observe that *z*(M)\\* increases nonlinearly with respect to the number of sensors. When *r* is small, the optimal monitoring strategy focuses on protecting the most vulnerable components (with a security level of 0.2). As the operator has access to more sensors, they randomize their placement to increase the post-security level of less critical components. With 19 sensors, the operator can monitor all components and ensure a post-security level of 1 for the entire network.\nNext, we compare the running times of the various approaches. Specifically, we illustrate the time to (a) solve and get an upper bound on *z*(M)\\*, (b) solve and construct a monitoring strategy using the coordination algorithm (CA), (c) solve using column generation (CG), and (d) solve using multiplicative weights updates (MWU) with *N* = 2, 480 iterations to ensure a theoretical (absolute) optimality gap of *ε* = 0.1. Their running times are illustrated in Fig. [fig:times].\n[scale=1]\n[ grid=major, grid style=dashed,gray!30, xlabel=Number of sensors *r*, ylabel=Running time [s], font=, xmin=1, xmax=17, ymin=0.01, ymax=1000, xtick=1,3,5,7,9,11,13,15,17, legend columns=4, legend style=/tikz/column 2/.style= column sep=0pt,, font =,legend cell align=left,at=(0.5,-0.18),anchor=north, ]\ntable[x=rs, y=Time Lower Bound Marg, col sep=comma, comment chars=%] ky5resultstimes.csv;\ntable[x=rs, y=Time Upper Bound, col sep=comma, comment chars=%] ky5resultstimes.csv;\ntable[x=rs, y=Time Optimal Value, col sep=comma, comment chars=%] ky5resultstimes.csv;\ntable[x=rs, y=Time Multiplicative Weight Update, col sep=comma, comment chars=%] ky5resultstimes.csv;\n[fig:times]\nFig. [fig:times] shows that the upper bound on *z*(M)\\* derived in Theorem [prop:bounds] can be efficiently computed; by solving, we obtain the upper bound in less than 0.2 seconds. We also observe that the number of sensors has limited impact on the running time.\nFurthermore, we find that CG is approximately 100 times faster than MWU. This is in part due to the number of iterations that are required for MWU to theoretically guarantee a small optimality gap. Interestingly, we find that CG can efficiently compute an optimal solution of when the number of sensors is small. However, as *r* increases, the running time of CG increases exponentially. Finally, we find that our approximate monitoring strategy, which achieves the lower bound in Theorem [prop:bounds] and is obtained by solving + CA, can be efficiently computed.\nFinally, we compare the implementability of the solutions generated by the three approaches by illustrating their node basis sizes in Fig. [fig:nodebasis].\n[scale=1]\n[ grid=major, grid style=dashed,gray!30, xlabel=Number of sensors *r*, ylabel=Node basis size ∣X(*σ*1)∣, font=, xmin=1, xmax=17, ymin=0, ymax=200, xtick=1,3,5,7,9,11,13,15,17, legend style=font =,draw=none,legend cell align=left,at=(0.8,0.37),anchor=north, ]\ntable[x=rs, y=Node Basis Size Marg, col sep=comma, comment chars=%] ky5results.csv;\ntable[x=rs, y=Node Basis Size at Optimality, col sep=comma, comment chars=%] ky5results.csv;\ntable[x=rs, y=Node Basis Size Multiplicative Weight Update, col sep=comma, comment chars=%] ky5results.csv;\n[fig:nodebasis]\nFig. [fig:nodebasis] clearly shows the implementability advantage of our monitoring strategy. In particular, we find that our solution requires monitoring at most 20 different locations and achieves an excellent worst-case post-security level (from Fig. [fig:bounds]). In contrast, the optimal post-security level achieved using CG or MWU requires monitoring up to 180 different locations. From an implementation perspective, our monitoring strategy is considerably easier to translate into a daily/weekly scheduling of monitoring operations.\nApplication II: Protecting Actuators from Extended Replay Attacks\n-----------------------------------------------------------------\nWe now show how our findings can be used to protect actuators in a networked control system from extended replay attacks. The networked system consists of a set of physical states X and actuators U, as well as fixed unprotected sensors. We consider an attacker who aims to conduct an undetectable attack on an actuator using an extended replay strategy. More precisely an extended replay attack on an actuator is undetectable if the sensor measurements remain in the steady state during the attack. To this end, the attacker must also compromise a subset of components (i.e., unprotected sensors and other actuators) to ensure that the attack is undetected. Such subset of components typically depends on the dynamics of the networked system, and can be efficiently determined in large-scale systems,.\nImportantly, the vulnerability of an actuator *u* with respect to extended replay attacks can be determined based on its security index *δ**u*, which represents the minimum number of sensors and actuators, including *u* itself, that the attacker must compromise to conduct an undetectable extended replay attack against *u*, in the absence of protected sensors. If no undetectable extended replay attack against *u* is possible, then *δ**u* = + ∞. We can then scale these security indices and define security levels for this application, following our modeling framework. Specifically, we adopt an identical scale as in the previous application, and define the security level of each actuator *u* ∈ U as follows:\n \n$$\\begin{aligned}\n\\varphi\\_u =&\\textstyle 0.2 \\cdot \\mathds{1}\\_{[\\delta\\_u \\in (0,5] ]} \n+ 0.4 \\cdot \\mathds{1}\\_{\\{\\delta\\_u \\in (5,15] \\}}+ 0.6 \\cdot \\mathds{1}\\_{\\{\\delta\\_u \\in (15,20] \\}} \\\\\n&\\textstyle\n+ 0.8 \\cdot \\mathds{1}\\_{\\{\\delta\\_u \\in (20,+\\infty)\\}}+ 1.0 \\cdot \\mathds{1}\\_{\\{\\delta\\_u= +\\infty\\}}.\\end{aligned}$$ \n[!htbp] [tab:ieee]\nccccccccccccc & & & & & & \n *r* & & + CA & CG & MWU & & + CA & CG & MWU & & + CA & CG & MWU \n50 & &0.0 & 0.0 & 0.3 & & 1.6 & 71.3 & 227.7 & & 223 & 373 & 430 \n100 & &0.0 & 0.4 & 0.4 & & 1.9 & 600 & 245.1 & & 318 & 868 & 737 \n150 & &0.0 & 1.1 & 0.4 & & 13.2 & 600 & 302.6 & & 328 & 981 & 824 \n200 & &0.0 & 8.0 & 0.3 & & 5.1 & 600 & 412.2 & & 328 & 1009 & 827 \n250 & &0.0 & 15.0 & 0.3 & & 174.2 & 600 & 581.2 & & 344 & 1024 & 1115 \n300 & &0.0 & 24.0 & 0.3 & & 39.2 & 600 & 591.0 & & 344 & 1086 & 612 \nTo defend the networked system, we assume the operator possesses protected sensors that cannot be compromised by the attacker and that can be positioned within the network. We assume that each positioned sensor can only measure one physical state, which is a commonly adopted model of large–scale systems . In particular, a protected sensor can detect if the state it measures deviates from its steady state value during an extended replay attack. Thus, actuator *u* ∈ U belongs to the monitoring set of state *x* ∈ X if placing a protected sensor at *x* prevents the attacker from conducting an undetectable extended replay attack on *u*.\nWe now consider the IEEE 2383 bus power system. We model the system using linearized swing equations where the generators are represented by two states (rotor angle and frequency) and load buses with one state (voltage angle) . We assume that all ∣X∣ = 3037 states are measurable, and that the attacker can conduct an attack using ∣U∣ = 1042 loads (we randomly selected 30% of the loads to be attackable). Table [tab:ieee] compares the performance of the different solution approaches on this networked system. Note that we set a time limit of 10 minutes for each approach.\nWe observe that for this network, our solution approach optimally solves the network monitoring problem. Indeed, the upper bound obtained by solving guarantees that the monitoring strategy computed from + CA is optimal for for every number of sensors. Although we observe some variance in the running time of our approach, the average running time for this network is 29 seconds. In general, we find that the hardest instances to solve are for which the number of sensors to position is high but not too close to the minimum set cover size. On the other hand, MWU achieves a very high performance, but requires 393 seconds on average. This illustrates the fact that the empirical performance of MWU is better than the theoretical guarantees. In contrast, we find that CG scales poorly with the number of sensors, which is interesting given that MWU and CG solve the very similar IPs and. Finally, we similarly note that the node basis (i.e., the locations that are monitored with positive probability) of our monitoring strategy is up to 3 times smaller than when utilizing CG or MWU, thus providing a solution that is more easily implementable in practice.\nSummary and Extension of Solution Methodology\n---------------------------------------------\nIn summary, we find that by extracting the structure of the network monitoring problem, we could derive an efficient solution approach. Our approach solves two IPs and runs the coordination algorithm to directly obtain a monitoring strategy with strong performance guarantees and that is easier to implement in practice. This contrasts with the classical CG and MWU algorithms that iteratively refine their solutions and generally run more slowly.\nInterestingly, more flexibility can be embedded in our solution approach. For instance, if the operator is interested in an even simpler monitoring strategy, we can add the cardinality constraint ∑*x* ∈ X*y**x* ≤ *s* in to find the best monitoring strategy of that monitors at most *s* locations. This permits the operator to achieve the desired tradeoff between solution implementability and guaranteed post-security level.\nFinally, we note that the three solution approaches can be combined for an even better performance. Indeed, for the instances for which our approach does not provide an optimal solution to the network monitoring problem, we can utilize our solution as a warm start for CG and/or MWU, which will then iteratively improve it.\nConclusion\n==========\nIn this article, we studied the problem of strategically monitoring large-scale networked systems with heterogeneous component criticalities using multiple sensors. Specifically, we formulated a large-scale maximin problem where the network operator selects a randomized placement of their protected sensors to improve the post-security level of the network’s most critical components against an attacker.\nWe developed a three-step solution approach that leverages the structure of the network monitoring problem. First, we solved a generalized covering set problem to determine the marginal probabilities to monitor each location. Then, we utilized a combinatorial algorithm to randomize the positioning of the multiple sensors and obtain a monitoring strategy that is consistent with the marginal probabilities. Finally, we solved a nonlinear set packing problem to evaluate our solution’s optimality gap. We also investigated problem instances with special structures, for which we refined our solution approach and optimality gaps.\nWe then adapted two classical solution methods for solving our monitoring problem based on column generation and multiplicative weights updates, and compared them on real-world benchmark water distribution and power systems. Our computational study revealed that our solution approach outperforms the classical methods as the size of the problem increases. In particular, we find that our approach can efficiently compute near optimal solutions for large instances and are significantly simpler to implement.\nAs part of future work, it would be interesting to investigate how the solution method developed in this article (combining smaller IPs and combinatorial algorithms) can be applied to solve problems where an attacker targets multiple components simultaneously, or where sensors are faulty.\nWe consider an optimal solution *σ*1\\* of and we set *C* = X(*σ*1\\*). Then, *z*(R*C*)\\* ≥ min*x* ∈ X(*σ*1\\*)E*X* ∼ *σ*1\\*[*f̄*(*X*, *x*)]. Furthermore, $\\min\\_{u \\notin U\\_C} \\varphi\\_u = \\min\\_{u \\notin U\\_{{\\mathcal{X}}(\\sigma^{1^\\*})}} \\varphi\\_u$. Therefore, $z^\\*\\_{({\\overline{\\mathcal{M}}})} \\leq \\max\\_{C \\subseteq {\\mathcal{X}}} \\min\\{z^\\*\\_{(\\mathcal{R}\\_C)},\\min\\_{u \\notin U\\_C} \\varphi\\_u\\}$.\nTo show the reverse inequality, let $C^\\* \\in \\arg\\max\\_{C \\subseteq {\\mathcal{X}}} \\min\\{z^\\*\\_{(\\mathcal{R}\\_C)},\\min\\_{u \\notin U\\_C} \\varphi\\_u\\}$. We also consider an optimal solution *σ*1\\* of (R*C*\\*). Without loss of generality, we can assume that X(*σ*1\\*) ⊆ *C*\\* (i.e., *σ*1\\* only places sensors within *C*\\*). Then, min*x* ∈ X(*σ*1\\*)E*X* ∼ *σ*1\\*[*f̄*(*X*, *x*)] ≥ *z*(R*C*\\*)\\*. Furthermore, let $u^\\prime \\notin U\\_{{\\mathcal{X}}(\\sigma^{1^\\*})}$. If *u*′ ∈ *U**C*\\* \\ *U*X(*σ*1\\*), let *x*′ ∈ *C*\\* \\ X(*σ*1\\*) such that *u*′ ∈ *U**x*′. We then obtain:\n \n$$\\begin{aligned}\n\\varphi\\_{u^\\prime} \\geq \\overline{\\varphi}\\_{x^\\prime} = \\mathbb{E}\\_{X \\sim \\sigma^{1^\\*}}[\\bar{f}(X,x^\\prime)] \\geq z^\\*\\_{(\\mathcal{R}\\_{C^\\*})}.\\end{aligned}$$ \nIf instead $u^\\prime \\notin U\\_{C^\\*}$, then $\\varphi\\_{u^\\prime} \\geq \\min\\_{u \\notin U\\_{C^\\*}} \\varphi\\_u$. In conclusion, $z^\\*\\_{({\\overline{\\mathcal{M}}})} = \\max\\_{C \\subseteq {\\mathcal{X}}} \\min\\{z^\\*\\_{(\\mathcal{R}\\_C)},\\min\\_{u \\notin U\\_C} \\varphi\\_u\\}$.\nWe first show that $z\\_{({\\mathcal{M}})}^\\* \\geq z\\_{({\\overline{\\mathcal{M}}})}^\\*$: Let *σ*1\\* ∈ Δ1 be an optimal solution of and let *u*′ ∈ U. If *u*′ ∈ *U*X(*σ*1\\*), then let *x*′ ∈ X(*σ*1\\*) such that *u*′ ∈ *U**x*′. Since $\\varphi\\_{u^\\prime} \\geq \\overline{\\varphi}\\_{x^\\prime}$ and for every *X* ∈ A, $ \\mathds{1}\\_{\\{x^\\prime \\notin X\\}} \\geq \\mathds{1}\\_{\\{u^\\prime \\notin U\\_X\\}}$, then\n \n$$\\begin{aligned}\n \\mathbb{E}\\_{X\\sim\\sigma^{1^\\*}}[f(X,u^\\prime)] &\\geq \\mathbb{E}\\_{X\\sim\\sigma^{1^\\*}}[1 - (1 - \\overline{\\varphi}\\_{x^\\prime}) \\mathds{1}\\_{\\{x^\\prime \\notin X\\}}]\\\\\n& \\geq \\min\\_{\\quad\\mathclap{x \\in {\\mathcal{X}}(\\sigma^{1^\\*})}\\quad} \\mathbb{E}\\_{X\\sim\\sigma^{1^\\*}}[1 - (1 - \\overline{\\varphi}\\_{x}) \\mathds{1}\\_{\\{x \\notin X\\}}].\\end{aligned}$$ \nIf instead $u^\\prime \\notin U\\_{{\\mathcal{X}}(\\sigma^{1^\\*})}$, then:\n \n$$\\begin{aligned}\n&\\mathbb{E}\\_{X\\sim\\sigma^{1^\\*}}[f(X,u^\\prime)] = \\varphi\\_{u^\\prime} \\geq \\min\\_{u \\notin U\\_{{\\mathcal{X}}(\\sigma^{1^\\*})} } \\varphi\\_u.\\end{aligned}$$ \nThus, $z^\\*\\_{(\\mathcal{M})} \\geq z^\\*\\_{({\\overline{\\mathcal{M}}})}$. We next show that for every *C* ⊆ X of size greater than *r*, $z^\\*\\_{(\\mathcal{R}\\_C)} = 1 - \\frac{k^\\*\\_C-{r}}{S^{k^\\*}\\_C}$.\nWe rewrite *C* = {*x*1, …, *x**n*} with *n* > *r* and indices satisfying $\\overline{\\varphi}\\_{x\\_1} \\leq \\cdots \\leq \\overline{\\varphi}\\_{x\\_n}$. Then necessarily, *k**C*\\* ≥ *r* + 1 since\n \n$$\\begin{aligned}\n1 - \\frac{r + 1 - r}{\\sum\\_{l=1}^{r+1} (1 - \\overline{\\varphi}\\_{x\\_l})^{-1}} \\geq 1 - (1 - \\overline{\\varphi}\\_{x\\_{r+1}}) = \\overline{\\varphi}\\_{x\\_{r+1}}.\\end{aligned}$$ \nLet *σ*1′ ∈ Δ1 be a monitoring strategy satisfying. The existence of such a monitoring strategy follows from Farkas’ lemma (for instance, see ), since\n \n$$\\begin{aligned}\n \\sum\\_{l=1}^{k^\\*\\_C} 1 - \\frac{k^\\*\\_C-{r}}{(1-\\overline{\\varphi}\\_{x})S^{k^\\*}\\_C} = r,\n \\end{aligned}$$ \nand for every *l* ∈ ⟦1, *k**C*\\*⟧,\n \n$$\\begin{aligned}\n1 \\geq 1 - \\frac{k^\\*\\_C-{r}}{(1-\\overline{\\varphi}\\_{x\\_l})S^{k^\\*}\\_C} \\geq 1 - \\frac{k^\\*\\_C-{r}}{(1-\\overline{\\varphi}\\_{x\\_{k^\\*\\_C}})S^{k^\\*}\\_C} \\geq 0.\n \\end{aligned}$$ \nWe next consider *x* ∈ *C*. If *x* ∈ {*x*1, …, *x**k**C*\\*}, then:\n \n$$\\begin{aligned}\n\\mathbb{E}\\_{X \\sim \\sigma^{1^\\prime}}[\\bar{f}(X,x)] &= \\overline{\\varphi}\\_x + (1-\\overline{\\varphi}\\_x)\\mathbb{P}\\_{X \\sim \\sigma^{1^\\prime}}(x \\in X) \\\\\n&= 1 - \\frac{k^\\*\\_C-{r}}{S^{k^\\*}\\_C}.\\end{aligned}$$ \nIf instead *x* ∈ *C* \\ {*x*1, …, *x**k**C*\\*}, then:\n \n$$\\begin{aligned}\n\\mathbb{E}\\_{X \\sim \\sigma^{1^\\prime}}[\\bar{f}(X,x)] &= \\overline{\\varphi}\\_x \\geq \\overline{\\varphi}\\_{x\\_{k^\\*\\_C+1}} \\\\\n&> 1 - \\frac{k^\\*\\_C + 1 - r}{S^{k^\\*}\\_C + (1 - \\overline{\\varphi}\\_{x\\_{k^\\*\\_C+1}})^{-1}},\\end{aligned}$$ \nwhich is equivalent to\n \n$$\\begin{aligned}\n\\mathbb{E}\\_{X \\sim \\sigma^{1^\\prime}}[\\bar{f}(X,x)] > 1 - \\frac{k^\\*\\_C - r}{S^{k^\\*}\\_C}.\\end{aligned}$$ \nThus, $z^\\*\\_{(\\mathcal{R}\\_C)} \\geq 1 - \\frac{k^\\*\\_C-{r}}{S^{k^\\*}\\_C}$.\nNow, let us assume by contradiction that $z^\\*\\_{(\\mathcal{R}\\_C)}> 1 - \\frac{k^\\*\\_C-{r}}{S^{k^\\*}\\_C}$. Let *σ*1\\* be an optimal solution of *z*(R*C*)\\*. Then, we deduce that:\n \n$$\\begin{aligned}\n{\\ForAll\\,}x \\in C, \\ \\mathbb{P}\\_{X\\sim \\sigma^{1^\\*}}(x \\in X) > 1 - \\frac{k^\\*\\_C-{r}}{(1-\\overline{\\varphi}\\_x)S^{k^\\*}\\_C},\\end{aligned}$$ \nwhich induces the following contradiction:\n \n$$\\begin{aligned}\nr &= \\sum\\_{X \\in {\\mathcal{A}}} \\sigma^{1^\\*}\\_X |X| = \\sum\\_{x \\in {\\mathcal{X}}} \\sum\\_{X \\in {\\mathcal{A}}} \\sigma^{1^\\*}\\_X \\mathds{1}\\_{\\{x \\in X\\}} \\\\\n&=\\sum\\_{x \\in {\\mathcal{X}}} \\mathbb{P}\\_{X\\sim \\sigma^{1^\\*}}(x \\in X) > \\sum\\_{l=1}^{k^\\*\\_C} 1 - \\frac{k^\\*\\_C-{r}}{(1-\\overline{\\varphi}\\_{x\\_l})S^{k^\\*}\\_C} = r.\\end{aligned}$$ \nIn conclusion, $z^\\*\\_{(\\mathcal{R}\\_C)}= 1 - \\frac{k^\\*\\_C-{r}}{S^{k^\\*}\\_C}$, and an optimal solution is given by *σ*1′ satisfying.\nWe note that for every *C* ⊆ X of size no more than *r*, *z*(R*C*)\\* = 1 (which is obtained by placing a sensor on each location in *C*). Also note that in such cases, U \\ *U**C* ≠ ∅ and $1 - \\frac{k^\\*\\_C-{r}}{S^{k^\\*}\\_C}\\geq 1 \\geq \\min\\_{u \\notin U\\_C} \\varphi\\_u$. Thus, from Lemma [lem:Angle], we deduce that \n$$z^\\*\\_{(\\overline{\\mathcal{M}})} = \\max\\_{C \\subseteq {\\mathcal{X}}} \\min\\bigg\\{1 - \\frac{k^\\*\\_C-{r}}{S^{k^\\*}\\_C},\\min\\_{\\mathclap{u \\notin U\\_C}}\\varphi\\_u\\bigg\\}.$$ \nFinally, we prove the upper bound on *z*(M)\\*. Since *f*(*X*, *u*) ≤ 1 for every *X* ∈ A and every *u* ∈ U, then *z*(M)\\* ≤ 1. Furthermore, by strong duality, we know that *z*(M)\\* = min*σ*2 ∈ Δ2max*X* ∈ AE*u* ∼ *σ*2[*f*(*X*, *u*)], where $\\Delta^2 \\coloneqq \\{\\sigma^2 \\in [0,1]^{|{\\mathcal{U}}|} \\ | \\ \\sum\\_{u \\in {\\mathcal{U}}}\\sigma^2\\_u = 1\\}.$ Then, given a set packing *T* ∈ T, we define *σ*2′ ∈ Δ2 as follows:\n \n$$\\begin{aligned}\n\\sigma^{2^\\prime}\\_u = \\begin{cases} \\frac{1}{(1-\\varphi\\_u)S\\_T}, & \\text{ if } u \\in T\\\\\n0, & \\text{ otherwise.}\\end{cases}\n \\end{aligned}$$ \nWe can then derive the following upper bound.\n \n$$\\begin{aligned}\n z^\\*\\_{({\\mathcal{M}})} &\\leq \\max\\_{X\\in {\\mathcal{A}}}\\mathbb{E}\\_{u\\sim \\sigma^{2^\\prime}}[f(X,u)] \\\\\n &= \\max\\_{X\\in {\\mathcal{A}}}\\sum\\_{u \\in T} \\sigma^{2^\\prime}\\_u (1 - (1-\\varphi\\_u)\\mathds{1}\\_{\\{u \\notin U\\_X\\}})\\\\\n & = 1 - \\frac{1}{S\\_T}\\min\\_{X\\in {\\mathcal{A}}}\\sum\\_{u \\in T}\\mathds{1}\\_{\\{u \\notin U\\_X\\}} \\leq 1 - \\frac{|T|-r}{S\\_T}\n \\end{aligned}$$ \nIn conclusion, $z\\_{({\\mathcal{M}})}^\\*\\leq \\min\\{1,\\min\\_{T \\in {\\mathcal{T}}} 1 - \\frac{|T|-{r}}{S\\_T}\\}$.\nLet (*y*\\*, *z*\\*, *ρ*\\*) be an optimal solution of, and we set *C* = supp(*y*\\*). Then, can be rewritten as follows:\n \n$$\\begin{aligned}\n{3}\n\\max\\_y &\\min\\bigg\\{\\max\\_\\rho \\min\\_{x \\in {\\operatorname{supp}}(y)} && \\overline{\\varphi}\\_x + (1- \\overline{\\varphi}\\_x) \\rho\\_x, \\min\\_{u \\notin U\\_{{\\operatorname{supp}}(y)}} \\varphi\\_u\\bigg\\}.\\nonumber\\\\\n \\text{s.t.} \\ & 0 \\leq \\rho\\_x \\leq y\\_x, \\ &&{\\ForAll\\,}x \\in {\\mathcal{X}}\\label{cons\\_1}\\\\\n & \\sum\\_{x \\in {\\mathcal{X}}}\\rho\\_x = r,\\label{cons\\_2}\\\\\n & y\\_x \\in \\{0,1\\}, \\ &&{\\ForAll\\,}x \\in {\\mathcal{X}}.\\nonumber\\end{aligned}$$ \nThe equivalence between and then follows from Lemma [lem:Angle], equality, and the existence of monitoring strategies satisfying constraints -.\nWe argued in the main text that Problem is indeed a MIP formulation of the following problem $\\min\\_{T \\in {\\mathcal{T}}} \\frac{S\\_T}{|T|-r}$. In particular, at optimality, constraints and *M*2 ensure that at optimality all *t**l* variables are equal to 0 except for the index corresponding to the set packing size, thus ensuring the validity of the formulation. Then, $\\min\\{1,\\min\\_{T \\in {\\mathcal{T}}} 1 - \\frac{|T|-{r}}{S\\_T}\\} = 1 - \\frac{1}{z^\\*\\_{(\\mathcal{P})}}.$ Note that we used the convention that if a is infeasible (which occurs if there is no set packing of size greater than *r*), then *z*(P)\\* = + ∞.\nWe consider an instance where *U**x* ∩ *U**x*′ = ∅ for every *x* ≠ *x*′ ∈ X. We index the locations in X = {*x*1, …, *x**n*} in nondecreasing order of their criticalities. Then, we consider the set packing *T*′ = {*u*1, …, *u**k*X\\*} ∈ T, where for every *l* ∈ ⟦1, *k*X\\*⟧, $\\varphi\\_{u\\_l} = \\overline{\\varphi}\\_{x\\_l}$. In other words, *T*′ selects the most critical component within each of the first *k*X\\* monitoring sets. As a consequence, ∣*T*′∣ = *k*X\\* and *S**T*′ = *S*X*k*\\*. Since *U*X = U, then from Theorem [prop:bounds], we obtain:\n \n$$\\begin{aligned}\nz^\\*\\_{(\\mathcal{R}\\_{\\mathcal{X}})} &= 1 - \\frac{k^\\*\\_{\\mathcal{X}}- r}{S^{k^\\*}\\_{\\mathcal{X}}} \\leq z\\_{({\\overline{\\mathcal{M}}})}^\\*\\leq z^\\*\\_{({\\mathcal{M}})} \\\\\n&\\leq 1 - \\frac{|T^\\prime| - r}{S\\_{T^\\prime}} = 1 - \\frac{k^\\*\\_{\\mathcal{X}}- r}{S^{k^\\*}\\_{\\mathcal{X}}}.\\end{aligned}$$ \nThus, $z^\\*\\_{({\\mathcal{M}})} = z^\\*\\_{({\\overline{\\mathcal{M}}})}$. Furthermore, any monitoring strategy *σ*1′ ∈ Δ1 satisfying for *C* = X, which is optimal for (RX), is then optimal for.\nWe suppose that for every component, the security levels are identical and equal to *φ*. Let *C* ⊆ X. Then, we find that *k**C*\\* = ∣*C*∣ since for every *k* ∈ ⟦1, ∣*C*∣⟧, $\\frac{k-r}{k}(1-\\varphi) \\leq 1- \\varphi$. Therefore, $z^\\*\\_{({\\overline{\\mathcal{M}}})} = \\max\\_{C \\subseteq {\\mathcal{X}}} \\min\\{1 - \\frac{|C| - r}{|C|}(1-\\varphi), \\min\\_{u \\notin U\\_C} \\varphi \\}$.\nNote that for any *C* ⊆ X, $1 - \\frac{|C| - r}{|C|}(1-\\varphi) = \\varphi + \\frac{r}{|C|} (1-\\varphi) \\geq \\varphi$. Therefore,\n \n$$\\begin{aligned}\nz^\\*\\_{({\\overline{\\mathcal{M}}})} = \\max\\_{C\\in \\mathcal{S}} \\varphi + \\frac{r}{|C|} (1-\\varphi) = \\varphi + \\frac{r}{n^\\*}(1-\\varphi),\\end{aligned}$$ \nwhere S is the collection of set covers. Similarly,\n \n$$\\begin{aligned}\n\\min\\_{T\\in{\\mathcal{T}}} 1 - \\frac{|T| - r}{S\\_T} = \\min\\_{T\\in{\\mathcal{T}}} 1 + \\frac{r}{|T|}(1-\\varphi) = 1 + \\frac{r}{m^\\*}(1-\\varphi).\\end{aligned}$$ \nJezdimir Milošević received the M.Sc. degree in electrical engineering and computer science from the University of Belgrade, Belgrade, Serbia, in 2015 and the Ph.D. in electrical engineering and computer science from the KTH Royal Institute of Technology, Stockholm, Sweden, in 2020.\nHe is currently a research and development engineer in Scania’s Autonomous Transport Solutions, Södertälje, Sweden. His research interests are within cyber-security of industrial control systems.\nMathieu Dahan received the M.S. and Ph.D. degrees in computational science and engineering from the Massachusetts Institute of Technology, Cambridge, MA, USA, in 2016 and 2019, respectively.\nHe is currently an Assistant Professor in the School of Industrial and Systems Engineering at the Georgia Institute of Technology, Atlanta, GA, USA. His research interests are in combinatorial optimization, game theory, and predictive analytics, with applications to service and healthcare operations, humanitarian systems, and logistics and supply chain management.\nSaurabh Amin received the Ph.D. degree in systems engineering from the University of California, Berkeley, Berkeley, CA, USA, in 2011.\nHe is currently a Professor with the Department of Civil and Environmental Engineering. He is a member of the Laboratory for Information and Decision Systems with the Massachusetts Institute of Technology, Cambridge, MA, USA. His fields of expertise include control and optimization, applied game theory, and networks. His research focuses on the design and implementation of resilient monitoring and control algorithms for networked infrastructure systems.\nHenrik Sandberg (F’23) received the M.Sc. degree in engineering physics and the Ph.D. degree in automatic control from Lund University, Lund, Sweden, in 1999 and 2004, respectively.\nHe is currently a Professor with the Division of Decision and Control Systems, KTH Royal Institute of Technology, Stockholm, Sweden. His current research interests include security of cyber-physical systems, power systems, model reduction, and fundamental limitations in control.\nDr. Sandberg was a recipient of the Best Student Paper Award from the IEEE Conference on Decision and Control in 2004, an Ingvar Carlsson Award from the Swedish Foundation for Strategic Research in 2007, and a Consolidator Grant from the Swedish Research Council in 2016. He has served on the editorial boards of IEEE Transactions on Automatic Control and the IFAC Journal Automatica. and the IFAC Journal Automatica.\n---\n1. Manuscript received April 6, 2023. The work of JM and HS was supported in part by the Swedish Civil Contingencies Agency (project CERCES2) and Digital Futures (project DEMOCRITUS). The work of MD was supported by the Georgia Tech new faculty start up grant. The work of SA was supported in part by AFOSR grant “Building Attack Resilience into Complex Networks” and NSF grant # 2039771.[↩](#fnref1)\n2. Jezdimir Milošević was with the School of Electrical Engineering and Computer Science, KTH Royal Institute of Technology, Stockholm, Sweden. He is now with Scania Autonomous Transport Solutions (e-mail: jezdimirmilosevic@gmail.com).[↩](#fnref2)\n3. Mathieu Dahan is with the School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332 USA (e-mail: mathieu.dahan@isye.gatech.edu).[↩](#fnref3)\n4. Saurabh Amin is with the Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139 USA (e-mail: amins@mit.edu).[↩](#fnref4)\n5. Henrik Sandberg is with the School of Electrical Engineering and Computer Science, KTH Royal Institute of Technology, Stockholm, Sweden (e-mail: hsan@kth.se).[↩](#fnref5)\n"},"data_id":{"kind":"string","value":"arxiv_0184745"}}},{"rowIdx":46191,"cells":{"content":{"kind":"string","value":"Three-boson bound states in Minkowski space with contact interactions\n=====================================================================\nThe structure of the three-boson bound state in Minkowski space is studied for a model with contact interaction. The Faddeev-Bethe-Salpeter equation is solved both in Minkowski and Euclidean spaces. The results are in fair agreement for comparable quantities, like the transverse amplitude obtained when the longitudinal constituent momenta of the light-front valence wave function are integrated out. The Minkowski space solution is obtained numerically by using a recently proposed method based on the direct integration over the singularities of the propagators and interaction kernel of the four-dimensional integral equation. The complex singular structure of the Faddeev components of the Bethe-Salpeter vertex function for space and time-like momenta in an example of a Borromean system is investigated in detail. Furthermore, the transverse amplitude is studied as a mean to access the double-parton transverse momentum distribution. Following that, we show that the two-body short-range correlation contained in the valence wave function is evidenced when the pair has a large relative momentum in a back-to-back configuration, where one of the Faddeev components of the Bethe-Salpeter amplitude dominates over the others. In this situation a power-law behavior is derived and confirmed numerically.\nIntroduction\n============\nThe Bethe-Salpeter (BS) approach is an important and efficient tool to investigate relativistic few-body systems. Solving the BS equation with a realistic interaction, especially for a three-body system, is technically a rather complicated problem. However, the principal qualitative properties can be understood through models that retain the main features of the physical system. One of these models, fundamental in nuclear physics and described (in the two-body case) in any textbook, is the zero-range interaction.\nThe three-body BS equation in Minkowski space with zero-range interaction was derived in Ref. in 1992. Later on, in 2017, the equation was solved in Euclidean space for the first time and then, quite recently, directly in Minkowski space. Concerning the Euclidean space solution, the reasons of this time lag was due to the fact that, though the BS equation was given in Ref. in a simple and transparent form, as it was there presented the equation did not allow to make the Wick rotation directly. Whether the rotating integration contour crosses the singularities or not, this depends on the point around which it is rotated. To determine the safe point one must make a shift of variables in the BS equation , as proposed in Ref. . This was the key to success. As for the Minkowski space solution , the methods were absent until recently. In Ref. the method developed in Ref. was used.\nThe aforementioned method is based on the direct integration of the singularities of the propagators and interaction kernel . It does not resort to the Nakanishi integral representation and light-front (LF) projection. In the present paper we will follow this method and explore it for obtaining information on the structure of relativistic three-body systems.\nHowever, the corresponding equation in the light-front dynamics (LFD) was derived and solved already in Ref. . Then the stability of the solution was thoroughly explored in Ref. . Finding the solution of the BS equation fully in Minkowski space is rather important for applications when used to calculate observables like parton distributions and electromagnetic form factors (see e.g. Ref. ). The Euclidean solution, though it provides the bound state spectrum, requires a careful analytical extension of the Euclidean BS amplitude, and for large enough momentum transfers overlapping cuts turns the task cumbersome, preventing to explore the whole range of momentum transfers. Besides that, the comparison between the BS and the LFD solutions provides valuable information about the structure of the system, i.e., regarding contribution of the higher Fock components etc.\nAs it was shown in Ref. , the effect coming from higher Fock components on the binding energy and transverse amplitude is huge, even for weakly bound states. This is different from the two-body case, where the truncation at the valence state does not present such a dramatic effect (see e.g. Refs. ). This difference in a three-body system is explained by the contribution of effective three-body forces of relativistic origin, as investigated in Ref. . Noteworthy that the BS equation for three bosons has a kernel analogous to the contribution provided by the quark exchange diagrams in quark-diquark models in the constituent quark picture , making even more appealing the outcomes of the Minkowski space approach to be presented as follows.\nThe conclusion that the effect coming from higher Fock components is sizable leads to raise doubts regarding the range of validity of valence inspired models, which are widely applied to hadron physics, as they might be inappropriate to describe certain features of the bound state dynamics, particularly for three-body systems. It is worth mentioning that even for two-boson bound states the contributions coming from higher Fock components, as shown by the calculations, can constitute more than 30% of the normalization. The BS equation and LFD approaches have already been used as a suitable framework in phenomenological applications. For instance, the calculation of the LF amplitudes in a simplified pion model with strongly bound constituent quarks was done through the solution of the BS equation directly in Minkowski space and also the final state interaction in heavy meson decays was studied using a relativistic LF model .\nAs mentioned, the comparison of the binding energies calculated within LFD and BS equation for a one-boson exchange kernel presents a significant discordance , unlike what happens for two-body systems. In Ref. there was found an increasing effect of the three-body forces as the exchanged boson mass *μ* grows, what is relevant for the zero-range case, which corresponds effectively to *μ* → ∞. Although that work was quite instructive, the three-body forces were taken into account only perturbatively, producing a significant contribution to the bound state energy, what indicates the necessity to go beyond perturbation theory. It is essential to obtain the non-perturbative solution of the three-body BS and LFD equations, including three-body forces, in order to have a thorough understanding of the physical system.\nIn the non-relativistic approach, within the Schrödinger equation, it is well known that the binding energy of a three-boson system with the two-body zero-range interaction is not bound from below, what is known as the Thomas collapse . As shown in Ref. , and further explored numerically in Ref. , the relativistic effects result in an effective repulsion at small distances that prevents the Thomas collapse in the relativistic case. This result was found for the valence truncation, within the LFD framework. Therefore, exploring the complete amplitude by means of the BS equation, which includes higher Fock contributions, is necessary to describe such a relativistic three-body system.\nFurthermore, the approach for three-boson systems allows one to explore within the relativistic context a wide and important field of research that is already very well established non-relativistically, known as the Efimov physics . The three-body approach developed here paves the way to explore many interesting relativistic phenomena and it is expected to bring more remarkable outcomes as further studies are done.\nThis paper is devoted to a detailed study of the Minkowski space solution of the three-boson Faddeev-BS equation in the case of the two-body zero-range interaction. As the goal is to address the zero-range interaction case, a major point is the influence of relativistic effects on the stability of the three-body system and the impact on its structure. To accomplish such a goal we focus on Borromean systems and the Faddeev-BS equation is solved both in Minkowski and Euclidean spaces. The choice made for Borromean states simplifies the computations in Minkowski space, as the bound state pole is absent in the two-body scattering amplitude, which is an input to the kernel of the Faddeev-BS equation.\nThe Minkowski space solution is obtained by the direct integration of the singularities of the propagators and interaction kernel , what allows to explore in the space and time-like momenta regions the complex singular structure of the Faddeev components of the BS vertex function of such a Borromean state. We study in detail the numerical solutions by showing that both methods produce results in fair agreement for the Faddeev component of the transverse amplitude obtained from the corresponding component of the valence wave function, after integration over the longitudinal LF momentum fractions. In addition, the double-parton content of the transverse amplitude is studied, and we evidenced the two-body short-range correlation contained in the valence wave function. The kinematical condition to expose the pair short-range correlation was set for large relative momentum in a back-to-back configuration, in such situation the Faddeev component of the BS amplitude that brings the pair interaction is the dominant one. We also found, as expected for large relative momentum, a power-law behavior, that was confirmed numerically.\nThe paper is organized as follows. The theoretical formalism for the two-body scattering amplitude, 3-body BS equation and transverse amplitudes is outlined in Secs. [Sec:twobodyamp]-[Sec:TA]. In Sec. [Sec:wickrotation] the Wick rotation of the three-body BS amplitude is revisited to clarify that the transverse amplitude is independent of that. In Sec. [Sec:Res] the numerical results are presented and discussed. The conclusions are then drawn in Sec. [Sec:Concl]. Some of the more lengthy derivations, and also a brief summary of the numerical methods, are available in appendices.\nTwo-body scattering amplitude\n=============================\nFor the contact interaction (with the four-leg vertex *i**λ*), the two-body amplitude ${\\cal F}(M^2\\_{12})$ is determined by the equation shown graphically in Fig. [Fig:two-body-amp] (see also Ref. ). Iterating, we find that the first contribution is simply *i**λ*, the second one is $(i \\lambda)^2{\\cal B}$, where ${\\cal B}$ is the amputated from (*i**λ*)2 the bubble graph, etc. That is:\n \n$$\\begin{aligned}\n\\label{eq37}\ni{\\cal F}(M^2\\_{12})&= i \\lambda+(i \\lambda)^2{\\cal B}+(i \\lambda)^3{\\cal B}^2+\\ldots=\\\\\n& \\frac{i \\lambda}{1-(i \\lambda){\\cal B}(M^2\\_{12})}= \\frac{1}{(i \\lambda)^{-1}-{\\cal B}(M^2\\_{12})},\n\\end{aligned}$$ \nor\n \n$$\\label{eq37a}\n{\\cal F}(M^2\\_{12})= \\frac{1}{i[(i \\lambda)^{-1}-{\\cal B}(M^2\\_{12})]},$$ \nwhere\n \n$$\\label{eq38}\n\\begin{aligned}\n{\\cal B}&(M^2\\_{12})=\\\\\n&\\int\\frac {d^4k}{(2\\pi)^4}\\frac{i}{(k^2-m^2+i\\epsilon)}\\frac{i}{[(k-P)^2-m^2+i\\epsilon]}. \n\\end{aligned}$$ \nHere *m* denotes the boson mass, and *P* is the total four-momentum of the two-body system, *P*2 ≡ *M*122, where *M*122 denotes the squared effective off-shell mass.\nDiagrammatic representation of the integral equation for the two-body scattering amplitude [Fig:two-body-amp] of Ref. .\nThe loop integration ([eq38]) has a log-type ultraviolet divergence that has to be regularized and renormalized by fixing the scattering amplitude at some physical value, which will be given by the bound state pole or scattering length. Although it is a well known and standard procedure, we will present it in the following for the sake of completeness as well as to fix our notation.\nNormalization of the scattering amplitude\n-----------------------------------------\nIn the derivation of the two-body amplitude, we follow the definitions and normalization of Ref. . According to it, the partial wave amplitude of angular momentum *L* is defined as\n \n$$\\label{fpw2}\n F\\_L(k\\_v)=\\frac{1}{32\\pi}\\int\\_{-1}^1dz \\; P\\_L(z) F(k\\_v, z)\\,,$$ \nwhere $k\\_v = |\\vec k|$ is the magnitude of the particle momentum in the rest-frame, *z* is the cosine of the center-of-mass (c.m.) scattering angle and *P**L*(*z*) is the Legendre polynomial. For a *z*-independent amplitude ([eq37a]):\n \n$$\\label{F0}\nF\\_0(k\\_v)=\\frac{1}{16\\pi} {\\cal F}(M^2\\_{12}).$$ \nIn the given normalization, the scattering amplitude is related to the phase shift by \n \n$$\\label{F0a}\nF\\_0(k\\_v)=\\frac{\\varepsilon\\_k}{k\\_v}\\exp(i\\delta\\_0)\\sin\\delta\\_0= \\varepsilon\\_k f\\_0(k\\_v),$$ \nwith $\\varepsilon\\_k=\\sqrt{k\\_v^2 + m^2}$. The S-matrix is unitary if the phase-shift *δ*0 is real and\n \n$$\\label{f0k}\nf\\_0(k\\_v)=\\frac{1}{k\\_v \\cot \\delta\\_0 -ik\\_v},$$ \nis the standard non-relativistic form of the scattering amplitude. The expansion in powers of *k**v*2 in the low energy region, gives:\n \n$$\\label{1a}\nk\\_v\\cot\\delta\\_0= - \\frac{1}{a}+\\frac{1}{2}r\\_0k\\_v^2+\\cdots,$$ \nwhere *a* is the scattering length and *r*0 the effective range. The relation to ${\\cal F}(M^2\\_{12})$ is\n \n$$\\label{fF}\nf\\_0(k\\_v)=\\frac{1}{16\\pi \\varepsilon\\_k}{\\cal F}(M^2\\_{12}),$$ \nand the two renormalization conditions to be used in the following, are based either on fixing the bound state pole or the scattering length. The latter condition reads\n \n$$\\label{ren}\nf\\_0(k\\_v=0)= \\frac{1}{16\\pi m}{\\cal F}(4m^2)=-a,$$ \nthat fixes the scattering amplitude at the continuum branch point.\nRenormalization via bound state pole\n------------------------------------\nOne way to calculate ${\\cal B}(M^2\\_{12})$ is to use the standard Feynman parametrization:\n \n$$\\label{Eq:Feynman\\_param}\n\\frac{1}{a\\,b}=\\int\\_0^1\\frac{du}{[ua+(1-u)b]^2},$$ \nwith *a* = *k*2 − *m*2 + *i**ε*, *b* = (*k* − *P*)2 − *m*2 + *i**ε*, and then compute the 4D integral in the Euclidean space. However, for *M*122 ≥ 4*m*2, the integrand of this integral becomes singular and this method is not so convenient.\nTherefore, to calculate the amplitude (especially for *M*122 > 4*m*2) we use the initial integral ([eq38]) (written in the c.m. frame $\\vec{P}=\\vec{0}$) and start by performing the integration over *k*0 by residues, i.e.\n \n$$\\label{eq41}\n\\begin{aligned}\n{\\cal B}(M^2\\_{12})&=-\\int\\frac{dk\\_0d^3k}{(2\\pi)^4}\\frac{1}{(k\\_0^2-k\\_v^2-m^2+i\\epsilon)}\\\\\n&\\times\\frac{1}{[(k\\_0-M\\_{12})^2-k\\_v^2-m^2+i\\epsilon]}=\\\\\n&2\\pi i(res\\_1(M\\_{12})+res\\_2(M\\_{12})).\n\\end{aligned}$$ \nHere *r**e**s*1, 2 are the residues of the integrand in one of the two poles in the upper half plane of the complex variable *k*0. The positions of the poles are \n*k*0(1) = − *ɛ**k* + *i**ε*, *k*0(2) = *M*12 − *ɛ**k* + *i**ε*, \n and the corresponding residues are given by\n \n$$\\begin{aligned}\nres\\_1(M^2\\_{12})&=&\\int\\_0^\\Lambda\\frac {k\\_v^2 dk\\_v}{(2\\pi)^3}\\frac{1}{\\varepsilon\\_k}\\frac{1}{[(\\varepsilon\\_k+M\\_{12})^2-\\varepsilon\\_k^2+i\\epsilon]},\n\\label{res1}\n\\\\\nres\\_2(M^2\\_{12})&=&\\int\\_0^\\Lambda\\frac {k\\_v^2 dk\\_v}{(2\\pi)^3}\\frac{1}{\\varepsilon\\_k}\\frac{1}{[M\\_{12}(M\\_{12}-2\\varepsilon\\_k)+i\\epsilon]},\n\\label{res2}\\end{aligned}$$ \nwhere the integrals are regularized by the momentum cut-off Λ. If − ∞ < *M*122 < 4*m*2, the integrals ([res1]) and ([res2]) are non-singular ones. Contrary to this, if *M*12 > 2*m*, the second residue is represented as a sum of two contributions: the principal value of the integral over *k**v* and the delta-function contribution, i.e.\n \n$$\\begin{aligned}\n\\; &res\\_2(M^2\\_{12})= res\\_{2a}(M^2\\_{12})+res\\_{2b}(M^2\\_{12}) \\\\\n&=\n\\;PV\\int\\_0^\\Lambda\\frac {k\\_v^2 dk\\_v}{(2\\pi)^3}\\frac{1}{\\varepsilon\\_k}\\frac{1}{M\\_{12}(M\\_{12}-2\\varepsilon\\_k)}\\\\\n&+\\;\\int\\_0^\\Lambda\\frac {k\\_v^2 dk\\_v}{(2\\pi)^3}\\frac{1}{\\varepsilon\\_k}(-i\\pi)\\delta[M\\_{12}(M\\_{12}-2\\varepsilon\\_k)]\\\\\n&= res\\_{2a}(M^2\\_{12})+\\frac{1}{2\\pi i}\\frac{y''}{16\\pi},\\end{aligned}\n\\label{eq42}$$ \nwhere the delta function is integrated out by taking Λ → ∞ and *y*ʺ is defined below, in Eq. ([eqy]).\nAs the contribution *r**e**s*2*a*(*M*122) in is divergent in the ultraviolet limit, it is necessary to perform a regularization process. By renormalizing we can express the bare parameters (in this work, the coupling constant *λ*) via observables (usually, in the field theory, via a ``physical\" coupling constant). From the condition that the two-body system has a bound state with the mass *M*2 and the amplitude ([eq37a]) has a pole at *M*12 = *M*2, one finds for the coupling constant *λ*\n \n$$(i \\lambda)^{-1}={\\cal B}(M^2\\_{2})=2\\pi i(res\\_1(M^2\\_{2})+res\\_2(M^2\\_{2})).\n\\label{lambda2}$$ \nThe denominator in ([eq37a]) then becomes\n \n$$\\begin{gathered}\ni[(i \\lambda)^{-1}-{\\cal B}(M^2\\_{12})]=i[{\\cal B}(M^2\\_{2B})-{\\cal B}(M^2\\_{12})]\\\\\n= i\\,PV\\int\\_0^{\\infty}\\frac{(M\\_{2}^2-M^2\\_{12}) }\n{32\\pi^2\\,\\varepsilon\\_k \\left[k\\_v^2-\\left(\\frac{1}{4}M^2\\_{12}-m^2\\right)\\right] }\\\\\n\\times\\frac{k\\_v^2 dk\\_v}{\\left[k\\_v^2+\\left(m^2-\\frac{1}{4}M^2\\_{2}\\right)\\right]} -\\frac{y''}{16\\pi}.\n\\label{den1}\\end{gathered}$$ \nIn Eq. , the principal value (PV) integral takes into account the singularity at $k\\_v=\\sqrt{\\frac{1}{4}M^2\\_{12}-m^2}$ and the limit of Λ → ∞ is taken since the integral is ultraviolet finite.\nNow the integral ([den1]), (in which the bare coupling constant *λ* is expressed via the two-body bound state mass *M*2) is finite and its calculation in different domains of the variable *M*12 results for F(*M*122) in:\n \n$$\\begin{aligned}\n&&\\mbox{({\\it i}) If $-\\infty < M\\_{12}^2\\leq 0$\\quad $(1\\geq y\\geq 0)$, then:}\n\\nonumber\\\\\n &&\\mathcal{F}(M^2\\_{12})=\n \\left[{\\frac{1}{16\\pi^2 y}\\log\\frac{1+y}{1-y}-\\frac{\\arctan y'\\_{M\\_{2}}}{8\\pi^2 y'\\_{M\\_{2}}}}\\right]^{-1}.\n \\label{Eq:F\\_amp1}\\\\\n&&\\mbox{({\\it ii}) If $0\\leq M\\_{12}^2 \\leq 4m^2 \\quad (0 \\leq y' < \\infty)$, then:} \n\\nonumber\\\\\n&& \\mathcal{F}(M^2\\_{12})=\\left[{\\frac{\\arctan y'}{8\\pi^2 y'}-\\frac{\\arctan y'\\_{M\\_{2}}}{8\\pi^2 y'\\_{M\\_{2}}}}\\right]^{-1}.\n \\label{Eq:F\\_amp2}\\\\\n&&\\mbox{({\\it iii}) If $ 4m^2\\leq M\\_{12}^2<\\infty \\quad (0 \\leq y'' \\leq 1)$, then:}\n\\nonumber\\\\ \n&& \\mathcal{F}(M^2\\_{12})=\\left[{\\frac{y''}{16\\pi^2}\\log\\frac{1+y''}{1-y''}-\\frac{\\arctan y'\\_{M\\_{2}}}{8\\pi^2 y'\\_{M\\_{2}}}-i\\frac{y''}{16\\pi}}\\right]^{-1}.\n \\label{Eq:F\\_amp3}\\end{aligned}$$ \nHere $ y'\\_{M\\_{2}}=\\frac{M\\_{2}}{\\sqrt{4m^2-M\\_{2}^2}} $ and\n \n$$\\label{eqy}\n\\begin{aligned}\n&y=\\frac{\\sqrt{-M\\_{12}^2}}{\\sqrt{4m^2-M\\_{12}^2}}, \n\\quad y'=\\frac{M\\_{12}}{\\sqrt{4m^2-M\\_{12}^2}},\\\\\n&y''=\\frac{\\sqrt{M\\_{12}^2-4m^2}}{M\\_{12}}.\n\\end{aligned}$$ \nWe have not yet introduced the scattering length *a* in F(*M*122) instead of the two-body bound state mass, which will allow to generalize the scattering amplitude for the case where no bound state exists. This will be discussed in detail in the next subsection. It should be also noticed that above the threshold *M*12 > 2*m*, the amplitude obtains in the denominator an imaginary part $-i\\frac{y''}{16\\pi}.$\nRenormalization via scattering length\n-------------------------------------\nThe two-body scattering amplitude - was obtained by fixing a bound state pole at *M*12 = *M*2. However, it can happen that the two-body bound state is absent, hence, the two-body scattering amplitude has no any bound state pole. Besides that, the three-body bound state may exist in absence of the two-body one. In this situation a different condition will be used: the requirement that the scattering amplitude at zero energy is equal to − *a*, where *a* is the two-body scattering length. The (non-renormalized) two-body amplitude ${\\cal F}(M^2\\_{12})$ still has the form of Eq. ([eq37a]). Its argument can be written as *M*122 = 4*ɛ**k*2. By using ([eq37a]) and ([ren]) we obtain\n \n$$\\frac{1}{16\\pi m} \\frac{1}{i[(i \\lambda)^{-1}-{\\cal B}(4m^2)]}=-a,$$ \nand therefore\n \n$$\\label{lambda\\_a}\n(i\\lambda)^{-1}={\\cal B}(4m^2)-\\frac{1}{16i\\pi m a}.$$ \nThe two-body amplitude is then given by\n \n$$\\label{Fa}\n{\\cal F}(M\\_{12}^2)=\\frac{1}{i\\Bigl[{\\cal B}(4m^2)-{\\cal B}(M^2\\_{12})\\Bigr]-\\frac{1}{16\\pi m a}}.$$ \nThe two-body scattering amplitude is obtained after substituting 2*m* for *M*2 in Eq. ([den1]), and in the different regions of *M*122 it reads:\n \n$$\\begin{aligned}\n&&\\mbox{({\\it i}) If $-\\infty < M\\_{12}^2\\leq 0$\\quad $(1\\geq y\\geq 0)$, then:}\n\\nonumber\\\\\n&&{\\cal F}(M^2\\_{12})=16\\pi\n\\left[{{\\frac{1}{\\pi y}\\log \\frac{1+y}\n{1-y} -\\frac{1}{ m a}}}\\right]^{-1}.\n\\label{eqfa1}\n\\\\\n&&\\mbox{({\\it ii}) If $0\\leq M\\_{12}^2 \\leq 4m^2 \\quad (0 \\leq y' < \\infty)$, then:} \n \\nonumber\\\\\n &&{\\cal F}(M^2\\_{12})= 16\\pi\n\\left[{{\\frac{2}{ \\pi}\\frac{\\arctan y'}{y'} -\\frac{1}{ m a}}}\\right]^{-1}.\n\\label{eqfa2}\n\\\\\n&&\\mbox{}\n\\nonumber\\\\\n&&\\mbox{({\\it iii}) If $ 4m^2\\leq M\\_{12}^2<\\infty \\quad (0 \\leq y'' \\leq 1)$, then:}\n\\nonumber\\\\\n&&{\\cal F}(M^2\\_{12})=16\\pi\n\\left[{\\frac{y''}{\\pi}\\log \\frac{1+y''}{1-y''}\n -\\frac{1}{ m a}-i y''}\\right]^{-1}.\n\\label{eqfa3}\\end{aligned}$$ \nFor negative scattering length *a* this amplitude has no poles.\nThe two-body amplitude can be written from Eq. in terms of the c.m. frame momentum as:\n \n$$\\label{Fb}\n{\\cal F}(M\\_{12}^2)=16\\pi\\left[{\\frac{k\\_v}{\\varepsilon\\_k\\pi}\\log\\frac{\\varepsilon\\_k+k\\_v}{\\varepsilon\\_k-k\\_v}-\\frac{1}{m a}-i\\frac{k\\_v}{\\varepsilon\\_k}}\\right]^{-1},$$ \nand then:\n \n$$k\\_v\\cot\\delta\\_0=\\frac{k\\_v}{\\pi}\\log\\frac{\\varepsilon\\_k+k\\_v}{\\varepsilon\\_k-k\\_v}-\\frac{\\varepsilon\\_k}{m a},$$ \nwhich is real showing the unitarity of the model amplitude. The power expansion for small *k**v* and comparison with allows to identify the effective range as\n \n$$r\\_0=\\frac{4}{m\\pi}-\\frac{1}{m^2a} \\,,$$ \nwhich is determined by the terms ∝ 1/*m*, 1/*m*2, reflecting the relativistic origin of the model.\nIn the case when the bound state exists, one finds that\n \n$$\\label{ar0}\n\\begin{aligned}\na&=\\frac{\\pi y'\\_{M\\_{2}}}{2m\\arctan(y'\\_{M\\_{2}})}, \\\\\n r\\_0&=\\frac{2\\Bigl[2y'\\_{M\\_{2}}-\\arctan(y'\\_{M\\_{2}})\\Bigr]}{\\pi m\\,y'\\_{M\\_{2}}},\n\\end{aligned}$$ \nand Eqs. - then coincide with -. Furthermore, for small binding energy *B* ≪ *m* the variable *y*ʹ*M*2 increases as $y'\\_{M\\_{2}}\\sim \\sqrt{m/B}$. The scattering length *a* also increases for *B* → 0 with\n \n$$\\label{ar1}\n a\\to\\frac{1}{\\sqrt{mB}}\\quad \\text{and} \\quad r\\_0\\to\\frac{4}{m\\pi},$$ \nwhereas the effective radius *r*0 tends to a constant.\nThree-body Bethe-Salpeter equation\n==================================\nThe solution of the zero-range three-body BS equation for three identical spinless particles using the Faddeev decomposition of the full BS amplitude can be reduced to the solution of one single integral equation for the spectator vertex function *v**M*(*q*, *p*) (external propagators are excluded). In the zero-range interaction case *v**M*(*q*, *p*) depends upon both the total momentum *p* and on the four-momentum of the spectator particle *q*. The equation reads :\n \n$$\\begin{aligned}\nv\\_M(q,p)&=2i\\mathcal{F}(M^2\\_{12})\\int \\frac{d^4 k}{(2\\pi)^4}\\frac{i}{[k^2-m^2+i\\epsilon]}\\\\\n&\\times\\frac{i}{[(p-q-k)^2-m^2+i\\epsilon]}v\\_M(k,p).\n\\label{Eq:BSE}\n\\end{aligned}$$ \nNotice that the momentum of the spectator particle, *q*, determines the effective mass of the two-boson subsystem, *M*12, due to the four-momentum conservation (see below). Therefore, the vertex function does not depend on other momenta besides *q* and the total four-momentum *p*. The other two components of the integral equation can be easily obtained through the cyclic permutation of the momentum of the constituent particles. The full BS amplitude in Minkowski space is recovered by multiplying the vertex function by the three external propagators and summing up the components, i.e.\n \n$$\\label{Eq:Phi\\_M}\n\\begin{aligned}\n&i\\Phi\\_M(k\\_1, k\\_2, k\\_3; p)=\\\\\n&i^3\\frac{v\\_M(k\\_1)+v\\_M(k\\_2)+v\\_M(k\\_3)}{(k^2\\_1-m^2+i\\epsilon)(k^2\\_2-m^2+i\\epsilon)(k^2\\_3-m^2+i\\epsilon)},\n\\end{aligned}$$ \nwhere *v**M*(*k*) ≡ *v**M*(*k*, *p*) (to simplify our notation) and the four-momenta obey the relation\n \n*k*1 + *k*2 + *k*3 = *p*. \nThe relativistic two-body zero-range scattering amplitude F(*M*122) in ([Eq:BSE]) was derived in Sec. [Sec:twobodyamp] and, being renormalized via scattering length, is given by equations, and. Its argument *M*122 is expressed via three-body momenta as *M*122 = (*p* − *q*)2. One major simplification in Eq. ([Eq:BSE]) happens due to the fact that the amplitude F(*M*122) does not depend on the loop integration variable *k* in the zero-range case. Due to that the two-body amplitude factors out in the integral equation, what does not happen for a finite-range interaction kernel like the one-boson exchange or the cross-ladder one.\nNotice that in Refs. the regime *M*122 > 4*m*2 of ${\\cal F}(M^2\\_{12})$ ([eqfa3]) was not presented, due to the range of the variables considered in those works. The amplitude in terms of the bound state mass, as presented in Refs. , is given in Eq. (i.e. in the physical domain, 0 ≤ *M*12 ≤ 2*m*).\nSuch a link is very important to understand the range covered by the results obtained previously, in Refs. , by considering only the situation where the two-body state is bound (i.e. *a* > 0) producing a real *M*2 through Eqs. ), and the full support covered by equations, and, including also virtual two-body bound states (i.e. *a* ∈ R). In other words, as mentioned, in the region for which *a* < 0 the amplitude F(*M*122) has no pole in the physical domain and, therefore, the two-body bound state does not exist. The three-body system can still be formed though, as a Borromean bound state.\nThe goal now is to solve the scalar three-body BS equation, derived in Ref. , for the lowest angular momentum bound state, with zero-range interaction, fully in Minkowski space and retaining implicitly the Fock-space composition beyond the valence truncation. The adopted method is the direct integration of the singularities of the four-dimensional integral equation, developed recently for the two-body BS equation in Ref. . The method does not rely on any ansatz, as e.g. the Nakanishi integral representation used for the solution of the two-body equation in Ref. . No three-dimensional reduction of the covariant 4D equation, as the one done by performing the projection onto the LF plane, is adopted. Part of what is exposed here was published in Ref. . Further comparisons with results obtained in the Euclidean space calculations will be provided to test the reliability of the method.\nOne interesting example of a calculation within the approach used here is the electromagnetic transition form factor , which quantifies the breakup of a two-body bound state. This highly complex calculation was performed through the direct integration method in Minkowski space, using as inputs the solutions, obtained by the same method , of the scattering and bound state BS equations. The transition form factor, including the final state interaction, was calculated in the whole kinematical region. It satisfied the non-trivial condition of current conservation explicitly verified numerically.\nEquation is a singular integral equation and solving it numerically is a very challenging task. For that reason, the equation requires a proper treatment to be rewritten in, at least, a less singular form before its numerical solution in the c.m. frame, $\\vec{p}=\\vec{0}$. The propagators, containing the strongest singularities of the BS equation kernel, are represented in the customary form \n \n$$\\label{Eq:prop\\_repr}\n\\begin{aligned}\n&\\frac{1}{k^2-m^2+i\\epsilon}=\\frac{1}{k\\_0^2-k^2\\_v-m^2+i\\epsilon}\n\\\\ &=PV\\frac{1}{k\\_0^2-\\varepsilon\\_k^2}\n-\\frac{i\\pi}{2\\varepsilon\\_k}[\\delta(k\\_0-\\varepsilon\\_k)+\\delta(k\\_0+\\varepsilon\\_k)],\n\\end{aligned}$$ \nwhere *P**V* denotes the principal value. The terms like $PV\\int\\ldots\\frac{dk\\_0}{k\\_0^2-\\varepsilon\\_k^2}$ contain the singularities at *k*0 = ± *ɛ**k* which are removed by subtracting integrals from the equation, with appropriate coefficients, in such a way that the final equation is not affected. For that, the following identities are used\n \n$$PV \\int\\_{-\\infty}^0 \\frac{dk\\_0}{k^2\\_0-\\varepsilon\\_k^2}=PV \\int\\_{0}^{\\infty} \\frac{dk\\_0}{k^2\\_0-\\varepsilon\\_k^2}=0.$$ \nThe second propagator in Eq. can be integrated over the angles analytically. Denoting $z=\\cos\\left(\\frac{\\vec{k}\\cdot\\vec{q}}{k\\_v q\\_v}\\right)$ and recalling that *d*3*k* = *k**v*2*d**k**v* *d**z* *d**φ*, one can write that\n \n$$\\begin{aligned}\n\\label{Eq:PI}\n&&\\Pi(q\\_0,q\\_v, k\\_0, k\\_v) =\\int\\frac{idzd\\varphi}{[(p-q-k)^2-m^2+i\\epsilon]} \\nonumber \\\\\n&&=\\frac{i \\pi}{q\\_v k\\_v} \\left\\{ \\log \\left| \\frac{(\\eta+1)}{(\\eta-1)} \\right| - i \\pi I(\\eta) \\right\\},\\end{aligned}$$ \nwith\n \n$$\\label{Eq:eta}\nI(\\eta)=\\left\\{\n\\begin{array}{lcrcl}\n1 & {\\rm if} & \\mid\\eta\\mid &\\leq& 1 \\cr\n0 & {\\rm if} & \\mid\\eta\\mid &> & 1\n\\end{array}\\right.,$$ \nand\n \n$$\\eta = \\frac{(M\\_3 - q\\_0 - k\\_0)^2 - k\\_v^2 - q\\_v^2 - m^2}{2q\\_v k\\_v},$$ \nwhere *M*3 denotes the bound state mass of the three-body system. The BS equation turns, after integration over the angles, into an integral equation with the kernel, that is still singular. However, these singularities are weakened by integration and become logarithmic ones and discontinuities, that will be treated numerically.\nOnce the propagators are expressed as in Eq. ([Eq:proprepr]), the principal value singularities are subtracted and the angular integrations are performed, Eq. acquires the following form:\n \n$$\\begin{aligned}\nv\\_M(q\\_0, q\\_v) \n&=& \\frac{\\mathcal{F}(M^2\\_{12})}{(2\\pi)^4} \\int\\_0^{\\infty} k^2\\_v dk\\_v\\left\\lbrace \\frac{2\\pi i}{2 \\varepsilon\\_k} \\left[\\Pi(q\\_0, q\\_v; \\varepsilon\\_k, k\\_v) v\\_M(\\varepsilon\\_k, k\\_v) + \\Pi(q\\_0, q\\_v; -\\varepsilon\\_k, k\\_v) v\\_M(-\\varepsilon\\_k, k\\_v)\\right]\\right. \n\\cr\n &-&2 \\int^0\\_{-\\infty} dk\\_0 \\left[ \\frac{ \\Pi(q\\_0, q\\_v; k\\_0, k\\_v) v\\_M(k\\_0, k\\_v) - \\Pi(q\\_0, q\\_v; -\\varepsilon\\_k, k\\_v) v\\_M(-\\varepsilon\\_k, k\\_v)}{ {k}\\_0^2-\\varepsilon\\_k^2 }\\right] \\cr\n&-& \\left. 2 \\int\\_0^{\\infty} dk\\_0 \\left[ \\frac{ \\Pi(q\\_0, q\\_v; k\\_0, k\\_v) v\\_M(k\\_0, k\\_v) - \\Pi(q\\_0, q\\_v; \\varepsilon\\_k, k\\_v) v\\_M(\\varepsilon\\_k, k\\_v)}{ {k}\\_0^2-\\varepsilon\\_k^2 }\\right]\\right\\rbrace.\n \\label{Eq:v}\\end{aligned}$$ \nThis equation has now, besides the unknown analytical behavior of *v**M*(*q*0, *q**v*) that will be discovered numerically, only weak singularities and discontinuities, but unlike ([Eq:BSE]) the singularities in *k*0 = ± *ɛ**k* no longer exist.\nThe logarithmic singularities of the kernel, Π(*q*0, *q**v*, *k*0, *k**v*), at *η* = ± 1 can be found for fixed values of *q*0, *q**v* and *k**v*, what makes the numerical treatment in *k*0 easier. Their positions with respect to the variable *k*0 are\n \n$$\\begin{aligned}\nk\\_0&=&(M\\_3-q\\_0)+\\sqrt{m^2+(k\\_v\\pm q\\_v)^2},\n\\nonumber\\\\\nk\\_0&=&(M\\_3-q\\_0)-\\sqrt{m^2+(k\\_v\\pm q\\_v)^2}.\n\\label{Eq:sing\\_k0}\\end{aligned}$$ \nAnalogously, the position of the singularities can be found for the variable *k**v*, so that the integration over this variable can be optimized numerically. The positions of the singularities of Π(*q*0, *q**v*, ± *ɛ**k*, *k**v*) as a function of *k**v* are given by\n \n$$k\\_v=\\frac{\\pm \\sqrt{M\\_{12}^2(M\\_{12}^2+q\\_v^2)(M\\_{12}^2-4m^2)}\\pm q\\_v M\\_{12}^2}{2M\\_{12}^2},\n\\label{Eq:sing\\_k}$$ \nwhere *M*122 = (*M*3 − *q*0)2 − *q**v*2. The expression under the square root is non-negative if\n \n*M*122 ≥ 4*m*2 or *M*122 ≤ 0, \nand, therefore, for existing real singularities in *k**v* one needs to ensure one of the following conditions for *q*0: $q\\_0<M\\_3-\\sqrt{q^2\\_v+4m^2}$ or *M*3 − *q**v* < *q*0 < *M*3 + *q**v* or $q\\_0>M\\_3+\\sqrt{q^2\\_v+4m^2}$. This means that the branching points that need to be considered while fixing the mesh numerically to separate the regions with and without singularities in *k**v* are\n \n$$\\label{Eq:peaks}\n\\begin{aligned}\nq^{(1)}\\_0=&M\\_3-\\sqrt{q^2\\_v+4m^2}, \\\\\nq^{(2)}\\_0=&M\\_3-q\\_v, \\\\\nq^{(3)}\\_0=&M\\_3+q\\_v, \\\\\nq^{(4)}\\_0=&M\\_3+\\sqrt{q^2\\_v+4m^2},\n\\end{aligned}$$ \nwith *q*0(1) < *q*0(2) < *q*0(3) < *q*0(4). As it can be seen from Eqs. -, these branching points are also present in the two-body amplitude F(*M*122). For more details on the behavior of the F(*M*122) amplitude, see Sec. [Sec:Fnum].\nRelation between the BS amplitude and LF wave function\n======================================================\nIn this section and in Appendix [App:BSALFWF], we establish the relation between the three-body BS amplitude and the three-body LF wave function (LFWF). It generalizes to the three-body system the relation (3.57) or (3.58) from Ref. for the two-scalar system and can be easily generalized to the arbitrary *n*-body case.\nThe three-body BS amplitude is defined analogously to the two-body one, namely:\n \n$$\\label{Phi0\\_1}\n\\Phi\\_M(x\\_1,x\\_2,x\\_3;p)=\\langle 0 \\left| T\\Bigl(\\varphi (x\\_1)\\varphi (x\\_2)\\varphi (x\\_3)\\Bigr)\\right| p\\rangle\\,.$$ \nAs is shown in Appendix [App:BSALFWF], the three-body LFWF can be related to the BS amplitude through\n \n$$\\label{bs7b\\_1}\n\\begin{aligned}\n\\psi (&\\vec{k}\\_{1\\perp},\\xi\\_1;\\vec{k}\\_{2\\perp},\\xi\\_2;\\vec{k}\\_{3\\perp},\\xi\\_3) = \\frac{(p^+)^2 }{\\sqrt{2\\pi}}\\, \\xi\\_1 \\xi\\_2 \\xi\\_3\n\\\\\n&\\times \\int dk^-\\_{1}\\, dk^-\\_{2}\n \\Phi\\_M(k\\_1,k\\_2,k\\_3;p),\n\\end{aligned}$$ \nwhere *ξ*1 + *ξ*2 + *ξ*3 = 1 and the plus and minus momentum components are given by *p*± = *p*0 ± *p*3, with analogous definitions for the other momenta.\nSimilarly, one has for the two-body case that\n \n$$\\label{bs8\\_1}\n\\begin{aligned}\n\\psi (\\vec{k}\\_{1\\perp},&\\xi\\_1;\\vec{k}\\_{2\\perp},\\xi\\_2)=\\\\\n &=\\frac{p^+}{\\sqrt{2\\pi}} \\, \\xi\\_1 \\xi\\_2 \\int dk^-\\_{1} \\Phi\\_M(k\\_1,k\\_2;p).\n\\end{aligned}$$ \nBy introducing the relative variable $k=\\frac{1}{2}(k\\_1-k\\_2)$, Φ*M* ≡ Φ*M*(*k*, *p*), Eq. ([bs81]) can be written on the form:\n \n$$\\label{bs9\\_1}\n\\psi (\\vec{k}\\_{\\perp},\\xi)=\\frac{p^+}{\\sqrt{2\\pi}} \\xi(1- \\xi)\\int dk^{-} \\Phi\\_M(\\vec{k}\\_{\\perp},k^+\\hspace{-.1cm},k^-;p).$$ \nIt differs from Eq. (3.57) of Ref. by the degrees of *π* due to the presence of the factor (2*π*)− 3/2 in Eq. (3.53) of Ref. which we did not introduce above.\nNoteworthy that the support of the function *ψ*(*k⃗*⊥, *ξ*) in the variable *ξ*, or of the integral \n∫*d**k*−Φ*M*(*k⃗*⊥, *k*+, *k*−; *p*), \n is 0 < *ξ* < 1, as it should be. This follows from the fact that Φ*M* is not an arbitrary function, but it is defined, in the coordinate space, by \n$$\\Phi\\_M(x\\_1,x\\_2;p)=\n\\langle 0 \\left| T\\Bigl(\\varphi (x\\_1)\\varphi (x\\_2)\\Bigr)\\right| p\\rangle\\,,$$ \n analogously to the three-boson BS amplitude defined in Eq. ([Phi01]). This support is also automatically obtained if one represents the BS amplitude in the Nakanishi form (see Ref. , Appendix D). The same conclusion is also valid for the three-body case, when using Eq. ([bs7b1]).\nNon-relativistic limit\n======================\nIn this section, the non-relativistic limits of the three-body Euclidean BS and valence LF equations, i.e. Eqs. (7) and (10) of Ref. , are considered. The Euclidean BS equation will be first analysed. Representing the three-body mass *M*3 as *M*3 = 3*m* − *B*3, with *B*3 denoting the three-body binding energy, and truncating the denominator in Eq. (7) of Ref. , and the terms in the fraction of the argument of the log in Eq. (8) of the aforementioned reference, to the leading terms of momenta and the binding energies, one gets\n \n$$\\label{K}\n\\begin{aligned}\nK&= \\frac {\\Pi\\_E(q\\_4,q\\_v,k\\_4,k\\_v)}{\\left(k\\_4-\\frac{i}{3}M\\_3\\right)^2+k\\_v^2+m^2}=\\\\\n&\\frac{\\frac{1}{2}\\log\\frac{\\left(k\\_4+q\\_4+\\frac{i}{3}M\\_3\\right)^2+(q\\_v+k\\_v)^2+m^2}{\\left(k\\_4+q\\_4+\\frac{i}{3}M\\_3\\right)^2+(q\\_v-k\\_v)^2+m^2}}\n{\\left(k\\_4-\\frac{i}{3}M\\_3\\right)^2+k\\_v^2+m^2}\n\\\\\n\\Rightarrow\\;K\\_{nr}& \\approx\\frac{\\frac{1}{2}\\log\\frac{\\frac{2}{3}B\\_3+\\frac{(k\\_v+q\\_v)^2}{2m}+i(k\\_4+q\\_4)}\n {\\frac{2}{3}B\\_3+\\frac{(k\\_v-q\\_v)^2}{2m}+i(k\\_4+q\\_4)}}{2m\\left(\\frac{1}{3}B\\_3-ik\\_4\\right)}.\n \\end{aligned}$$ \nAt the first glance, one could neglect the terms $\\frac{(k\\_v\\pm q\\_v)^2}{2m}$ in comparison to (*k*4 + *q*4), however, this would result in *K**n**r* ≡ 0, and therefore it is necessary to keep them.\nFollowing Ref. , one can write *E*2 through *M*122 = (2*m* − *E*2)2, the two-body bound state mass as *M*2 = 2*m* − *B*2 and introduce them in the two-body amplitude ${\\cal F}(M\\_{12}^2)$ in the physical domain (0 ≤ *M*122 ≤ 4*m*2) (see Eq. ([Eq:Famp2])) to elaborate the non-relativistic limit. In such case, *m* → ∞, and the ${\\cal F} (M\\_{12}^2)$ amplitude becomes\n \n$$\\label{Fnr}\n{\\cal F}(M^2\\_{12})=\\frac{16\\pi\\sqrt{m}}{\\sqrt{E\\_2}-\\sqrt{B\\_2}},$$ \nor, alternatively,\n \n$$\\label{Fnr1}\n{\\cal F}(-M^2\\_{12})=\\frac{16\\pi\\sqrt{m}}{\\sqrt{2m-\\sqrt{-M^2\\_{12}}}-\\sqrt{B\\_2}}.$$ \nSince ${M}\\_{12}^2= \\left(\\frac{2}{3}p-iq\\_4\\right)^2-q^2\\_v = -(\\frac{2}{3}iM\\_3+q\\_4)^2-q^2\\_v$ in the limit *m* → ∞ one gets\n \n$$\\begin{aligned}\nE\\_2&=2m-\\sqrt{-\\left[\\frac{2}{3}i(3m-B\\_3)+q\\_4\\right]^2-q^2\\_v}\\\\\n&\\approx \\frac{2}{3}B\\_3+iq\\_4+\\frac{q\\_v^2}{4m}.\n\\end{aligned}$$ \nSubstituting it in ([Fnr]), one finds for the scattering amplitude\n \n$$\\label{Fnr2}\n{\\cal F}(M^2\\_{12})=\\frac{16\\pi\\sqrt{m}}{\\sqrt{ \\frac{2}{3}B\\_3+iq\\_4+\\frac{q\\_v^2}{4m}}-\\sqrt{B\\_2}}.$$ \nAfter these manipulations, Eq. (7) of Ref. obtains the form\n \n$$\\label{eqnr}\n\\begin{aligned}\n\\tilde{v}'\\_E(&q\\_4,q\\_v)= \\frac{1}{\\pi^2\\sqrt{m}}\\frac{1}{\\sqrt{\\frac{2}{3}B\\_3+iq\\_4+\\frac{q\\_v^2}{4m}}-\\sqrt{B\\_2}} \\\\\n&\\times\\int\\_0^{\\Lambda} dk\\_v \\int\\_{-\\infty}^{\\infty}\\frac{dk\\_4}{\\left(\\frac{1}{3}B\\_3-ik\\_4\\right)}\\\\\n&\\times\\log\\frac{\\frac{2}{3}B\\_3+\\frac{(k\\_v+q\\_v)^2}{2m}+i(k\\_4+q\\_4)}\n {\\frac{2}{3}B\\_3+\\frac{(k\\_v-q\\_v)^2}{2m}+i(k\\_4+q\\_4)}\\;\\tilde{v}'\\_E(k\\_4,k\\_v),\n\\end{aligned}$$ \nwhere it was introduced a cutoff Λ to prevent the Thomas collapse . In order to obtain the time independent equation, the integration over *k*4 needs to be performed. Since this is a lengthy derivation, it will not be done here explicitly.\nFor the three-body LF equation given by Eq. (10) of Ref. , the non-relativistic limit, obtained by following the same steps as before, reads\n \n$$\\label{LFnr}\n\\begin{aligned}\n\\Gamma\\_{nr}(&\\vec{q})=\\frac{1}{\\pi^2m^{3/2}}\\frac{1}{\\sqrt{E\\_2}-\\sqrt{B\\_2}}\\\\\n&\\times\\int\\frac{\\Gamma\\_{nr}(\\vec{k})d^3k}{{B\\_3+\\frac{q\\_v^2}{2m}+\\frac{k\\_v^2}{2m}+\n\\frac{(\\vec{q}+\\vec{k})^2}{2m}}},\n\\end{aligned}$$ \nwhere\n \n$$\\begin{aligned}\nE\\_2&=2m-M\\_{12}\\approx B\\_3+\\frac{3}{4}\\frac{q\\_v^2}{m}. \n\\end{aligned}$$ \nHere the factor $\\frac{1}{\\sqrt{E\\_2}-\\sqrt{B\\_2}}$ is originating from the two-body amplitude ([Fnr]) when *m* → ∞.\nEq. ([LFnr]) is the same as Eq. (18) of Ref. . This equation is known as the Skornyakov-Ter-Martirosyan equation . The non-relativistic equation can be also written in the form\n \n$$\\label{nr18}\n\\begin{aligned}\n\\Gamma\\_{nr}(&\\vec{q})=\\frac{1}{\\pi^2\\sqrt{m}}\n\\frac{1}{\\sqrt{B\\_3+\\frac{3}{4}\\frac{q\\_v^2}{m}}-\\sqrt{B\\_{2}}}\\\\\n&\\times\\int \\frac{\\Gamma\\_{nr}(\\vec{k})d^3k}\n{k\\_v^2+\\vec{k}{\\makebox[0.08cm]{$\\cdot$}}\\vec{q}+q\\_v^2+mB\\_3},\n\\end{aligned}$$ \nand, for the s-wave, after integrating over the angles, it reads\n \n$$\\label{nr20}\n\\begin{aligned}\n\\Gamma&\\_{nr}(q\\_v)=\\frac{2}{\\pi\\sqrt{m}}\n\\frac{1}{\\sqrt{B\\_3+\\frac{3}{4}\\frac{q\\_v^2}{m}}-\\sqrt{B\\_2}}\\\\\n&\\times\\int\\_0^{\\Lambda} \\log\\left(\\frac{k\\_v^2+k\\_vq\\_v+q\\_v^2+mB\\_3}{k\\_v^2-k\\_vq\\_v+q\\_v^2+mB\\_3}\\right)\n\\Gamma\\_{nr}(k\\_v)\\frac{k\\_vdk\\_v}{q\\_v}.\n\\end{aligned}$$ \nThe above equation, like ([eqnr]), and in contrast to, requires a cutoff in order to allow a physical solution avoiding the Thomas collapse .\nTransverse amplitudes\n=====================\nThe vertex function *v*(*q*0, *q**v*) is fundamentally dependent on the metric (Euclidean or Minkowski one) adopted to define the integral equation. The transverse amplitude is, instead, an useful quantity for comparison between calculations performed in Euclidean and Minkowski spaces. Furthermore, it gives information on the valence wave function integrated in the longitudinal momenta in the present model, where an infinite number of Fock-components are taken into account implicitly by the Bethe-Salpeter framework. The rich structure of the three-boson bound state transverse amplitude is investigated numerically in Sec. [Sec:ResTA], as it is a mean to exploit the double parton momentum dependence of the valence wave function, and also gives access to the dynamical correlation between the constituents.\nThe derivation of the expressions for the Minkowski transverse amplitude is presented below in Sec. [Ssec:transvmink]. The final amplitude, computed with the BS amplitude obtained from the solution of the BS equation in Minkowski space ([Eq:v]), is expected to coincide with the one defined in Euclidean space. The expressions for the latter one will be derived in Sec. [Ssec:transveucl].\nMinkowski space\n---------------\nAs mentioned, the BS amplitude Φ*M* can be written in terms of the three vertex components by introducing the external propagators. It is given above by Eq. ([Eq:PhiM]).\nThe transverse amplitude can be defined via Φ*M* as\n \n$$\\label{Eq:L\\_Mink}\n\\begin{aligned}\nL(&\\vec{k}\\_{1\\perp}, \\vec{k}\\_{2\\perp})=\n\\\\&L\\_1(\\vec{k}\\_{1{\\perp}}, \\vec{k}\\_{2{\\perp}})+L\\_2(\\vec{k}\\_{1{\\perp}}, \\vec{k}\\_{2{\\perp}})+L\\_3(\\vec{k}\\_{1{\\perp}}, \\vec{k}\\_{2{\\perp}})=\\\\\n& \\int\\_{-\\infty}^{\\infty}dk\\_{10} \\int\\_{-\\infty}^{\\infty} dk\\_{1z} \\int\\_{-\\infty}^{\\infty} dk\\_{20} \\int\\_{-\\infty}^{\\infty} dk\\_{2z}\\\\\n&\\times i\\Phi\\_M(k\\_{10}, k\\_{1z}, k\\_{20}, k\\_{2z}; \\vec{k}\\_{1\\perp}, \\vec{k}\\_{2\\perp}).\n\\end{aligned}$$ \nHowever, in the three identical boson case, only one of its Faddeev components *L**i* is enough for the comparison with the transverse amplitude derived from the Euclidean BS solution. The first Faddeev component is given by\n \n$$\\begin{aligned}\nL\\_1(\\vec{k}\\_{1\\perp}, \\vec{k}\\_{2\\perp})&=i\\int\\_{-\\infty}^{\\infty}dk\\_{10}\\int\\_{-\\infty}^{\\infty}dk\\_{1z}\\;\\frac{v\\_M(k\\_{10}, k\\_{1v})}{k\\_1^2-m\\_1^2+i\\epsilon}\\\\\n&\\times\\chi(k\\_{10}, k\\_{1z}; \\vec{k}\\_{1\\perp}, \\vec{k}\\_{2\\perp}),\n\\end{aligned}$$ \nwith\n \n$$\\label{Eq:chi}\n\\begin{aligned}\n\\chi(& k\\_{10}, k\\_{1z};\\vec{k}\\_{1\\perp}, \\vec{k}\\_{2\\perp})=\\\\\n&i^2\\int\\frac{d^2k\\_2}{(k^2\\_2-m\\_2^2+i\\epsilon)[(p'-k\\_2)^2-m\\_3^2+i\\epsilon]},\n\\end{aligned}$$ \nwhere the following quantities enter: *k**i* = (*k**i*0, *k**i**z*) and *d*2*k**i* = *d**k**i*0*d**k**i**z* with *i* = 1, 2. Moreover,\n \n*m*22 = *m*2 + ∣*k⃗*2 ⊥∣2, *m*32 = *m*2 + (*p⃗*⊥ − *k⃗*1 ⊥ − *k⃗*2 ⊥)2, \nand *p*ʹ = (*p*ʹ0, *p*ʹ*z*) = *p* − *k*1 = (*p*0 − *k*10, *p**z* − *k*1*z*).\nThe two-dimensional integral in can be \nperformed by first introducing the Feynman \nparametrization and then making the transformation *k*2 → *k*2 + (1 − *u*)*p*ʹ. After integration over *k*2, using a Wick rotation as *k*0 = *i**k*4, we find\n \n$$\\label{Eq:chi\\_3}\n\\begin{aligned}\n\\chi(k\\_{10}, k\\_{1z}; &\\vec{k}\\_{1\\perp}, \\vec{k}\\_{2\\perp}) \n=i^2\\int\\_{0}^1 du\\int\\frac{d^2k\\_2}{(k^2\\_2+D+i\\epsilon)^2} \\\\\n&= -\\pi i^3\\int\\_{0}^{1}\\frac{du}{D+i\\epsilon},\n\\end{aligned}$$ \nwith\n \n*D* = *u*(1 − *u*)*p*ʹ2 − *m*22*u* − (1 − *u*)*m*32. \nThe denominator *D* is zero at\n \n$$\\begin{aligned}\n\\label{upum}\nu&\\_{\\mp}=\\frac{1}{2p'^2}\\left[p'^2-m\\_2^2+m^2\\_3\\right.\\\\\n&\\mp\\left.\\sqrt{((m\\_2-m\\_3)^2-p'^2)((m\\_2+m\\_3)^2-p'^2)}\\right],\n\\end{aligned}$$ \nbut for *p*ʹ2 < (*m*2 + *m*3)2, the equality *D* = 0 is never satisfied in the interval 0 < *u* < 1, so the term *i**ε* can be dropped out in Eq. and the integral over the Feynman parameter *u* can be performed safely analytically, giving the following\n \n$$\\begin{aligned}\n\\chi(k\\_{10}&, k\\_{1z}; \\vec{k}\\_{1\\perp}, \\vec{k}\\_{2\\perp}) = \\frac{\\pi i^3}{p'^2(u\\_- -u\\_+)}\\\\\n&\\times\\int\\_{0}^{1}du\\biggl[\\frac{1}{u-u\\_-}-\\frac{1}{1-u\\_+} \\biggr]=\\\\\n& -\\frac{i\\pi}{p'^2(u\\_{-}-u\\_+)}[\\log(1-u\\_-)\\\\\n&-\\log(-u\\_-)-\\log(-1+u\\_+)+\\log(u\\_+)],\n\\end{aligned}$$ \nwith *u*± defined in.\nIn the situation where *p*ʹ2 > (*m*2 + *m*3)2, the zeroes of the denominator, *u*±, are placed on the real axis for the interval *u* ∈ [0, 1]. For that reason, one can separate *χ* in two terms, analogously to what was done in, i.e.\n \n$$\\begin{aligned}\n&\\chi(k\\_{10}, k\\_{1z}; \\vec{k}\\_{1\\perp}, \\vec{k}\\_{2\\perp}) = \\\\\n&\\chi'(k\\_{10}, k\\_{1z}; \\vec{k}\\_{1\\perp}, \\vec{k}\\_{2\\perp}) + \\chi''(k\\_{10}, k\\_{1z}; \\vec{k}\\_{1\\perp}, \\vec{k}\\_{2\\perp}),\n\\end{aligned}$$ \nwhere\n \n$$\\begin{aligned}\n\\label{Eq:chip}\n&&\\chi'(k\\_{10}, k\\_{1z}; \\vec{k}\\_{1\\perp}, \\vec{k}\\_{2\\perp}) = \\frac{\\pi i^3}{p'^2(u\\_- -u\\_+)}\n\\nonumber \\\\\n&&\\times\\biggl[\\text{PV}\\int\\_{0}^{1}\\frac{du}{u-u\\_-}- \\text{PV}\\int\\_{0}^{1}\\frac{du}{u-u\\_+} \\biggr],\\end{aligned}$$ \nand\n \n$$\\begin{aligned}\n\\chi''(&k\\_{10}, k\\_{1z};\\vec{k}\\_{1\\perp}, \\vec{k}\\_{2\\perp}) = \\frac{\\pi i^3}{p'^2(u\\_- -u\\_+)}\\\\\n& \\times\\biggl[-i\\pi\\int\\_{0}^{1}du \\delta(u-u\\_-)-i\\pi\\int\\_{0}^{1}du \\delta(u-u\\_+) \\biggr]\\\\\n&=\\frac{2\\pi^2}{\\sqrt{[p'^2-(m\\_2-m\\_3)^2][p'^2-(m\\_2+m\\_3)^2]}}.\n\\end{aligned}$$ \nThe principal value integrals in Eq. can be carried out analytically and one obtains for *χ*ʹ the following expression\n \n$$\\begin{aligned}\n&\\chi'(k\\_{10},k\\_{1z};\\vec{k}\\_{1\\perp},\\vec{k}\\_{2\\perp})=\\\\\n&i\\pi\\frac{\\log\\frac{m^2\\_2+m^2\\_3-p'^2-\\sqrt{[p'^2-(m\\_2-m\\_3)^2][p'^2-(m\\_2+m\\_3)^2]}}{m^2\\_2+m^2\\_3-p'^2+\\sqrt{[p'^2-(m\\_2-m\\_3)^2][p'^2-(m\\_2+m\\_3)^2]}}}{\\sqrt{[p'^2-(m\\_2-m\\_3)^2][p'^2-(m\\_2+m\\_3)^2]}}.\n\\end{aligned}$$ \nThe contribution *L*1(*k⃗*1 ⊥, *k⃗*2 ⊥) can subsequently be written in the form\n \n$$\\begin{gathered}\nL\\_1(\\vec{k}\\_{1{\\perp}}, \\vec{k}\\_{2{\\perp}})=\\\\ \n-i\\int\\_{-\\infty}^{\\infty}dk\\_{1z}\\Bigg\\{ \\frac{i\\pi}{2\\tilde{k}\\_{10}}\\left[\\chi(\\tilde{k}\\_{10}, k\\_{1z}; \\vec{k}\\_{1\\perp},\\vec{k}\\_{2\\perp})v\\_M(\\tilde{k}\\_{10},k\\_{1v})\\right.\\\\ \\left.\n+\\chi(-\\tilde{k}\\_{10}, k\\_{1z}; \\vec{k}\\_{1\\perp},\\vec{k}\\_{2\\perp})v\\_M(-\\tilde{k}\\_{10},k\\_{1v})\\right]\\\\\n-\\int\\_0^{\\infty}dk\\_{10}\\left[\\frac{\\chi(-k\\_{10},k\\_{1z};\\vec{k}\\_{1\\perp},\\vec{k}\\_{2\\perp})v\\_M(-k\\_{10},k\\_{1v})}{k^2\\_{10}-\\tilde{k}^2\\_{10}}\\right.\\\\\n\\left.-\\frac{\\chi(-\\tilde{k}\\_{10},k\\_{1z};\\vec{k}\\_{1\\perp},\\vec{k}\\_{2\\perp})v\\_M(-\\tilde{k}\\_{10},k\\_{1v})}{k^2\\_{10}-\\tilde{k}^2\\_{10}}\\right]\\\\\n -\\int\\_0^{\\infty}dk\\_{10}\\left[\\frac{\\chi(k\\_{10},k\\_{1z};\\vec{k}\\_{1\\perp},\\vec{k}\\_{2\\perp})v\\_M(k\\_{10},k\\_{1v})}{k^2\\_{10}-\\tilde{k}^2\\_{10}}\n \\right. \\\\ \\left.\n -\\frac{\\chi(\\tilde{k}\\_{10},k\\_{1z};\\vec{k}\\_{1\\perp},\\vec{k}\\_{2\\perp})v\\_M(\\tilde{k}\\_{10},k\\_{1v})}{k^2\\_{10}-\\tilde{k}^2\\_{10}}\\right]\\Bigg\\}, \n\\label{Eq:transverse\\_final}\\end{gathered}$$ \nwhere\n \n$$\\tilde{k}\\_{10}=\\sqrt{k^2\\_{1z}+|\\vec{k}\\_{1\\perp}|^2+m^2}.$$ \nSimilarly to the treatment of the BS equation in Sec. [Sec:BSE], propagators like [*k*12 − *m*12 + *i**ε*]− 1 were expressed in the form and subtractions were made to eliminate the principal value singularities at *k*0 = ± *k̃*10.\nIt should be noticed that the function *χ* in Eq. has square-root singularities at *p*ʹ2 = (*m*2 ± *m*3)2. The functions *χ*( ± *k̃*10, *k*1*z*; *k⃗*1 ⊥, *k⃗*2 ⊥) are thus singular at\n \n$$\\begin{gathered}\n\\label{kroots}\nk\\_{1z}=\\pm\\frac{2}{M\\_3}\\sqrt{(M\\_3+m\\_1)^2-(m\\_2+m\\_3)^2} \\\\ \n\\times \\sqrt{(M\\_3-m\\_1)^2-(m\\_2+m\\_3)^2}.\\end{gathered}$$ \nFurthermore, for fixed *k*1*z*, the positions of the singular points of the functions *χ*( − *k*10, *k*1*z*; *k⃗*1 ⊥, *k⃗*2 ⊥) and *χ*(*k*10, *k*1*z*; *k⃗*1 ⊥, *k⃗*2 ⊥) are given by\n \n$$k\\_{10}=-M\\_3+\\sqrt{k^2\\_{1z}+(m\\_2+m\\_3)^2},$$ \nand\n \n$$k\\_{10}=M\\_3\\pm \\sqrt{k\\_{1z}^2+(m\\_2+m\\_3)^2},$$ \nrespectively. In this case only the singular points located on the positive *k*0 axis need to be considered (see Eq. ). In fact, it turns out that the integrands in Eq. are symmetric with respect to *k*1*z* → − *k*1*z*. Therefore, one needs to consider only the region where *k*1*z* > 0 and multiply the equation by a factor 2. Furthermore, only the positive solutions of Eq. are needed.\nEuclidean space\n---------------\nThe expressions for the transverse amplitudes in the Euclidean space, presented in Ref. , will be derived in detail in this section and in Appendix [App:transveuclkernel]. As mentioned in that paper, the following change of variables in the original equation ([Eq:BSE]) was performed,\n \n$$\\label{shift}\nk\\_i = k'\\_i+\\frac{p}{3},\\quad (i=1, 2, 3),$$ \nin order to allow the Wick rotation without crossing any singularities. The primed momenta satisfy the relation\n \n*k*ʹ1 + *k*ʹ2 + *k*ʹ3 = 0. \nThe BS amplitude in Minkowski space can be written as\n \n$$\\label{Eq:phi\\_M}\n\\begin{aligned}\n&i\\tilde{\\Phi}\\_M(k'\\_1,k'\\_2,k'\\_3;M\\_3)=i^3\\frac{\\tilde{v}\\_M(k'\\_1)+\\tilde{v}\\_M(k'\\_2)+\\tilde{v}\\_M(k'\\_3)}{(k'\\_1+\\frac{p}{3})^2-m^2+i\\epsilon}\\\\\n&\\times\\frac{1}{[(k'\\_2+\\frac{p}{3})^2-m^2+i\\epsilon][(k'\\_3+\\frac{p}{3})^2-m^2+i\\epsilon]}=\\\\\n&i^3\\frac{\\tilde{v}\\_M(k'\\_1)+\\tilde{v}\\_M(k'\\_2)+\\tilde{v}\\_M(-k'\\_1-k'\\_2)}{[(k'\\_1+\\frac{p}{3})^2-m^2+i\\epsilon][(k'\\_2+\\frac{p}{3})^2-m^2+i\\epsilon]}\\\\\n&\\times\\frac{1}{(k'\\_1+k'\\_2-\\frac{p}{3})^2-m^2+i\\epsilon},\n\\end{aligned}$$ \nwhere\n \n$$\\tilde{\\Phi}\\_M(k'\\_1,k'\\_2,k'\\_3;p)=\\Phi\\_M\\bigl(k'\\_1+\\frac{p}{3},k'\\_2+\\frac{p}{3},k'\\_3+\\frac{p}{3};p\\bigr),$$ \nand\n \n$$\\label{Eq:vtilde}\n\\tilde{v}\\_M(k'\\_i) = v\\_M\\bigl(k'\\_i+\\frac{p}{3}\\bigr).$$ \nNow, in new (shifted) integration variable one can perform the Wick rotation in the equation ([Eq:BSE]) and transform this equation into the Euclidean space. For the full Euclidean BS amplitude one gets:\n \n$$\\label{Eq:phi\\_E}\n\\begin{aligned}\ni\\tilde{\\Phi}\\_E(&k'\\_{14}, k'\\_{1z}, \\vec{k}'\\_{1\\perp};k'\\_{24}, k'\\_{2z}, \\vec{k}'\\_{2\\perp})=\\\\\n-&i^3\\frac{\\tilde{v}\\_E(k'\\_{14},k'\\_{1v})+\\tilde{v}\\_E(k'\\_{24},k'\\_{2v})+\\tilde{v}\\_E(k'\\_{34},k'\\_{3v})}{(k'\\_{14}-i\\frac{M\\_3}{3})^2+k'^2\\_{1z}+m^2\\_1}\\\\\n&\\times\\frac{1}{(k'\\_{24}-i\\frac{M\\_3}{3})^2+k'^2\\_{2z}+m^2\\_2}\\\\\n&\\times \\frac{1}{(k'\\_{14}+k'\\_{24}+i\\frac{M\\_3}{3})^2+(k'\\_{1z}+k'\\_{2z})^2+m^2\\_3},\n\\end{aligned}$$ \nwhere\n \n$$\\label{m1m2m3}\n\\begin{aligned}\nk'\\_{iv} &= \\sqrt{|\\vec{k}'\\_{i\\perp}|^2+k'^2\\_{iz}}, \\\\\nm^2\\_i &= |\\vec{k}'\\_{i\\perp}|^2 + m^2, \\qquad (i=1,2,3) \\\\\n\\vec{k}'\\_{3\\perp} &= -(\\vec{k}'\\_{1\\perp}+\\vec{k}'\\_{2\\perp}).\n\\end{aligned}$$ \nThe full Euclidean transverse amplitude, corresponding to the Minkowski one given by, reads\n \n$$\\label{Eq:L\\_E}\n\\begin{aligned}\nL(&\\vec{k}'\\_{1\\perp}, \\vec{k}'\\_{2\\perp})=\n\\\\&L\\_1(\\vec{k}'\\_{1{\\perp}}, \\vec{k}'\\_{2{\\perp}})+L\\_2(\\vec{k}'\\_{1{\\perp}}, \\vec{k}'\\_{2{\\perp}})+L\\_3(\\vec{k}'\\_{1{\\perp}}, \\vec{k}'\\_{2{\\perp}})=\\\\\n& -\\int\\_{-\\infty}^{\\infty}dk'\\_{14} \\int\\_{-\\infty}^{\\infty} dk'\\_{1z} \\int\\_{-\\infty}^{\\infty} dk'\\_{24} \\int\\_{-\\infty}^{\\infty} dk'\\_{2z}\\\\\n&\\times i\\tilde{\\Phi}\\_E(k'\\_{14},k'\\_{1z},k'\\_{24},k'\\_{2z}; \\vec{k}'\\_{1\\perp}, \\vec{k}'\\_{2\\perp}).\n\\end{aligned}$$ \nBy insertion of Eq. in, it is found that one of the contributions to the transverse amplitude is given by\n \n$$\\label{Eq:L1\\_E\\_final}\n\\begin{aligned}\nL\\_1(\\vec{k}'\\_{1\\perp}, &\\vec{k}'\\_{2\\perp})= -\\int\\_{-\\infty}^{\\infty} dk'\\_{1z} \\int\\_{-\\infty}^{\\infty}dk'\\_{14}\\\\\n&\\times\\chi(k'\\_{14},k'\\_{1z}; \\vec{k}'\\_{1\\perp}, \\vec{k}'\\_{2\\perp})\\tilde{v}(k'\\_{1v},k'\\_{14})\\\\\n& \\times \\frac{i}{(k'\\_{14}-i\\frac{M\\_3}{3})^2+k'^2\\_{1z}+m^2\\_1},\n\\end{aligned}$$ \nwhere the function *χ* is derived in Appendix [App:transveuclkernel] and reads\n \n$$\\label{Eq:chi\\_E\\_2}\n\\begin{aligned}\n\\chi(k'\\_{14},&k'\\_{1z}; \\vec{k}'\\_{1\\perp}, \\vec{k}'\\_{2\\perp}) = -\\pi\\int\\_{0}^1 \\frac{du}{A}. \n\\end{aligned}$$ \nHere the denominator *A* is given by\n \n*A* = *a**u*2 + *b**u* + *c*, \nwith\n \n$$\\label{abc}\n\\begin{aligned}\na&=-k'^2\\_{1z}-\\Bigl(k'\\_{14}+\\frac{2}{3}iM\\_3\\Bigr)^2,\\\\\nb&=k'^2\\_{1z}+\\Bigl(k'\\_{14}+\\frac{2}{3}iM\\_3\\Bigr)^2+m^2\\_2-m^2\\_3,\\\\\nc&=m^2\\_3,\n\\end{aligned}$$ \nwhere the *m**i*’s are defined by Eqs. ([m1m2]) and ([m1m2m3]).\nWick rotation in the three-body BS equation\n===========================================\nIt was shown in Ref. that the three-body BS equation, after the introduction of the shifted variables allows the Wick rotation without crossing any singularities. The validity of this rotation in the complex plane is a key for the equivalence between the Minkowski- and Euclidean-space transverse amplitudes. Therefore, we outline in this section some of the main points related to the Wick rotation of.\nThe BS equation enclosing the shifted variables takes the form\n \n$$\\label{Eq:BSE\\_shifted}\n \\begin{aligned}\n \\tilde{v}\\_M(q',p) = &2i\\mathcal{F}(M'^2\\_{12})\\int \\frac{d^4 k'}{(2\\pi)^4}\\frac{i}{[(k' + \\frac{p}{3})^2 - m^2 + i\\epsilon]} \n \\\\ & \\times \\frac{i}{[(\\frac{p}{3} - q' - k')^2 - m^2 +i\\epsilon]}\\tilde{v}\\_M(k',p),\n \\end{aligned}$$ \nwhere $M'^2\\_{12} = (\\frac{2}{3}p - q')^2$ and *ṽ**M* is defined by.\nIn the center-of-mass frame the pole in the upper half of the *k*ʹ0 complex plane of the first propagator is located at\n \n$$\\label{Eq:sing\\_kpp1}\n k'^{(+)}\\_{01} = -\\frac{M\\_3}{3} - \\sqrt{k'^2\\_v + m^2} + i\\epsilon,$$ \nand for the second one the position of the pole is\n \n*k*ʹ02( + ) = *η*ʹ − *q*ʹ0 + *i**ε*, \nwith\n \n$$\\label{Eq:eta\\_p}\n \\eta' = \\frac{M\\_3}{3}-\\sqrt{(\\vec{k}\\,'+\\vec{q}\\,^\\prime)^2 + m^2}.$$ \nConsequently, as seen from, the first propagator does not have any poles in the first quadrant. Moreover, since *M*3 < 3*m* one has that *η*ʹ < 0. The Wick rotation of can thus be done safely without crossing any singularities.\nContrary to this, for the BS equation in the form, written in terms of the unshifted variables (i.e. *k* and *q*), the second propagator has a pole at\n \n*k*02( + ) = *η* − *q*0 + *i**ε*, \nwhere\n \n$$\\label{Eq:eta\\_new}\n \\eta = M\\_3-\\sqrt{(\\vec{k}+\\vec{q})^2 + m^2},$$ \nand if *M*3 > *m*, there exist *k⃗* and *q⃗* such that *η* > 0, and the Wick rotation is thus not permitted.\nEach of the three Faddeev components of the transverse amplitute, given by, is like the right-hand side of the BS equation, an integral over a vertex function times a product of propagators. It is therefore clear that the Wick rotation (after introduction of the shifted variables) also should hold for the transverse amplitude. The propagators entering the definition of the BS amplitude have poles of the form, and should therefore not cause any additional problems in this respect. The expected equivalence between the transverse amplitudes computed in Minkowski and Euclidean spaces respectively, is confirmed by the numerical results to be presented in Sec. [Sec:ResTA]. Though, in this paper the transverse amplitudes, depending on *k**x*, *k**y*, are compared, we want to stress that the aforementioned arguments are also applicable to the more general amplitudes, not integrated over *k**z*, i.e., depending on *k**x*, *k**y*, *k**z*.\nResults and Discussion\n======================\nVertex function in Minkowski space\n----------------------------------\nThe Faddeev component of the vertex function is quantitatively studied in Minkowski space, as it carries the dynamical content of the relativistic three-body model. From it the full BS amplitude of the system can be constructed. In Minkowski space it has a nontrivial analytic structure since several branch points given by ([Eq:peaks]) are present in the kernel of Eq. , and reflected in the cusps appearing in the vertex function, as will be presented in what follows. Evidently, the Euclidean equation, obtained after the Wick rotation (see Ref. ) does not present in its integration path any singularity as well as the corresponding solution. Despite of this, both solutions can be compared as we are going to present.\nWe solved Eq. adopting a spline decomposition of the vertex function *v**M*(*q*, *p*) used in Ref. , see Appendix [Sec:NM]. The inputs are the scattering length *a* and the three-body binding energy *B*3, the same as in the Euclidean calculations performed in Ref. . In the numerical solution of, we multiplied the right-hand side of it by a parameter (eigenvalue) *λ*. A consistent solution then corresponds to an eigenvalue of *λ* = 1.0.\nThree results for the eigenvalue are given in Table [Tab:results], for the following values of the two-body scattering length: *a**m* = − 1.280, *a**m* = − 1.500 and *a**m* = − 1.705. The results for the eigenvalue *λ*, expected to be real and equal to one, present small deviations from the unity and also an imaginary part.\nNevertheless, it is important to mention another potential source of error: cutoffs were introduced to constrain the domains of the variables *q**v* and *q*0. It is very difficult to reach a reasonable convergence considering the full domains, as the size of the region where the singularities (given by Eqs. and ) appear is enlarged along the axes. Moreover, the asymptotic regions start at larger momenta.\nThe actual values used to truncate the variables were *q**v*max/*m* = 6.0 and *q*0max/*m* = 13.0, for the two smallest binding energies, or *q*0max/*m* = 15.0, for the case where *B*3/*m* = 1.001. Regardless, the convergence was reached within about 10% for the worst case. On the other hand, in the Euclidean calculations it is possible to take into account the whole range of the involved variables *q**v*, *q*4, *k**v* and *k*4 through a mapping procedure. The fact that in the Minkowski approach cutoffs were applied while the whole domains were used in the Euclidean calculations makes the results not fully comparable. This might be one of the reasons why in *λ* small non-zero imaginary parts appear and for the deviations from 1 obtained in the real part.\n[!htbp]\nc c c *B*3/*m* & *a**m* & *λ* \n0.006 & − 1.280 & 0.999 − 0.054*i* \n0.395 & − 1.500 & 1.000 + 0.002*i* \n1.001 & − 1.705 & 0.997 + 0.106*i* \nThe rest of this section will be devoted to a detailed study of the representative case with *B*/*m* = 0.395 and *a**m* = − 1.5. Though, important to point out that many of the stated conclusions are valid also for the other cases in Table [Tab:results].\nIn Fig. [Fig:v] it is shown the calculated real and imaginary parts of the vertex function *v**M*(*q*0, *q**v*) versus *q*0 for three fixed values of *q**v*. For all three cases there is a quite good agreement between the numerical peak positions and the analytical ones, given by Eq. . Though, worth to mention that, due to the scale of the figure, some of the peaks for the case *q**v*/*m* = 2.5 are not clearly visible. The peaks in Fig. [Fig:v] appear as branching points of the kernel Π(*q*0, *q**v*, ± *ɛ**k*, *k**v*), defining its singularities, as discussed in Sec. [Sec:BSE]. Interestingly, the aforementioned positions correspond to *M*122 = 0 and *M*122 = 4 *m*2, which give the branching points of the two-body scattering amplitude F(*M*122). In Fig. [Fig:v] it is seen that for small values of *q**v* a singularity appears at *q*0 ≈ *M*3. The distance between the external peaks, corresponding to *M*122 = 4 *m*2, is equal to $2\\sqrt{q\\_v^2+4m^2}$, an increasing function with respect to *q**v*. This fact makes things more complicated from the numerical point of view, as for large values of *q**v* a very wide region of *q*0 has to be covered. This imposes the need of cutoffs for the variables.\n \nSimilarly, in Fig. [Fig:vq] we present the real and imaginary parts of *v**M*(*q*0, *q**v*) with respect to *q**v*, for *q*0 = *m*. In the figure is seen a peak (both in the real and imaginary part) which corresponds to the branching point *q**v* = *M*3 − *q*0. However, for the other points given by Eq. , no solution exists such that *q*0 = *m* and *q**v* ≥ 0. It can be seen in the figure that the amplitude asymptotically goes to zero for large values of *q**v*, as expected.\nReal and imaginary parts of the vertex function, *v**M*(*q*0 = *m*, *q**v*) with respect to *q**v*. [Fig:vq]\nFurthermore, in Fig. [Fig:vqMinkEucl], we compare for *q*0 = *M*3/3 the calculated real and imaginary parts of *v**M*(*q*0, *q**v*) versus *q**v* with the corresponding Euclidean results for *ṽ**E*(*q*ʹ, *p*) = *v**E*(*q*ʹ + *p*/3, *p*), i.e. *q*ʹ4 = 0. It is seen that the results for both real and imaginary parts are practically the same for small values of *q**v*. However, the Minkowskian amplitude has a peak at the branching point *q**v* = 2*M*3/3 and the amplitudes also differ significantly at larger values of *q**v*, a difference coming from the contribution of the cut necessary to be taken into account to match the limit of *q*ʹ4 → 0 with the Minkowski point. For the sake of comparison both amplitudes are normalized to 1 for *q*ʹ4 = 0 and *q**v* = 0.\nReal and imaginary parts of the Minkowski space vertex function, *v**M*(*q*0 = *M*3/3, *q**v*) versus *q**v*, compared with the corresponding Euclidean results for *ṽ**E*(*q*ʹ4, *q*ʹ*v*). [Fig:vqMinkEucl]\nTransverse amplitude\n--------------------\nAlthough the transverse amplitude has a different meaning with respect to the standard distribution amplitude (see Refs. ), it shares in common with the distribution amplitude the valence wave function content of the full LFWF in Fock space. However, the transverse amplitude has a distinctive feature with respect to the distribution amplitude as it can be obtained as well from the BS amplitude in Euclidean space, making it a useful quantity to compare with the corresponding Minkowski space quantity. The demonstration of equivalence between the transverse amplitude obtained in Minkowski and Euclidean spaces has been given in Ref. resorting to the Nakanishi integral representation of the BS amplitude in a two-boson system. Furthermore, as discussed in Sec. [Sec:wickrotation], the Wick rotation of the BS equation can be performed without crossing any singularities and should also hold for the transverse amplitude. Consequently, the transverse amplitudes computed in Minkowski and Euclidean spaces should agree with each other also for the three-body system, which is confirmed by the results presented in this section.\nIn what follows we will exploit the structure of the transverse amplitude associated with one Faddeev component. The reader has to keep in mind that the full transverse amplitude from the three-body wave function is a coherent sum of the three components. However, to expose the consequences of our dynamical assumptions it is simpler to look individually to each Faddeev component, as the sum of *L**i*(∣*k⃗**i* ⊥∣, ∣*k⃗**j* ⊥∣, *θ**i**j*) where *θ**i**j* denotes the angle between *k⃗**i* ⊥ and *k⃗**j* ⊥, will present a much more complex 3D landscape as a function of the two independent transverse momenta.\nIn Fig. [Fig:transverse] it is displayed the modulus of the contribution *L*1(∣*k⃗*1 ⊥∣, ∣*k⃗*2 ⊥∣, *θ*) to the transverse amplitude versus ∣*k⃗*1 ⊥∣, calculated from Eq. . It is also shown, for comparison, the corresponding Euclidean results calculated through Eq. . It is seen that the Minkowski and Euclidean results are in fair agreement with each other. The non-smooth behavior of the BS solution in Minkowski space, shown in Fig. [Fig:v], is washed out and makes the agreement between the Euclidean and Minkowski space transverse amplitudes even more remarkable.\nTransverse amplitude modulus, ∣*L*1∣ for *θ* = 0 versus ∣*k⃗*1 ⊥∣/*m* for ∣*k⃗*2 ⊥∣/*m* = 0.0, 0.5, 1.0, obtained in Minkowski space compared with the one calculated in Euclidean space. [Fig:transverse]\nTransverse amplitude modulus, ∣*L*1∣ as a function of cos(*θ*) for (∣*k⃗*1 ⊥∣/*m*, ∣*k⃗*2 ⊥∣/*m*) = (0.5, 0.5), (2.0, 0.5), obtained in Minkowski space compared with the one computed in Euclidean space. [Fig:transversetheta]\nThe dependence of the modulus of the transverse amplitude ∣*L*1∣ on the angle *θ* between *k⃗*1 ⊥ and *k⃗*2 ⊥ is also displayed, in Fig. [Fig:transversetheta]. The modulus of *L*1 is a slowly decreasing function with respect to cos*θ*. As seen in the figure, a satisfactory agreement is again found between the Euclidean (lines) and Minkowski (symbols) calculations. Although the interaction is active in the s-wave and the vertex function is dependent only on the time component and the modulus of the spatial momentum in the rest-frame, the Faddeev component of the BS amplitude acquired a weak angular dependence, as we illustrated, due to the presence of the individual propagators and momentum conservation, essentially containing both s- and p-wave dependencies. The mixing of higher waves appears as the binding energy increases, which is expected as the bound state mass becomes smaller increasing the sensitivity of the denominator in Eq. ([Eq:chiE2]) to variations of the relative angle, as it is transparent in the case of the transverse amplitude in Euclidean space.\nThe three-dimensional structure of the transverse amplitude is further exposed in Fig. [Fig:transverse3d], where we plot the absolute value of the amplitude *L*1(∣*k⃗*1 ⊥∣, ∣*k⃗*2 ⊥∣, *θ*) with respect to ∣*k⃗*1 ⊥∣ and ∣*k⃗*2 ⊥∣ for different fixed values of *θ*. The computations were for simplicity performed in Euclidean space, since it has already been shown above that the Euclidean and Minkowski calculations give the same results for the transverse amplitude. One can conclude from the figure that the transverse amplitude becomes wider when the value of is *θ* is increased, due to correlations proportional to *k⃗*1 ⊥ ⋅ *k⃗*2 ⊥. This effect is clearly visible, in particular, for the anti-aligned case (*θ* = *π*), i.e. when the transverse momenta obey the relation ∣*k⃗*3 ⊥∣2 = (∣*k⃗*1 ⊥∣ − ∣*k⃗*2 ⊥∣)2. For *θ* = *π* and along ∣*k⃗*1 ⊥∣ = ∣*k⃗*2 ⊥∣ there is a clear enhancement of the transverse amplitude which exhibits a bump due to the vanishing value of $\\vec k\\_{3\\perp}$. The development of this pattern is seen by inspecting the evolution of the amplitude with *θ* and comparing the results for the *π*/2 and *π* angles. This feature is further enhanced when the three-body bound state mass decreases for a strongly bound system.\n \nShort-range correlation\n-----------------------\nThe two-body short-range correlation (in the context of non-relativistic nuclear physics, see e.g. Ref. ) is exhibited by the model for large relative momentum, ∣*k⃗*2 ⊥ − *k⃗*3 ⊥∣, and a back-to-back momentum configuration of particles 2 and 3, where the transverse amplitude is dominated by the component *L*1. Note that this amplitude is also symmetric by the exchange of *k⃗*2 ⊥ and *k⃗*3 ⊥ due to the bosonic nature of the system with equal masses. In the relativistic context the concept of the short-range two-body correlation has not yet been developed, but its emergent imprints are observed in the transverse amplitude.\nTransverse amplitude modulus, ∣*L*1∣ as a function of ∣*k⃗*2 ⊥ − *k⃗*3 ⊥∣ for ∣*k⃗*1 ⊥∣ = 0. In the inset of the figure is shown ∣*k⃗*2 ⊥ − *k⃗*3 ⊥∣2∣*L*1∣ versus ∣*k⃗*2 ⊥ − *k⃗*3 ⊥∣ with a dashed line indicating the asymptotic behavior. [correl]\nThe plot for *L*1 as a function of ∣*k⃗*2 ⊥ − *k⃗*3 ⊥∣ for the back-to-back configuration (∣*k⃗*1 ⊥∣ = 0) is shown in Fig. [correl]. In the situation when ∣*k⃗*2 ⊥ − *k⃗*3 ⊥∣ is large, only *L*1 dominates in the full three-body transverse amplitude, being the counterpart of the factorization of the non-relativistic wave function as a two-body term depending on the relative distance between the two particles times a function of the relative coordinates of the other *N* − 2 particles. In the model with contact interaction, the denominator in Eq. ([Eq:chiE2]) provides the momentum dependence of the ``wave function\" of the short-range correlated pair in the valence LFWF of the three-body system. The relative momentum behavior shown in the figure reflects the free propagators of the bosons, as the contact interaction does not bring any momentum dependence. This property is shared by the non-relativistic three-body model with a zero-range potential. Furthermore, for ∣*k⃗*1 ⊥∣ = 0 the denominator of Eq. ([Eq:chiE2]) provides the large momentum behavior as *L*1 ∼ ∣*k⃗*3 ⊥∣− 2 that leads to *L*1 ∼ ∣*k⃗*2 ⊥ − *k⃗*3 ⊥∣− 2 in the back-to-back configuration, as confirmed by our results. The scale for the asymptotic behavior is naturally fixed by the individual boson mass, as it can be seen in Fig. [correl], where *L*1 shows an accentuated drop in this momentum range. In the inset plot the asymptotic behavior for large relative momentum is shown.\nThe asymptotic property of *L*1 also follows from the structure of the valence LFWF, which has the three-body LF propagator as the dominant factor for the contact interaction and at large momentum is just the inverse of the free three-body mass, i.e. \n$$M\\_0^2=\\sum\\_i\\frac{\\vec{k}^2\\_{i\\perp}}{\\xi\\_i}\\,.$$ \n Note that *M*0− 2 ∼ ∣*k⃗*2 ⊥ − *k⃗*3 ⊥∣− 2 for ∣*k⃗*1 ⊥∣ = 0, which is the power-law behavior seen in Fig. [correl]. Such asymptotic form is expected to change for models with finite range interactions.\nConclusions\n===========\nThe three-body BS equation has been solved, directly in Minkowski space, by standard analytical and numerical methods, where (*i*) no ansatz or assumption has been introduced to represent the BS amplitude and (*ii*) the singularities from the kernel are treated analytically and numerically directly in the four-dimensional equation. The application of the direct integration method to the three-body Faddeev-BS equation was already presented in Ref. . However, in the present paper the Minkowski-space structure of the three-body system has been analyzed in far greater detail and brings up the structure of the short-range correlated pair in the Borromean system.\nThe computed amplitude turns out to be highly peaked, indicating the presence of a singular behavior, as shown in Sec. [Sec:TA]. This is very different from what was found for the amplitude computed through the Wick-rotated equation in Ref. , due to the presence of branching points and the associated cuts. In one example we expose the cut contribution to the amplitude comparing results from Minkowski and Euclidean calculations.\nAlthough the BS amplitudes obtained from the solution in Euclidean and Minkowski spaces are fundamentally different, they can be compared by means of the transverse amplitude. The comparison shows a notable agreement, giving more confidence on the reliability of the direct integration method. Furthermore, the transverse amplitude reveals the structure of the short-range correlated pair in the valence wave function, which was found when the pair has large relative momentum in a back-to-back configuration. We found that the Faddeev component of the BS amplitude defined with the pair interaction dominates over the others, and a power-law behavior of the type $\\sim |\\vec k\\_{i\\perp}-\\vec k\\_{j\\perp}|^{-2}$ is found for the associated transverse amplitude and confirmed by our numerical results.\nIn this work, we show that the results obtained by the direct integration in Minkowski space agree well with the Euclidean results for comparable quantities, however this method is quite demanding from the numerical point of view. One possible way to improve on this and additionally be able to treat more realistic kernels and/or the spin degree of freedom is to transform the BS equation into a non-singular form by using the Nakanishi integral representation . The formulation of the BS approach to the three-body problem via Nakanishi integral representation is already in progress and computations based on this method will be undertaken in the near future. Once the BS amplitude of the three-body state is known in Minkowski space it can be used to investigate electromagnetic form factors, the diversity of parton longitudinal and transverse momentum distributions, as well as the space-time structure of the pair short-range correlation.\nFurthermore, one interesting direction for future explorations of the three-body system is to consider particles with non-equal masses, as a framework, for example, to study baryons with a heavy-light content. Of course still many steps have to be considered to include the subtle physics of Quantum-Chromodynamics (QCD) in a continuum model (see e.g. Refs. ), but now in Minkowski space. We expect that the formulation will also allow to explore excited states and, through it, in the low energy region, the Efimov phenomena relativistically.\nWe are grateful to Jaume Carbonell for stimulating discussions. We thank for the support from Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) of Brazil. J.H.A.N. acknowledges the support of the grant #2014/19094-8 and V.A.K. of the grant #2015/22701-6 from Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP). E.Y. thanks for the financial support of the grants #2016/25143-7 and #2018/21758-2 from FAPESP. T.F. thanks the FAPESP Thematic Project grant #17/05660-0. V.A.K. is also sincerely grateful to group of theoretical nuclear physics of ITA, São José dos Campos, Brazil, for kind hospitality during his visits.\nDerivation of the relation between the BS amplitude and LFWF [App:BSALFWF]\n==========================================================================\nIn this Appendix, we derive in detail the relation between the three-body BS amplitude and the three-body LFWF.\nThe three-body BS amplitude is defined by Eq. . Let us define the integral\n \n$$\\label{I3}\n\\begin{aligned}\nI\\_3&=\\int d^4x\\_1\\, d^4x\\_2\\, d^4x\\_3\\,\\delta (\\omega {\\makebox[0.08cm]{$\\cdot$}}x\\_1)\\ \\delta (\\omega{\\makebox[0.08cm]{$\\cdot$}}x\\_2)\\ \\delta (\\omega{\\makebox[0.08cm]{$\\cdot$}}x\\_3)\\\n\\\\&\\times\\Phi\\_M(x\\_1,x\\_2,x\\_3;p)\\exp (ik\\_1{\\makebox[0.08cm]{$\\cdot$}}x\\_1+ik\\_2{\\makebox[0.08cm]{$\\cdot$}}x\\_2+ik\\_3{\\makebox[0.08cm]{$\\cdot$}}x\\_3)\\,\n\\end{aligned}$$ \nwhere *ω* = (*ω*0, *ω⃗*), *ω*2 = 0 and *k*1, 2, 3 are the on-shell momenta, i.e. *k*12 = *k*22 = *k*32 = *m*2. The delta functions in restrict the variation of the arguments of the coordinate space BS amplitude to the LF hyperplane $\\omega{\\makebox[0.08cm]{$\\cdot$}}x=0$.\nWe now represent the *δ*-functions in ([I3]) in the integral form\n \n$$\\label{delta}\n\\delta(\\omega{\\makebox[0.08cm]{$\\cdot$}}x\\_i)=\\frac{1}{2\\pi}\\int \\exp(-i\\omega{\\makebox[0.08cm]{$\\cdot$}}x\\_i\\,\\tau\\_i)d\\tau\\_i.$$ \nDue to translation invariance, when all *x**i*’s are shifted by *a*: *x* → *x* + *a*, the BS amplitude obtains a factor $\\exp(-ia{\\makebox[0.08cm]{$\\cdot$}}p)$:\n \n$$\\begin{aligned}\n\\label{Phi1}\n&&\\Phi\\_M(x\\_1+a,x\\_2+a,x\\_3+a;p)= \\nonumber\n\\\\\n&&=\\exp(-ia{\\makebox[0.08cm]{$\\cdot$}}p)\\Phi\\_M(x\\_1,x\\_2,x\\_3;p),\\end{aligned}$$ \nlike the non-relativistic wave function.\nWe introduce then the BS amplitude in momentum space. We define it, extracting the delta function, responsible for conservation of momenta:\n \n$$\\label{Phix}\n\\begin{aligned}\n\\Phi\\_M(&x\\_1,x\\_2,x\\_3;p)=\\frac{(2\\pi)^4}{(2\\pi)^{12}}\\int d^4k'\\_1\\, d^4k'\\_2\\, d^4k'\\_3\\, \\\\\n& \\times\\exp (-ik'\\_1{\\makebox[0.08cm]{$\\cdot$}}x\\_1-ik'\\_2{\\makebox[0.08cm]{$\\cdot$}}x\\_2-ik'\\_3{\\makebox[0.08cm]{$\\cdot$}}x\\_3)\\\\\n&\\times\\delta^{(4)}(k'\\_1+k'\\_2+k'\\_3-p)\\, \\Phi\\_M(k'\\_1,k'\\_2,k'\\_3;p).\n\\end{aligned}$$ \nBecause of the delta-function in ([Phix]) the amplitude Φ*M*(*x*1, *x*2, *x*3; *p*) satisfies the relation ([Phi1]). We emphasize that here, in contrast to Eq. ([I3]), all the arguments *k*ʹ1, 2, 3 of the BS amplitude are the off-mass-shell momenta.\nWe substitute ([Phix]) and ([delta]) in ([I3]) and integrate over *x*1, 2, 3 and *k*ʹ1, 2, 3. The result reads:\n \n$$\\label{I3a}\n\\begin{aligned}\nI\\_3&=\\frac{1}{(2\\pi)^{3}}\\int d\\tau\\_1 d\\tau\\_2 d\\tau\\_3\\,(2\\pi)^4 \\\\\n&\\times\\delta(k\\_1+k\\_2+k\\_3-p-\\omega\\tau\\_1-\\omega\\tau\\_2-\\omega\\tau\\_3)\\\\\n&\\times\\Phi\\_M(k\\_1-\\omega\\tau\\_1,k\\_2-\\omega\\tau\\_2,k\\_3-\\omega\\tau\\_3;p) \\,,\n\\end{aligned}$$ \nwhere the variables *τ*1, 2, 3 ∈ ] − ∞, ∞[. To avoid confusion, we emphasize again: the four-momenta *k*1, 2, 3 in this formula are the on-shell momenta, according to the definition of ([I3]), whereas, the arguments of the BS amplitude are the off-shell momenta, like *k*ʹ1, 2, 3 in ([Phix]); namely: $(k\\_1-\\omega\\tau\\_1)^2=m^2-2(\\omega{\\makebox[0.08cm]{$\\cdot$}}k\\_1)\\tau\\_1\\neq m^2$ since *τ*1 ≠ 0 and similarly for other arguments except for *p*2 = *M*32.\nWe subsequently introduce the variable *τ* = *τ*1 + *τ*2 + *τ*3 and represent the integral ([I3a]) in the following three equivalent forms:\n \n$$\\label{I3b}\n\\begin{aligned}\n&I\\_{3a}=\\frac{(2\\pi)^4}{(2\\pi)^{3}}\\int d\\tau\\, \\delta(k\\_1+k\\_2+k\\_3-p-\\omega\\tau)\\\\\n& \\times\\int d\\tau\\_1 d\\tau\\_2 \\\\\n&\\times\\Phi\\_M(k\\_1-\\omega\\tau\\_1,k\\_2-\\omega\\tau\\_2,k\\_3-\\omega(\\tau-\\tau\\_1-\\tau\\_2);p),\\\\\n&I\\_{3b}=\\frac{(2\\pi)^4}{(2\\pi)^{3}}\\int d\\tau\\, \\delta(k\\_1+k\\_2+k\\_3-p-\\omega\\tau)\\\\\n&\\times\\int d\\tau\\_2 d\\tau\\_3 \\\\\n&\\times\\Phi\\_M(k\\_1-\\omega(\\tau-\\tau\\_2-\\tau\\_3),k\\_2-\\omega\\tau\\_2,k\\_3-\\omega\\tau\\_3;p),\\\\\n&I\\_{3c}=\\frac{(2\\pi)^4}{(2\\pi)^{3}}\\int d\\tau\\,\\delta(k\\_1+k\\_2+k\\_3-p-\\omega\\tau)\\\\\n&\\times\\int d\\tau\\_1 d\\tau\\_3 \\\\\n&\\times\\Phi\\_M(k\\_1-\\omega\\tau\\_1,k\\_2-\\omega(\\tau-\\tau\\_1-\\tau\\_3),k\\_3-\\omega\\tau\\_3;p).\n\\end{aligned}$$ \nIt can be also represented as:\n \n$$\\label{I3c}\n\\begin{aligned}\nI\\_3&=\\frac{(2\\pi)^4}{(2\\pi)^{3}}\\int d\\tau\\, \\delta(k\\_1+k\\_2+k\\_3-p-\\omega\\tau)\\\\\n&\\times\\int d\\tau\\_1 d\\tau\\_2 d\\tau\\_3\\;\\delta(\\tau\\_1+\\tau\\_2+\\tau\\_3-\\tau)\\\\\n&\\times \\Phi\\_M(k\\_1-\\omega\\tau\\_1,k\\_2-\\omega\\tau\\_2,k\\_3-\\omega\\tau\\_3;p),\n\\end{aligned}$$ \nwhere by means of the delta function *δ*(*τ*1 + *τ*2 + *τ*3 − *τ*) one can exclude any *τ**i* and obtain Eq. ([I3b]). Depending on the convenience, one can chose any of the forms in ([I3b]) to calculate the double integral over *τ**i*, *τ**j*. With the standard choice *ω**μ* = (1, 0, 0, − 1), i.e., in the LF coordinates, *ω⃗*⊥ = 0, *ω*+ = 0, *ω*− = 2, these integrals are reduced to the integrals over the *k**i*− components. The value of *τ* is determined from the conservation law *k*1 + *k*2 + *k*3 = *p* + *ω**τ*. For example, squaring this equation, we find\n \n$$\\begin{aligned}\n\\tau&=\\frac{(k\\_1+k\\_2+k\\_3)^2-M\\_3^2}{2(\\omega{\\makebox[0.08cm]{$\\cdot$}}p)}=\\\\\n&\\frac{1}{2(\\omega{\\makebox[0.08cm]{$\\cdot$}}p)}\n\\left(\\frac{k\\_{1\\perp}^2+m^2}{x\\_1}\n+\\frac{k\\_{2\\perp}^2+m^2}{x\\_2}+\\frac{k\\_{3\\perp}^2+m^2}{x\\_3}-M\\_3^2\\right)\\\\\n&\\equiv\\frac{1}{2(\\omega{\\makebox[0.08cm]{$\\cdot$}}p)}\\left(M\\_0^2-M\\_3^2\\right).\n\\end{aligned}$$ \nOn the other hand, the integral ([I3]) can be expressed in terms of the three-body LFWF. We assume that the LF plane is the limit of a space-like plane, therefore the operators *φ*(*x*1), *φ*(*x*2), *φ*(*x*3), commute with each other, and, hence, the symbol of the *T* product in ([Phi01]) can be omitted. In the considered representation, the Heisenberg operators *φ*(*x*) in ([Phi01]) are identical on the light front $\\omega{\\makebox[0.08cm]{$\\cdot$}}x=0$ to the Schrödinger ones (just as in the ordinary formulation of field theory the Heisenberg and Schrödinger operators are identical for *t* = 0). The Schrödinger operator *φ*(*x*) (for the spinless case, for simplicity), which for $\\omega{\\makebox[0.08cm]{$\\cdot$}}x=0$ is the free field operator, is given by:\n \n$$\\label{bs5}\n\\begin{aligned}\n\\varphi(x)&=\\frac 1{(2\\pi )^{3/2}}\\int \\frac{d^3k}{\\sqrt{2\\varepsilon\\_k}}\\\\\n&\\times\\left[a(\\vec k)\n\\exp (-ik{\\makebox[0.08cm]{$\\cdot$}}x)+a^{\\dagger}(\\vec k)\\exp (ik{\\makebox[0.08cm]{$\\cdot$}}x)\\right].\n\\end{aligned}$$ \nWe represent the state vector ∣*p*⟩ ≡ *ϕ*(*p*) in ([I3]) in the form of the expansion via the Fock states:\n \n$$\\label{wfp1}\n\\begin{aligned}\n|p\\rangle&=(2\\pi)^{3/2}\\int\\psi(k\\_1,k\\_2,k\\_3,p,\\omega\\tau)\\\\\n&\\times\\delta^{(4)}(k\\_1+k\\_2+k\\_2-p-\\omega\\tau)2(\\omega{\\makebox[0.08cm]{$\\cdot$}}p)d\\tau \\\\\n&\\times\n \\frac{d^3k\\_1}{(2\\pi)^{3/2}\\sqrt{2\\varepsilon\\_{k\\_1}}}\n\\frac{d^3k\\_2}{(2\\pi)^{3/2}\\sqrt{2\\varepsilon\\_{k\\_2}}}\n\\frac{d^3k\\_3}{(2\\pi)^{3/2}\\sqrt{2\\varepsilon\\_{k\\_3}}}\\\\\n&\\times a^\\dagger(\\vec{k}\\_1)a^\\dagger(\\vec{k}\\_2)\na^\\dagger(\\vec{k}\\_3)|0\\rangle + \\cdots\\.\n\\end{aligned}$$ \nIn ([bs5]) and ([wfp1]) the four-momenta *k*1, 2, 3 are on mass-shells. We substitute this expression in \nΦ*M*(*x*1, *x*2, *x*3; *p*), Eq. ([Phi01]). Since the vacuum state on the light front is always “bare”, the creation operator, applied to the vacuum state ⟨0∣ gives zero, and in the operators *φ*(*x*) the part containing the annihilation operators only survives. This cuts out the three-body Fock component in the state vector. We thus obtain:\n \n$$\\begin{aligned}\n\\Phi\\_M(&x\\_1,x\\_2,x\\_3;p)=(2\\pi)^{3/2}\\int\\psi(k\\_1,k\\_2,k\\_3,p,\\omega\\tau)\n\\\\\n&\\times\\delta^{(4)}(k'\\_1+k'\\_2+k'\\_2-p-\\omega\\tau)2(\\omega{\\makebox[0.08cm]{$\\cdot$}}p)d\\tau\\\\\n&\\times\\exp(-ik'\\_1x\\_1-ik'\\_2x\\_2-ik'\\_3x\\_3)\\\\\n&\\times\\frac{d^3k'\\_1}{(2\\pi)^{3}2\\varepsilon\\_{k'\\_1}}\\frac{d^3k'\\_2}{(2\\pi)^{3}2\\varepsilon\\_{k'\\_2}}\\frac{d^3k'\\_3}{(2\\pi)^{3}2\\varepsilon\\_{k'\\_3}}.\n\\end{aligned}$$ \nThen we substitute this Φ*M*(*x*1, *x*2, *x*3; *p*) in ([I3]) and integrate over *x*1, 2, 3. The integration, for example, over *x*1 and then over *k*ʹ1 is fulfilled as follows\n \n$$\\begin{aligned}\n&\\int \\frac{d^4x\\_1 d^3k'\\_1}{(2\\pi)^{4}2\\varepsilon\\_{k'\\_1}} \\exp(-ik'\\_1x\\_1)\\exp(-i\\omega{\\makebox[0.08cm]{$\\cdot$}}x\\_1\\tau\\_1)d\\tau\\_1 \\\\\n&\\times\\exp(ik\\_1x\\_1)\\\\\n&=\\int \\delta^{(4)}(k\\_1-k'\\_1-\\omega\\tau\\_1)d\\tau\\_1\\frac{d^3k'\\_1}{2\\varepsilon\\_{k'\\_1}}\n\\\\\n&=\\int \\delta^{(4)}(k\\_1-k'\\_1-\\omega\\tau\\_1)d\\tau\\_1d^4k'\\_1\\theta(\\omega{\\makebox[0.08cm]{$\\cdot$}}k'\\_1)\\delta({k'}\\_1^2-m^2)\\\\\n&=\\int \\theta(\\omega{\\makebox[0.08cm]{$\\cdot$}}k\\_1)\\delta((k\\_1+\\omega\\tau\\_1)^2-m^2)\\\\\n&=\\int d\\tau\\_1 \\delta(2(\\omega {\\makebox[0.08cm]{$\\cdot$}}k\\_1)\\tau\\_1)\n=\\frac{1}{2(\\omega {\\makebox[0.08cm]{$\\cdot$}}k\\_1)},\n\\end{aligned}$$ \nand similarly for the integrations over *x*2, *k*ʹ2 and *x*3, *k*ʹ3. We used here that $\\theta(k'\\_{10})=\\theta(\\omega{\\makebox[0.08cm]{$\\cdot$}}k'\\_1)$ for *k*ʹ10 > 0. Then for *I*3 we get (cf. Eq. (3.56) from Ref. ):\n \n$$\\label{bs6}\n\\begin{aligned}\nI\\_3&=\\frac{(2\\pi )^{3/2}2(\\omega{\\makebox[0.08cm]{$\\cdot$}}p)}{2(\\omega{\\makebox[0.08cm]{$\\cdot$}}k\\_1)2(\\omega{\\makebox[0.08cm]{$\\cdot$}}k\\_2)2(\\omega{\\makebox[0.08cm]{$\\cdot$}}k\\_3)}\n\\int d\\tau\\psi (k\\_1,k\\_2,k\\_3;p,\\omega \\tau )\\\\\n&\\times\\delta^{(4)}(k\\_1+k\\_2+k\\_3-p-\\omega \\tau )\\.\n\\end{aligned}$$ \nComparing ([bs6]) and ([I3c]), we find:\n \n$$\\label{bs7}\n\\begin{aligned}\n\\psi (k\\_1&,k\\_2,k\\_3,p,\\omega \\tau )=\\frac{1}{\\sqrt{2\\pi}}\\frac{2(\\omega{\\makebox[0.08cm]{$\\cdot$}}k\\_1)2(\\omega{\\makebox[0.08cm]{$\\cdot$}}k\\_2)2(\\omega{\\makebox[0.08cm]{$\\cdot$}}k\\_3)}{2(\\omega {\\makebox[0.08cm]{$\\cdot$}}p)}\\\\\n&\\times\\int d\\tau\\_1 d\\tau\\_2 d\\tau\\_3\\;\\delta(\\tau\\_1+\\tau\\_2+\\tau\\_3-\\tau)\\\\\n&\\times \\Phi\\_M(k\\_1-\\omega\\tau\\_1,k\\_2-\\omega\\tau\\_2,k\\_3-\\omega\\tau\\_3;p).\n\\end{aligned}$$ \nAs mentioned, in ordinary LFD, Eq. ([bs7]) corresponds to the integration over *k*−. This equation makes the link between the three-body BS amplitude Φ*M* and the wave function *ψ* defined on the light front specified by *ω*. As it is seen from the above derivation, it is generalizable (with the same coefficient $1/\\sqrt{2\\pi})$ for arbitrary number of particles. Simply the number of the factors $2(\\omega{\\makebox[0.08cm]{$\\cdot$}}k\\_i)$ and of the arguments increases. In the LF coordinates Eq. ([bs7]) obtains the form:\n \n$$\\begin{gathered}\n \\label{bs7a}\n\\psi (\\vec{k}\\_{1\\perp},\\xi\\_1;\\vec{k}\\_{2\\perp},\\xi\\_2;\\vec{k}\\_{3\\perp},\\xi\\_3)=\n \\frac{1}{\\sqrt{2\\pi}}\\frac{(p^+)^2}{2} 2\\xi\\_1 2\\xi\\_2 2\\xi\\_3 \\\\ \\times \\int d\\tau\\_1 d\\tau\\_2 d\\tau\\_3\\delta(\\tau\\_1+\\tau\\_2+\\tau\\_3-\\tau)\n\\Phi\\_M(\\tilde k\\_1;\\tilde k\\_2;\\tilde k\\_3;p).\\end{gathered}$$ \nwhere $\\tilde k\\_i\\equiv\\{ \\vec{k}\\_{i\\perp},k^+\\_{i},k^-\\_{i}-2\\tau\\_i\\}$ and 0 < *ξ**i* < 1 (*ξ*1 + *ξ*2 + *ξ*3 = 1) denotes the longitudinal momentum fraction of particle *i*.\nWe introduce now new integration variables: \n*k*ʹ1− = *k*1− − 2*τ*1, etc, and then\n \n$$\\label{bs7b}\n\\begin{aligned}\n&\\psi (\\vec{k}\\_{1\\perp},\\xi\\_1;\\vec{k}\\_{2\\perp},\\xi\\_2;\\vec{k}\\_{3\\perp},\\xi\\_3)=\\frac{(p^+)^2}{\\sqrt{2\\pi}} \\, \\xi\\_1 \\xi\\_2 \\xi\\_3\\\\\n&\\times\\int dk^{-}\\_{1}\\, dk^{-}\\_{2}\\, \n\\Phi\\_M(k\\_1,k\\_2,k\\_3;p)\\\\\n&=\\frac{(p^+)^2}{\\sqrt{2\\pi}} \\xi\\_1 \\xi\\_2 \\xi\\_3\n\\int dk^-\\_{1}dk^-\\_{2} \n\\Phi\\_M(k\\_1,k\\_2,k\\_3;p).\n\\end{aligned}$$ \nIn the last line, we omitted the integration over the 3rd argument since it is not independent. In the above formula it is understood that *k*3 = *p* − *k*1 − *k*2. One can chose any pair of arguments: (12), (13) or (23), depending on the convenience.\nCalculating the Euclidean transverse amplitude\n==============================================\nIn this Appendix we derive in detail the function *χ* occurring in the expression for the Euclidean transverse amplitude, Eq. .\nFrom Eqs. and, we define *χ* as the integral\n \n$$\\label{Eq:chi\\_E}\n\\begin{aligned}\n\\chi(k'\\_{14},&k'\\_{1z}; \\vec{k}'\\_{1\\perp}, \\vec{k}'\\_{2\\perp}) = \\\\\n&\\int\\_{-\\infty}^{\\infty} dk'\\_{20} \\int\\_{-\\infty}^{\\infty} dk'\\_{2z} \\frac{i}{(k'\\_{24}-i\\frac{M\\_3}{3})^2+k'^2\\_{2z}+m^2\\_2}\\\\\n&\\times\\frac{i}{(k'\\_{14}+k'\\_{24}+i\\frac{M\\_3}{3})^2+(k'\\_{1z}+k'\\_{2z})^2+m^2\\_3}. \n\\end{aligned}$$ \nThe two propagators in can then be put together by using the Feynman parametrization leading to the result\n \n$$\\label{Eq:denom\\_prod}\n\\begin{aligned}\n&\\frac{i}{(k'\\_{24}-i\\frac{M\\_3}{3})^2+k'^2\\_{2z}+m^2\\_2}\\\\\n&\\times\\frac{i}{(k'\\_{14}+k'\\_{24}+i\\frac{M\\_3}{3})^2+(k'\\_{1z}+k'\\_{2z})^2+m^2\\_3}\\\\\n&= - \\int\\_{0}^1 \\frac{du}{D^2},\n\\end{aligned}$$ \nwhere the denominator reads\n \n$$\\label{Eq:D\\_eucl}\n\\begin{aligned}\nD &= k'^2\\_{24}+k'^2\\_{2z} + (1-u)\\bigl[k'^2\\_{14} + k'^2\\_{1z}\\bigr] + \\frac{2}{3}iM\\_3k'\\_{24} \\\\\n& + 2(1-u)k'\\_{1z}k'\\_{2z} + \\frac{2}{3}(1-u)k'\\_{14}(3k'\\_{24}+iM\\_3) \\\\\n& - \\frac{4}{3}iuM\\_3k'\\_{24} + (1-u)m^2\\_3 + u\\,m^2\\_2 - \\frac{M^2\\_3}{9} = \\\\\n& k'^2\\_{24}+k'^2\\_{2z} + (1-u)\\bigl[k'^2\\_{14} + k'^2\\_{1z}\\bigr] + \\\\\n& 2\\Bigl[(1-u)k'\\_{14}-\\frac{iM\\_{3}}{3}(-1+2u) \\Bigr]k'\\_{24} \\\\\n& + 2(1-u)k'\\_{1z}k'\\_{2z} + \\frac{2}{3}iM\\_3(1-u)k\\_{14} + (1-u)m^2\\_3 \\\\\n&+u\\,m^2\\_2 - \\frac{M^2\\_3}{9}.\n\\end{aligned}$$ \nWe subsequently eliminate the terms linear in *k*ʹ24 and *k*ʹ2*z*, by performing in Eqs. , and the transformations\n \n$$\\begin{aligned}\nk'\\_{24} &\\longrightarrow k'\\_{24} - \\alpha,\\\\\nk'\\_{2z} &\\longrightarrow k'\\_{2z} -\\beta,\n\\end{aligned}$$ \nwith\n \n$$\\alpha = (1-u)k'\\_{14}-\\frac{iM\\_3}{3}(-1+2u),$$ \nand\n \n*β* = (1 − *u*)*k*ʹ1*z*. \nBy these transformations the denominator is changed into\n \n$$\\begin{aligned}\nD\\longrightarrow\\tilde{D} = k'^2\\_{24}+k'^2\\_{2z} + A, \n\\end{aligned}$$ \nwhere\n \n$$\\begin{aligned}\nA &= u(1-u)\\bigl[k'^2\\_{14} + k'^2\\_{1z}\\bigr] + (1-u)m^2\\_3 + um^2\\_2 \\\\\n& + \\frac{4}{3}iM\\_3u(1-u)k'\\_{14} - \\frac{4}{9}M^2\\_3u(1-u). \n\\end{aligned}$$ \nThe integrals over *k*ʹ24 and *k*ʹ2*z* in can now be performed analytically, and the result is\n \n$$\\label{Eq:chi\\_E\\_2\\_new}\n\\begin{aligned}\n\\chi(&k'\\_{14},k'\\_{1z}; \\vec{k}'\\_{1\\perp}, \\vec{k}'\\_{2\\perp}) = -\\int\\_{0}^1 du\\\\\n&\\times\\int\\_{-\\infty}^{\\infty} dk'\\_{20} \\int\\_{-\\infty}^{\\infty} \\frac{dk'\\_{2z}}{(k'^2\\_{24}+k'^2\\_{2v}+A)^2}= \\\\\n& -2\\pi \\int\\_{0}^1 du \\int\\_{0}^{\\infty} \\frac{k'dk'}{(k'^2+A)^2}=-\\pi\\int\\_{0}^1 \\frac{du}{A}. \n\\end{aligned}$$ \nAlternatively, one can write the quantity *A* in the form\n \n*A* = *a**u*2 + *b**u* + *c*, \nwith\n \n$$\\label{abc\\_2}\n\\begin{aligned}\na&=-k'^2\\_{1z}-\\Bigl(k'\\_{14}+\\frac{2}{3}iM\\_3\\Bigr)^2,\\\\\nb&=k'^2\\_{1z}+\\Bigl(k'\\_{14}+\\frac{2}{3}iM\\_3\\Bigr)^2+m^2\\_2-m^2\\_3,\\\\\nc&=m^2\\_3.\n\\end{aligned}$$ \nNumerical methods\n=================\nWe solve in this work Eq. by expanding the amplitude *v**M*(*q*0, *q**v*) in a bicubic spline basis, on a finite domain Ω = *I**q*0 × *I**q**v* = [ − *q*0max, *q*0max] × [0, *q**v*max], i.e. \n \n*v**M*(*q*0, *q**v*) = ∑*k* = 02*N**q*0 + 1∑*l* = 02*N**q**v* + 1*A**i**j**S**k*(*q*0)*S**l*(*q**v*), \nwhere the unknown coefficients *A**i**j* are to determined. In the numerical implementation, the interval *I**x* (*x* = *q*0, *q**v*) is partioned into *N**x* subintervals, so that good convergence was reached. The adopted spline functions, *S**j*(*x*) are given by \n \n$$\\begin{aligned}\nS\\_{2i}(x)&=\\left\\lbrace \\begin{array}{l l}\n3\\left(\\frac{x-x\\_{i-1}}{h\\_i}\\right)^2-2\\Bigl(\\frac{x-x\\_{i-1}}{h\\_i}\\Bigr)^3, \n\\\\[1.2\\medskipamount] \\quad \\text{if } x\\in[x\\_{i-1},x\\_i]\\\\[1.7\\medskipamount]\n3\\left(\\frac{x\\_{i+1}-x}{h\\_{i+1}}\\right)^2-2\\left(\\frac{x\\_{i+1}-x}{h\\_{i+1}}\\right)^3, \\\\[1.2\\medskipamount]\n\\quad \\text{if } x\\in[x\\_i,x\\_{i+1}]\\\\[1.7\\medskipamount]\n0, \\quad \\text{if } x\\not\\in[x\\_{i-1},x\\_{i+1}]\n\\end{array}\\right.\\\\[1.2\\medskipamount]\nS\\_{2i+1}(x)&=\\left\\lbrace \\begin{array}{l l}\n\\left[-\\left(\\frac{x-x\\_{i-1}}{h\\_i}\\right)^2+\\left(\\frac{x-x\\_{i-1}}{h\\_i}\\right)^3\\right]h\\_i, \\\\[1.2\\medskipamount]\n\\quad \\text{if } x\\in[x\\_{i-1},x\\_i]\\\\[1.7\\medskipamount]\n\\left[\\left(\\frac{x\\_{i+1}-x}{h\\_{i+1}}\\right)^2-\\left(\\frac{x\\_{i+1}-x}{h\\_{i+1}}\\right)^3\\right]h\\_{i+1}, \\\\[1.2\\medskipamount]\n\\quad \\text{if } x\\in[x\\_i,x\\_{i+1}]\\\\[1.7\\medskipamount]\n0, \\quad \\text{ if } x\\not\\in[x\\_{i-1},x\\_{i+1}]\n\\end{array}\\right.\\\\\n\\end{aligned}$$ \nwith *h**i* = *x**i* − *x**i* − 1.\nBy using, Eq. can be transformed to a generalized eigenvalue problem of the form\n \n∑*i*ʹ*j*ʹ*F**i**j**i*ʹ*j*ʹ*A**i*ʹ*j*ʹ = *λ*(*M*3)∑*i*ʹ*j*ʹ*V**i**j**i*ʹ*j*ʹ*A**i*ʹ*j*ʹ, \nwhere\n \n*F**i**j**i*ʹ*j*ʹ = *S**i*ʹ(*q*0(*i*))*S**j*ʹ(*q**v*(*j*)), \nand the array *V**i**j**i*ʹ*j*ʹ is the right-hand side of with *v**M* replaced by *S**i*ʹ(*q*0(*i*))*S**j*ʹ(*q**v*(*j*)). The variable *q*0 (*q**v*) has here been discretized on a mesh consisting of 2*N**q*0 + 2 (2*N**q**v* + 2) points. The three-body mass *M*3, or equivalently the three-body binding energy *B*3, can subsequently be obtained from the condition\n \n*λ*(*M*3) = 1. \nEq. constitutes a non-linear equation relative to *M*3 and is rather time-consuming to solve. For simplicity, we use thus instead as inputs in the calculations the scattering length *a* and the *M*3, obtained from the solution of the Euclidean BS equation. Eq. is then solved for the eigenvalue *λ* and the coefficients *A**i**j*.\nThe kernel Π(*q*0, *q**v*, *k*0, *k**v*) (see Eq. ), which enters Eq. has logarithmic singularities, and the analytic expressions for the singular points are given by Eqs. and. In the present work, the integrals over *k*0 and *k**v* are computed by dividing a given integration interval into subintervals *I**i* = [*a**i*, *b**i*], so that each subinterval contains at most one singular point which is just one of the end points of the subinterval. For each subinterval, the integrand singularity is subsequently weakened by adopting a change of variables of the form\n \n$$\\int\\_{a\\_i}^{b\\_i}f(x)dx=\\int\\_0^{\\sqrt{b\\_i-a\\_i}}2tf(a\\_i+t^2)dt,$$ \nfor a subinterval with a singularity at *a**i*, and\n \n$$\\int\\_{a\\_i}^{b\\_i}f(x)dx=\\int\\_0^{\\sqrt{b\\_i-a\\_i}}2tf(b\\_i-t^2)dt,$$ \nif the singularity is at the end point *b**i*. The resulting integrals involving smooth functions can then be performed by Gauss-Legendre integration.\n \nNumerical convergence\n---------------------\nAs mentioned, in this work the three-body BS equation is solved by using an expansion of the amplitude *v**M*(*q*0, *q**v*) in terms of a finite number of spline functions. Evidently, it is important to check that the adopted number basis functions is enough.\nFor this purpose, we show in Fig. [Fig:Convergence] the real and imaginary parts of *v**M*(*q*0, *q**v* = 0.5*m*), computed by using different number of subintervals *N**q**v* and *N**q*0, corresponding to the variables *q**v* and *q*0. In the calculations we used the parameters *a**m* = − 1.5 and *B*3/*m* = 0.395. It is seen in the figure that for *N**q**v* ≥ 40 and *N**q*0 ≥ 80, the solution is well-converged.\n \nBehavior of ${\\cal F}(M^2\\_{12})$\n---------------------------------\nFor negative *"},"data_id":{"kind":"string","value":"arxiv_0184754"}}},{"rowIdx":46192,"cells":{"content":{"kind":"string","value":"addtoresetequationsection\nNIKHEF-2015-027 \nCPHT-RR025.0815\n **Dyonic ISO(7) supergravity and the duality hierarchy**\nAdolfo Guarino1 and Oscar Varela2, 3\n*1Nikhef Theory Group, Science Park 105, 1098 XG Amsterdam, The Netherlands* \n*2 Center for the Fundamental Laws of Nature, \nHarvard University, Cambridge, MA 02138, USA* \n*3Centre de Physique Théorique, Ecole Polytechnique, \nCNRS UMR 7644, 91128 Palaiseau Cedex, France* \n**Abstract**\n> \n> Motivated by its well defined higher dimensional origin, a detailed study of *D* = 4 ${{\\cal N}}=8$ supergravity with a dyonically gauged $\\textrm{ISO}(7) = \\textrm{SO}(7) \\ltimes \\mathbb{R}^7$ gauge group is performed. We write down the Lagrangian and describe the tensor and duality hierarchies, focusing on an interesting subsector with closed field equations and supersymmetry transformations. We then truncate the ${{\\cal N}}=8$ theory to some smaller sectors with ${{\\cal N}}=2$ and ${{\\cal N}}=1$ supersymmetry and SU(3), G2 and SO(4) bosonic symmetry. Canonical and superpotential formulations for these sectors are given, and their vacuum structure and spectra is analysed. Unlike the purely electric ISO(7) gauging, the dyonic gauging displays a rich structure of vacua, all of them AdS. We recover all previously known ones and find a new ${{\\cal N}}=1$ vacuum with SU(3) symmetry and various non-supersymmetric vacua, all of them stable within the full ${{\\cal N}}=8$ theory.\n> \n> \n> \nIntroduction\n============\nMaximal gauged supergravity in four dimensions often admits continuous or discrete symplectic deformations that respect ${{\\cal N}}=8$ supersymmetry and the gauge group. The simplest type of deformation introduces a dependence on a dimensionless parameter *c* in the gauging-dependent couplings of the theory. The covariant derivatives, for example, acquire a new coupling to the magnetic vectors proportional to *c* ,\n \n$$\\label{DSO8\\_intro}\nD = d - g \\, ({{\\cal A}}^{\\Lambda} - c \\, \\tilde{{{\\cal A}}}\\_{ \\Lambda}) \\,$$ \nthus leading to a dyonic gauging. The role of this parameter, in a passive picture, is to tune the electric/magnetic symplectic frame prior to introducing the gauging. In the ungauged limit, *c* can be set to zero without loss of generality by a symplectic transformation. At finite gauge coupling *g*, however, electric/magnetic duality is broken and the theory typically becomes sensitive to the symplectic frame specified by *c*. Various aspects of this deformation for different gauge groups have now been studied, including its effect on the vacuum structure, on domain-wall and black hole solutions, or on inflationary models.\nAn immediate question is whether these ${{\\cal N}}=8$ dyonic gaugings descend from higher dimensions. This was recently answered positively when the gauge group is chosen to be ISO(7)*c* ≡ CSO(7, 0, 1)*c* $\\equiv \\textrm{SO}(7) \\ltimes \\mathbb{R}\\_{c}^7$. Here and often in the following, we have followed the notation of and have sticked in a subscript *c* to denote that ISO(7) (more precisely, only its seven translations) is gauged dyonically. In we showed that *D* = 4 ${{\\cal N}}=8$ ISO(7)-dyonically-gauged supergravity arises as a consistent truncation of massive type IIA supergravity on the six-sphere, with the magnetic coupling constant *m* ≡ *g**c* identified upon reduction with the Romans mass, $\\hat F\\_\\0 = m$. All solutions of the *D* = 4 theory uplift to solutions of massive type IIA by the consistency of the truncation. In particular, its vacua (all known ones are AdS) give rise to AdS4 backgrounds of massive type IIA string theory. Quantitative evidence was also given in that these AdS4 vacua are dual to the simplest type of Chern-Simons theories with a single gauge group and adjoint matter. The answer to the question of the higher-dimensional origin of these dyonic gaugings is of course gauge group dependent. Arguments have been recently given against an M-theory origin of the dyonic deformation of the SO(8) gauging.\nThe distinct higher-dimensional origin of the dyonic ISO(7) gauging singles it out and makes it worth of further detailed investigation. This is what we set up to do in this paper from a purely four-dimensional perspective, leaving further research on the precise connection with ten dimensions for separate publications. Various aspects of the ISO(7) gauging have already been studied. The purely electric, *c* = 0 (*i.e.* *m* = 0), ISO(7)-gauged theory was constructed long ago from the SO(8)-gauged theory by a limiting procedure that implements the Inönü-Wigner contraction from SO(8) to ISO(7) directly in the supergravity. The symplectic deformations corresponding to various gauge groups, including ISO(7), were studied in. The ISO(7)*c* family of gaugings was found to be discrete, containing only two members: the purely electric *c* = 0 theory, and the dyonic *c* ≠ 0 theory. All non-vanishing values of *c* lead to equivalent theories.\nThe basic formalism to deal with generic gaugings of *D* = 4 ${{\\cal N}}=8$ supergravity has been laid out in, see also. The gauging is encoded in an embedding tensor that governs both the non-Abelian coupling of the vectors to themselves and to the rest of the supergravity fields, and the embedding of the gauge group into the global U-duality group, E7(7). Gaugings that involve minimal couplings to the magnetic vectors in a given symplectic frame necessarily require the presence of two-form potentials. These appear both sourcing the field strengths of the vectors and in new topological terms in the Lagrangian, without upsetting the count of degrees of freedom. More generally, a larger set of *p*-form potentials, *p* = 1, …, 4, of a so-called tensor hierarchy can be considered. These include all vectors in the theory, a larger set of two-forms than a given gauging would typically require, and three- and four-forms, all of them in irreducible E7(7) representations. Except for the four-form potentials, the fields up the tensor hierarchy are definitely dynamical as they typically cannot be gauged away: they do carry degrees of freedom, albeit not independent ones. Indeed, bringing the metric and scalars into the picture, the higher rank forms can be Hodge-dualised into (of course, dynamical) combinations of scalars and their derivatives. The ${{\\cal N}}=8$ tensor hierarchy equipped with these dualisations has been referred to as the ‘duality hierarchy’. See for the hierarchies in less supersymmetric contexts. In this paper we will specify the Lagrangian for the ISO(7)*c* gaugings following the embedding tensor formalism. We will also be interested in the duality hierarchy, paying particular attention to a subsector with closed field equations and supersymmetry transformations. This subsector arises upon suitable restriction of the full E7(7)-covariant duality hierarchy. Although it is only SL(7)-covariant, rather than E7(7), this subsector is still ${{\\cal N}}=8$.\nWe will also study the vacuum structure of the ISO(7)*c* gaugings. More concretely, we provide a systematic classification of the critical points of the scalar potential that preserve at least SU(3) and at least a particular SO(4) within SO(7) ⊂ ISO(7). We do this by working out the truncations of the ${{\\cal N}}=8$ theory to the SU(3)- and that particular SO(4)-invariant sectors, and then extremising the resulting potentials. Although the G2-invariant sector is contained within the SU(3) sector, we find it useful to provide a separate treatment for it too. All these sectors are supersymmetric and, as a crosscheck on our calculations, we cast them in the corresponding ${{\\cal N}}=2$ or ${{\\cal N}}=1$ canonical form. We provide explicit parameterisations for the scalars in these subsectors. This allows us to give the location of the critical points in the full scalar space E7(7)/SU(8) relative to those parameterisations. This result is new even for the critical points that were already known (see below), which had been found using a method whose power resides, precisely, in its being insensitive to their actual location.\nQuite surprisingly, the vacuum structure of the electric and dyonic ISO(7) gaugings turns out to be very different. In fact, while the former has no known vacua, the latter displays a rich (AdS) vacuum structure. Some of these critical points were already known, including points with ${{\\cal N}}=1$, G2, and ${{\\cal N}}=3$, SO(4) symmetry, and non-supersymmetric points with SO(7), SO(6) and G2 symmetry. Among the non-supersymmetric points, only the latter is stable, at least within the full ${{\\cal N}}=8$ theory. Our classification recovers all these extrema and finds new ones with ${{\\cal N}}=2$, SU(3) × U(1) symmetry (which we already reported on in ), a point with SU(3), ${{\\cal N}}=1$ symmetry, and stable non-supersymmetric points with SU(3) and SO(4) symmetry. Some, but not all, of these points have counterparts in either the electric or dyonic SO(8) gauging, with the same residual supersymmetry, bosonic symmetry and mass spectrum. See table [Table:SU3&SO4Points] for a summary of the known critical points of the dyonic ISO(7) supergravity.\n[t!] [Table:SU3&SO4Points]\nIn section [sec:ISO(7)], we construct the ISO(7)*c* theory using the embedding tensor formalism, and specify the bosonic Lagrangian, an ${{\\cal N}}=8$ subsector of the duality hierarchy and the supersymmetry transformations. In the rest of the paper we flesh out some interesting subsectors with less supersymmetry and bosonic symmetry: see sections [sec:SU(3)-sector], [sec:G2-sector] and [sec:SO(4)-sector] for discussions of the SU(3), G2 and an SO(4)-invariant sectors, respectively. Canonical supersymmetric formulations are given and the critical points of the scalar potential in these sectors are computed. Four appendices close the paper. The first two offer further discussion. Appendix [app:Z2xSO(3)-sector] contains the truncation of the ${{\\cal N}}=8$ theory to yet another subsector, with $\\,{{\\cal N}}=1\\,$ supersymmetry and Z2 × SO(3) bosonic symmetry, relevant to non-geometric type IIA orientifold reductions. Appendix [app:MTheoryonSE7] comments on the relation of the SU(3)-invariant sector of the ISO(7)*c* theory to the $\\,{{\\cal N}}=2\\,$ supergravity that arises from consistent truncation of M-theory on an arbitrary Sasaki-Einstein manifold. The last two are technical: appendix [app:ConstN=8] gives some details of the construction of the ${{\\cal N}}=8$ ISO(7)*c* theory, while appendix [app:ScalarMatrix] gives explicit parameterisations for the supergravity scalar kinetic matrix in the invariant sectors discussed in the main text.\nMaximal supergravity with dyonic ISO(7) gauging\n===============================================\nWe will now present the *D* = 4 ${{\\cal N}}=8$ supergravity theory with a dyonically-gauged ISO(7) gauge group, focusing on its bosonic sector. We review the embedding tensor and the field content, including the tensor hierarchy, in section [subsec:FCET]. An interesting subsector of the latter is discussed in [sec:TruncTensorHierarchy]. The bosonic Lagrangian and supersymmetry transformations can be found in [subsec:LST] and [sec:susyvars]. See also appendix [app:ConstN=8] for some details of the construction of the theory from the general formalism of.\nTensor hierarchy and ISO(7) embedding tensor\n--------------------------------------------\nThe bosonic field content of maximal supergravity in four dimensions includes the vielbein *e**μ**α*, scalars that parameterise a coset representative ${\\cal V}\\_\\mathbb{M}{}^{ij}$, M = 1, …, 56, *i* = 1, …, 8, of E7(7)/SU(8) and vectors ${{\\cal A}}^\\mathbb{M}$, M = 1, …, 56, in the fundamental representation of the U-duality group E7(7), with two-form field strengths ${{\\cal H}}\\_\\2^\\mathbb{M}$. In the presence of magnetic charges, as it will be the case in this work, a set of two-form potentials[1](#fn1) ${{\\cal B}}\\_\\alpha$, *α* = 1, …, 133, in the adjoint of E7(7) and with three-form field strengths ${{\\cal H}}\\_{\\3 \\alpha}$, is generically required by gauge invariance. A gauging-dependent projection of the two-form potentials ${{\\cal B}}\\_\\alpha$ typically enters the *D* = 4 Lagrangian and the field strengths ${{\\cal H}}\\_\\2^\\mathbb{M}$ of the vectors.\nMore generally, these $\\bm{56}$ vectors ${{\\cal A}}^\\mathbb{M}$ and $\\bm{133}$ two-forms ${{\\cal B}}\\_\\alpha$ are the first two sets of fields in an E7(7)-covariant tensor hierarchy that further includes $\\bm{912}$ three-form potentials ${{\\cal C}}\\_{\\alpha}{}^{\\mathbb{M}}$ with four-form field strengths ${{{\\cal H}}\\_{\\4 \\alpha}}^\\mathbb{M}$, and $\\bm{133} + \\bm{8645}$ four-form potentials. Like for the lower rank forms, certain gauging-dependent projections of the three-form potentials ${{\\cal C}}\\_{\\alpha}{}^{\\mathbb{M}}$ enter the three-form field strengths ${{\\cal H}}\\_{\\3 \\alpha}$, and so on. Obviously, not all the fields in the tensor hierarchy carry independent degrees of freedom: the higher rank forms can be dualised into scalars and their derivatives. This was discussed at length in, where the tensor hierarchy equipped with these dualisations was dubbed the ‘duality hierarchy’. It is possible to write a generic ${{\\cal N}}=8$ gauged supergravity Lagrangian that includes higher-rank fields in the E7(7) tensor hierarchy. This Lagrangian reduces, after imposing the duality relations, to the conventional Lagrangian containing only the metric, scalars, vectors and the two-forms switched on by magnetic gaugings. In section [subsec:LST] we will write the Lagrangian for the dyonic ISO(7) gauging in the formulation of, although we will still find it useful to consider, in section [sec:TruncTensorHierarchy], a (restricted) duality hierarchy containing forms of higher rank.\nTo conclude this summary of the ${{\\cal N}}=8$ field content, recall that the fermionic sector contains the gravitino *ψ**μ**i* and spin 1/2 fields *χ**i**j**k*, in the $\\overline{\\textbf{8}}$ and $\\overline{\\textbf{56}}$ of the R-symmetry group SU(8), respectively. Both fermions are chiral, *e.g.*, *γ*5 *ψ**μ**i* = *ψ**μ**i*, with *γ*5 = *i**γ*0*γ*1*γ*2*γ*3 the chirality operator and *γ**α* the Cliff(1, 3) matrices. Recall that, in four dimensions, charge conjugation reverses the fermion chirality. Following convention, we denote negative chirality spinors with lower SU(8) indices, *γ*5 *ψ**μ**i* = − *ψ**μ**i*.\nIn order to formulate the ISO(7) gauging, it is natural to branch out the above E7(7)-covariant bosonic field content into representations of SL(7), given that ISO(7) is contained in E7(7) through the chain[2](#fn2)\n \n$$\\begin{aligned}\n \\label{eq:chainE7}\n\\textrm{ISO(7)} \\equiv \\textrm{SO}(7) \\ltimes \\mathbb{R}^7\n \\, \\subset \\, \n \\textrm{SL}(7) \\ltimes \\mathbb{R}^7 \n \\, \\subset \\, \n \\textrm{GL}(7) \\ltimes \\mathbb{R}^7 \n \\, \\subset \\, \n \\textrm{SL}(8) \\subset \\textrm{E}\\_{7(7)} \\;.\\end{aligned}$$ \nFor this purpose, we find it useful to introduce fundamental SL(8) indices *A*, *B* = 1, …, 8, and a collective index Λ ≡ [*A**B*] = 1, …, 28. For SL(7), we only need to introduce fundamental indices, *I* = 1, …, 7. The $\\bm{56}$ vectors, for example, branch as\n \n$$\\begin{aligned}\n\\label{vectors56}\n{{\\cal A}}^\\mathbb{M} = ( {{\\cal A}}^\\Lambda \\,,\\, \\tilde{{{\\cal A}}}\\_\\Lambda ) = ( {{\\cal A}}^{AB} \\,,\\, \\tilde{{{\\cal A}}}\\_{AB} ) = ( {{\\cal A}}^{IJ}, {{\\cal A}}^{I} \\,,\\, \\tilde{{{\\cal A}}}\\_{IJ}, \\tilde{{{\\cal A}}}\\_{I} ) \\.\\end{aligned}$$ \nWe have dropped the ‘8’ label in ${{\\cal A}}^{I8} $ and $\\tilde{{{\\cal A}}}\\_{I8}$, and have put tildes on the magnetic vectors. Although the tildes are redundant with the lower position of the indices, we find this emphatic notation visually useful. Similarly, the $\\bm{133}$ two-form, ${{\\cal B}}\\_\\alpha$, and $\\bm{912}$ three-form, ${{\\cal C}}\\_\\alpha{}^\\mathbb{M}$, potentials branch as well into SL(7) representations: see equation ([eq:Branchings]) for the relevant decompositions.\nIn ${{\\cal N}}=8$ supergravity, gaugings are completely specified by the embedding tensor ΘM*α* . This determines the embedding of the gauge group into the E7(7) duality group. Linear constraints enforce ΘM*α* to lie generically in the **912** of E7(7), and quadratic constraints (see equation ([QC])) ensure the consistency of the gauging. Fixing the gauge group to be ISO(7), the linear constraint reduces the embedding tensor to lie in the $\\bm{28} + \\bm{1}$ of SL(7) and the quadratic constraint allows for the following non-vanishing components of ΘM*α* = ( ΘΛ*α* , ΘΛ *α* ) only:\n \n$$\\label{Theta-compSL7}\n\\Theta\\_{[IJ]\\phantom{K}L}^{\\phantom{[IJ]}K} = 2\\, \\delta\\_{[I}^{K} \\, \\delta\\_{J]L} \n\\hspace{5mm}, \\hspace{5mm}\n\\Theta\\_{[I8] \\phantom{8} K}^{\\phantom{[I8]} 8} = -\\delta\\_{I K}\n\\hspace{5mm} \\textrm{ and } \\hspace{5mm}\n{\\Theta^{[I8] \\, 8}}\\_{K} = c \\, \\delta^{I}\\_{K} \\,$$ \nsee. Here, *c* is an arbitrary real constant. It was shown in that all non-vanishing values of *c* lead to equivalent theories up to a rescaling of the gauge coupling *g*. Therefore, for *g* ≠ 0, there exist two possible ISO(7) gaugings of *D* = 4 ${{\\cal N}}=8$ supergravity : *c* = 0 and *c* ≠ 0. The first two components in ([Theta-compSL7]), associated to the $\\bm{28}$, couple to the electric vectors, while the last component, related to the singlet, couples to the magnetic vectors. Strictly speaking, only SO(7) singlets enter ([Theta-compSL7]). In particular, in the first two components, only the singlet in the decomposition of the $\\bm{28}$ of SL(7) under SO(7) is involved, and is realised as a Kronecker delta with two lower indices. We nevertheless find it useful to refer to the electric, ΘΛ*α*, and magnetic, ΘΛ *α*, components of ([Theta-compSL7]) as the $\\bm{28}$ and singlet of SL(7), respectively.\nThe physical difference between the *c* = 0 and *c* ≠ 0 ISO(7) gaugings is most easily seen by looking at the covariant derivatives. Denoting by *g* the (electric) gauge coupling and introducing a magnetic gauge coupling *m* through\n \n*m* ≡ *g* *c* , \nthe covariant derivatives induced by the ISO(7) embedding tensor ([Theta-compSL7]) are\n \n$$\\label{DISO7}\nD = d \\, - g \\, {{\\cal A}}^{IJ} \\, {t\\_{[I}}^{K} \\, \\delta\\_{J]K} + \\big( g \\, \\delta\\_{IJ} \\, {{\\cal A}}^{I} - m \\, \\tcA\\_{J} \\big) \\, {t\\_{8}}^{J} \\.$$ \nThe ${{\\cal A}}^{IJ}$ terms can be equivalently written using the $\\bm{48} $ SL(7) generators ${t\\_{I}}^{J} - \\tfrac17 \\, t\\_K{}^K \\, \\delta\\_{I}^J $. These, together with the $\\bm{7}^\\prime$ generators *t*8*J*, generate the $\\textrm{SL}(7) \\ltimes \\mathbb{R}^7$ subgroup of E7(7) in ([eq:chainE7]). See ([Gener63]), ([Gener70]) for the expressions of the E7(7) generators (*t**α*)MN in the fundamental representation, in the SL(8) basis. In agreement with the table on page 37 of, the embedding tensor components in the $\\bm{28}$ couple the $\\bm{21}^\\prime$ electric vectors ${{\\cal A}}^{IJ}$ to the $\\bm{48}$ generators ${t\\_{I}}^{J} - \\tfrac17 \\, t\\_K{}^K \\, \\delta\\_{I}^J $, and the $\\bm{7}^\\prime$ electric vectors ${{\\cal A}}^I$ to the $\\bm{7}^\\prime$ generators *t*8*J*, while the singlet component of the embedding tensor couples the $\\bm{7}$ magnetic vectors $\\tilde{{{\\cal A}}}\\_I$ to the $\\bm{7}^\\prime$ generators *t*8*J* whenever *c* ≠ 0. The choice *c* = 0 in ([Theta-compSL7]) thus leads to the purely electric ISO(7) gauging constructed in by other methods. For *c* ≠ 0, the gauging is dyonic in the symplectic frame where ([Theta-compSL7]) is expressed: the R7 translations of ISO(7) are gauged dyonically. The rotations SO(7) are only gauged electrically, though: the constraints on the ISO(7) embedding tensor set to zero the $\\bm{28}^\\prime$ components that would induce a magnetic gauging of SO(7), as well as the $\\bm{7}^\\prime$, see the table in. Thus, the $\\bm{21}$ magnetic vectors $\\tilde{{{\\cal A}}}\\_{IJ}$ do not participate in the gauging. Observe, finally, that the combinations *T**I**J* ≡ 2 *t*[*I**K* *δ**J*]*K* and *T**I* ≡ *t*8*J* *δ**J**I* in ([DISO7]) correspond to the SO(7) and R7 generators of the gauge group $\\,\\textrm{ISO}(7)=\\textrm{SO}(7) \\ltimes \\mathbb{R}^{7}\\,$, see ([ISO7brackets]).\nIndices of SL(7) cannot be raised or lowered. For the ISO(7) gauging, these can be identified with SO(7) indices upon contraction with the embedding tensor. Even in this case, we will refrain from raising and lowering them with the SO(7) metric *δ**I**J*.\nA restricted duality hierarchy\n------------------------------\nIn this section, we consider a certain subset of fields in the SL(7)-branched out tensor hierarchy that includes all $\\bm{56} \\rightarrow (\\bm{21}^\\prime + \\bm{7}^\\prime) + ( \\bm{21}+ \\bm{7}) $ electric and magnetic vectors, but excludes all of the four-forms and most of the SL(7)-covariant two-forms and three-forms that respectively arise in the branching of the $\\bm{133}$ and $\\bm{912}$ of E7(7) under SL(7). It only includes the two-forms associated to the generators of $\\textrm{SL}(7) \\ltimes \\mathbb{R}^7$ and the three-forms in the conjugate representation of the electric part of the embedding tensor. Specifically, we wish to consider the following ${{\\cal N}}=8$ bosonic field content, in SL(7) representations,\n \n$$\\begin{aligned}\n \\label{eq:SL7fieldcontent4D}\n\\bm{1} & \\textrm{metric}: & \\; ds\\_4^2 \\nonumber \\\\\n\\bm{21}^\\prime + \\bm{7}^\\prime + \\bm{21}+ \\bm{7} & \\textrm{coset representatives}: & \\; {\\cal V}^{IJ \\, ij} \\;, \\; {\\cal V}^{I8 \\, ij} \\;, \\; \\tilde{{\\cal V}}\\_{IJ}{}^{ij} \\;, \\; \\tilde{{\\cal V}}\\_{I8}{}^{ij} \\;, \\nonumber \\\\\n\\bm{21}^\\prime + \\bm{7}^\\prime + \\bm{21}+ \\bm{7} & \\textrm{vectors}: & \\; {{\\cal A}}^{IJ} \\;, \\quad {{\\cal A}}^{I} \\;, \\quad \\; \\tilde{{{\\cal A}}}\\_{IJ} \\;, \\quad \\tilde{{{\\cal A}}}\\_{I} \\;, \\nonumber \\\\\n\\bm{48} + \\bm{7}^\\prime & \\textrm{two-forms}: & \\; {{\\cal B}}\\_{I}{}^J \\;, \\quad {{\\cal B}}^I \\;, \\nonumber \\\\\n\\bm{28}^\\prime & \\textrm{three-forms}: & \\; {{\\cal C}}^{IJ} \\;,\\end{aligned}$$ \nalong with the fermions *ψ**μ**i* and *χ**i**j**k* in the $\\overline{\\textbf{8}}$ and $\\overline{\\textbf{56}}$ of SU(8). Note that ${{\\cal A}}^{IJ} \\equiv {{\\cal A}}^{[IJ]}$, but ${{\\cal C}}^{IJ} \\equiv {{\\cal C}}^{(IJ)}$. The vectors ${{\\cal A}}^{IJ}$ and ${{\\cal A}}^I$ can alternatively be considered to lie respectively in the adjoint and fundamental of SO(7), as they must for the $\\textrm{ISO}(7) = \\textrm{SO}(7) \\ltimes \\mathbb{R}^7$ gauging. The representations shown for the coset representatives correspond to their SL(7) indices *I* = 1, …, 7. Unlike for the vectors and two-forms, we have kept the label ‘8’ in them that comes from the branching ([eq:chainE7]) through SL(8). Their antisymmetric upper (lower) indices *i**j* label the $\\,\\bm{\\overline{28}}\\,$ ($\\bm{28}$) of SU(8).\nConsidering the field content ([eq:SL7fieldcontent4D]) requires some justification, since it contains more fields than necessary to write the ISO(7)-gauged Lagrangian in the formulation of (see section [subsec:LST]), yet does not include all the fields in the full tensor hierarchy. The relevance of this field content will only become apparent when we discuss the full embedding of the ISO(7) gauging into type IIA. It is nevertheless still possible to justify the self-consistency of the field content ([eq:SL7fieldcontent4D]) from a purely four-dimensional perspective. As we will next show, for the *g**m* ≠ 0 ISO(7) gauging, ([eq:SL7fieldcontent4D]) defines a consistent subsector of the full E7(7) duality hierarchy, in the conventional sense. Namely, the Bianchi identities of the *p*-forms, *p* = 1, 2, 3 in ([eq:SL7fieldcontent4D]), the duality relations that these forms satisfy together with the metric and scalars, their equations of motion and supersymmetry variations, all close among themselves. This restricted field content preserves, of course, full ${{\\cal N}}=8$ supersymmetry since we are also keeping the $\\bm{\\overline{8}}$ gravitini. The rest of this subsection will be devoted to show the closure of the Bianchi identities and duality relations, while the closure of the supersymmetry variations will be verified in section [sec:susyvars].\nIn order to show the closure of the Bianchi identities, we first compute the field strengths of the *p*-form potentials in ([eq:SL7fieldcontent4D]) specified by the ISO(7)*c* gauging ([Theta-compSL7]). The two-form field strengths of the vectors are given by\n \n$$\\label{eqO:2FormFieldStrengths}\n\\begin{array}{lll}\n{\\cal H}^{IJ}\\_\\2 &=& d {{\\cal A}}^{IJ} - g \\, \\delta\\_{KL} \\, {{\\cal A}}^{IK} \\wedge {{\\cal A}}^{LJ} \\, \\\\[7pt]\n{\\cal H}^I\\_\\2 &=& d {{\\cal A}}^I -g \\, \\delta\\_{JK} \\, {{\\cal A}}^{IJ} \\wedge {{\\cal A}}^K +\\tfrac12 m \\, {{\\cal A}}^{IJ} \\wedge \\tilde {{\\cal A}}\\_J + m \\, {{\\cal B}}^I \\,\\\\[2mm]\n\\tilde{{\\cal H}}\\_{\\2 IJ} &=& d \\tilde{{{\\cal A}}}\\_{IJ} + g \\, \\delta\\_{K[I} \\, {{\\cal A}}^{KL} \\wedge \\tilde{{{\\cal A}}}\\_{J]L} + g \\, \\delta\\_{K[I} \\, {{\\cal A}}^{K} \\wedge \\tilde{{{\\cal A}}}\\_{J]} - m \\, \\tilde{{{\\cal A}}}\\_{I} \\wedge \\tilde{A}\\_{J} +2 g \\, \\delta\\_{K[I} \\, {{\\cal B}}\\_{J]}{}^K \\, \\\\[2mm]\n\\tilde{{\\cal H}}\\_{\\2 I} &=& d \\tilde{{{\\cal A}}}\\_I - \\tfrac12 g \\, \\delta\\_{IJ} \\, {{\\cal A}}^{JK} \\wedge \\tilde{{{\\cal A}}}\\_K + g \\, \\delta\\_{IJ} \\, {{\\cal B}}^J \\,\n\\end{array}$$ \nthe three-form field strengths of the two-form potentials are\n \n$$\\label{eqO:3FormFieldStrengths}\n\\begin{array}{lll}\n{\\cal H}\\_{\\3 I}{}^J &=& D {{\\cal B}}\\_I{}^J + \\tfrac12 {{\\cal A}}^{JK} \\wedge d \\tilde{{{\\cal A}}}\\_{IK} + \\tfrac12 {{\\cal A}}^{J} \\wedge d \\tilde{{{\\cal A}}}\\_{I} + \\tfrac12 \\tilde{{{\\cal A}}}\\_{IK} \\wedge d {{\\cal A}}^{JK} + \\tfrac12 \\tilde{{{\\cal A}}}\\_{I} \\wedge d {{\\cal A}}^{J} \\\\[2mm] \n&& - \\tfrac 12 g \\, \\delta\\_{KL} \\, {{\\cal A}}^{JK} \\wedge {{\\cal A}}^{LM} \\wedge \\tilde{{{\\cal A}}}\\_{IM} - \\tfrac 12 g \\, \\delta\\_{KL} \\, {{\\cal A}}^{JK} \\wedge {{\\cal A}}^{L} \\wedge \\tilde{{{\\cal A}}}\\_{I} \\\\[2mm]\n&& + \\tfrac 16 g \\, \\delta\\_{IK} \\, {{\\cal A}}^{JL} \\wedge {{\\cal A}}^{KM} \\wedge \\tilde{{{\\cal A}}}\\_{LM} - \\tfrac 13 g \\, \\delta\\_{IK} \\, {{\\cal A}}^{(J} \\wedge {{\\cal A}}^{K)L} \\wedge \\tilde{{{\\cal A}}}\\_{L} \\\\[2mm] \n&& - \\tfrac12 m \\, {{\\cal A}}^{JK} \\wedge \\tilde{{{\\cal A}}}\\_I \\wedge \\tilde{{{\\cal A}}}\\_K - 2 g \\, \\delta\\_{IK} \\, {{\\cal C}}^{JK} - \\tfrac17 \\, \\delta\\_I^J \\, (\\textrm{trace})\n \\, \\\\[4mm]\n{\\cal H}\\_{\\3}^I &=& D {{\\cal B}}^I -\\tfrac12 {{\\cal A}}^{IJ} \\wedge d \\tilde{{{\\cal A}}}\\_J -\\tfrac12 \\tilde{{{\\cal A}}}\\_J \\wedge d {{\\cal A}}^{IJ} +\\tfrac12 g \\, \\delta\\_{JK} \\, {{\\cal A}}^{IJ} \\wedge {{\\cal A}}^{KL} \\wedge \\tilde{{{\\cal A}}}\\_L \\;,\n\\end{array}$$ \nand the four-form field strengths of the three-form potentials read\n \n$$\\label{eqO:4FormFieldStrengths}\n\\begin{array}{lll}\n{\\cal H}\\_{\\4}^{IJ} & = & D {{\\cal C}}^{IJ} - {\\cal H}\\_\\2^{K(I} \\wedge {{\\cal B}}\\_K{}^{J)} + {\\cal H}\\_\\2^{(I} \\wedge {{\\cal B}}^{J)} - \\tfrac{1}{2} m \\, {{\\cal B}}^I \\wedge {{\\cal B}}^J - \\tfrac16 {{\\cal A}}^{K(I} \\wedge \\tilde{{{\\cal A}}}\\_{KL} \\wedge d {{\\cal A}}^{J)L} \\\\[2mm]\n&& + \\tfrac16 {{\\cal A}}^{IK} \\wedge {{\\cal A}}^{JL} \\wedge d \\tilde{{{\\cal A}}}\\_{KL} - \\tfrac16 {{\\cal A}}^{K(I} \\wedge \\tilde{{{\\cal A}}}\\_{K} \\wedge d {{\\cal A}}^{J)} - \\tfrac13 {{\\cal A}}^{K(I} \\wedge {{\\cal A}}^{J)} \\wedge d \\tilde{{{\\cal A}}}\\_{K} \\\\[2mm]\n&& - \\tfrac16 {{\\cal A}}^{(I} \\wedge \\tilde{{{\\cal A}}}\\_{K} \\wedge d {{\\cal A}}^{J)K} -\\tfrac16 g\\, \\delta\\_{KL} \\, {{\\cal A}}^{K(I} \\wedge {{{\\cal A}}}^{J)M} \\wedge {{\\cal A}}^{LN} \\wedge \\tilde{{{\\cal A}}}\\_{MN} \\\\[2mm]\n&& + \\tfrac16 g\\, \\delta\\_{KL} \\, {{\\cal A}}^{K(I} \\wedge {{{\\cal A}}}^{J)} \\wedge {{\\cal A}}^{LM} \\wedge \\tilde{{{\\cal A}}}\\_{M} - \\tfrac16 g\\, \\delta\\_{KL} \\, {{\\cal A}}^{K(I} \\wedge {{{\\cal A}}}^{J)M} \\wedge {{\\cal A}}^{L} \\wedge \\tilde{{{\\cal A}}}\\_{M} \\\\[2mm]\n&& - \\tfrac18 m\\, {{\\cal A}}^{IK} \\wedge {{{\\cal A}}}^{JL} \\wedge \\tilde{{{\\cal A}}}\\_{K} \\wedge \\tilde{{{\\cal A}}}\\_{L} \\.\n\\end{array}$$ \nFollowing ([DISO7]), in ([eqO:3FormFieldStrengths])–([eqO:4FormFieldStrengths]) we have defined the covariant derivatives\n \n$$\\label{eqO:covariantDerFormPotis}\n\\begin{array}{lll}\nD {{\\cal B}}\\_I{}^J & \\equiv & d {{\\cal B}}\\_I{}^J - g \\, \\delta\\_{KL} \\, {{\\cal A}}^{JK} \\wedge {{\\cal B}}\\_I{}^L - g \\, \\delta\\_{IK} \\, {{\\cal A}}^{KL} \\wedge {{\\cal B}}\\_L{}^J - g \\, \\delta\\_{IK} \\, {{\\cal A}}^K \\wedge {{\\cal B}}^J +m \\, \\tilde{{{\\cal A}}}\\_I \\wedge {{\\cal B}}^J \\\\[2pt]\n&& - \\tfrac17 \\, \\delta\\_I^J \\, (\\textrm{trace}) \\, \\\\[6pt]\nD {{\\cal B}}^I & \\equiv & d {{\\cal B}}^I - g \\, \\delta\\_{JK} \\, {{\\cal A}}^{IJ} \\wedge {{\\cal B}}^K \\, \\\\[6pt]\nD {{\\cal C}}^{IJ} & \\equiv & d {{\\cal C}}^{IJ} +2 g \\, \\delta\\_{KL} \\, {{\\cal A}}^{K(I} \\wedge {{\\cal C}}^{J)L} \\.\n\\end{array}$$ \nWe have obtained the two- and three-form field strengths ([eqO:2FormFieldStrengths]), ([eqO:3FormFieldStrengths]) by particularising to the ISO(7)*c* embedding tensor ([Theta-compSL7]) the generic expressions dictated by the *D* = 4 embedding tensor formalism (see appendix [app:ConstN=8]). On the other hand, we obtained the four-form field strength ([eqO:4FormFieldStrengths]) from the IIA truncation formulae. This expression is also compatible with that dictated by the *D* = 4 embedding tensor formalism, see appendix B of. Note the pure Yang-Mills form of the electric field strengths ${\\cal H}^{IJ}\\_\\2$, in agreement with the purely electric gauging of the SO(7) subgroup of ISO(7) when *g* ≠ 0. The electric field strengths ${\\cal H}\\_\\2^{I}$ contain the contribution expected from the semidirect action of the electric SO(7) rotations on the electric abelian translations R7, plus contributions of the magnetic vectors $\\,\\tcA\\_{I}\\,$ and the two-forms $\\,{{\\cal B}}^{I}\\,$ due to the dyonic gauging when *m* ≠ 0.\nIntroducing, from ([DISO7]) with the generators in the appropriate representation, the following covariant derivatives of the two-form field strengths\n \n$$\\label{DH2}\n\\begin{array}{lll}\nD {\\cal H}\\_\\2^{IJ} & \\equiv & d {\\cal H}\\_{\\2}^{IJ} - 2 \\, g \\, \\delta\\_{KL} \\, {{\\cal A}}^{K[I} \\wedge {\\cal H}\\_\\2^{J]L} \\, \\\\[6pt]\nD {\\cal H}\\_\\2^I & \\equiv & d {\\cal H}\\_\\2^I - g \\, \\delta\\_{JK} {{\\cal A}}^{IJ} \\wedge {{\\cal H}}\\_\\2^K + g \\, \\delta\\_{JK} \\, {{\\cal A}}^J \\wedge {{\\cal H}}\\_\\2^{IK} - m \\, \\tilde{{{\\cal A}}}\\_J \\wedge {{\\cal H}}\\_\\2^{IJ} \\, \\\\[6pt]\nD \\tilde{{\\cal H}}\\_{\\2 IJ} & \\equiv & d \\tilde{{\\cal H}}\\_{\\2 IJ} + 2 \\, g \\, \\delta\\_{K[I} {{\\cal A}}^{KL} \\wedge \\tilde{{{\\cal H}}}\\_{\\2 J]L} + 2 \\, g \\, \\delta\\_{K[I} {{\\cal A}}^{K} \\wedge \\tilde{{{\\cal H}}}\\_{\\2 J]} - 2 \\, m \\, \\tcA\\_{[I} \\wedge \\tilde{{{\\cal H}}}\\_{\\2 J]} \\, \\\\[6pt]\nD \\tilde{{\\cal H}}\\_{\\2 I} & \\equiv & d \\tilde{{\\cal H}}\\_{\\2 I} - g \\, \\delta\\_{IJ} \\, {{\\cal A}}^{JK} \\wedge \\tilde{{{\\cal H}}}\\_{\\2 K} \\,\n\\end{array}$$ \nand of the three-form field strengths,\n \n$$\\label{DH3}\n\\begin{array}{lll}\nD {\\cal H}\\_{\\3 I}{}^J & \\equiv & d {\\cal H}\\_{\\3 I}{}^J - g \\, \\delta\\_{KL} \\, {{\\cal A}}^{JK} \\wedge {{\\cal H}}\\_{\\3 I}{}^L- g \\, \\delta\\_{IK} \\, {{\\cal A}}^{KL} \\wedge {{\\cal H}}\\_{\\3 L}{}^J - g \\, \\delta\\_{IK} \\, {{\\cal A}}^{K} \\wedge {{\\cal H}}\\_{\\3}^J \\\\[4pt]\n&& + \\, m \\, \\tilde{{{\\cal A}}}\\_I \\wedge {{\\cal H}}\\_\\3^J - \\tfrac17 \\, \\delta\\_I^J \\, (\\textrm{trace}) \\, \\\\[8pt]\nD {\\cal H}\\_\\3^I & \\equiv & d {\\cal H}\\_\\3^I - g \\, \\delta\\_{JK} \\, {{\\cal A}}^{IJ} \\wedge {{\\cal H}}\\_{\\3}^K \\, \n\\end{array}$$ \nwe find that the Bianchi identities corresponding to the form potentials in ([eq:SL7fieldcontent4D]) can be written as\n \n$$\\label{eq:TruncBianchis}\n\\begin{array}{l}\nD {\\cal H}\\_{\\2}^{IJ} = 0 \\hspace{2mm}, \\hspace{2mm}\nD {\\cal H}\\_\\2^{I} = m \\, {{\\cal H}}^{I}\\_{\\3} \\hspace{2mm}, \\hspace{2mm}\nD {\\tilde{\\cal H}}\\_{\\2 IJ} = - 2 \\, g \\, {{{\\cal H}}\\_{\\3 [I}}^{K}\\,\\delta\\_{J]K} \\hspace{2mm}, \\hspace{2mm}\nD {\\tilde{\\cal H}}\\_{\\2 I} = g \\, \\delta\\_{IJ} \\, {{\\cal H}}^{J}\\_{\\3} \\, \\\\[3mm]\nD {\\cal H}\\_{\\3 I}{}^J = {{\\cal H}}\\_\\2^{JK} \\wedge \\tilde{{{\\cal H}}}\\_{\\2 IK} + {{\\cal H}}\\_\\2^{J} \\wedge \\tilde{{{\\cal H}}}\\_{\\2 I} -2 g\\, \\delta\\_{IK} \\, {{\\cal H}}\\_\\4^{JK} - \\tfrac17 \\, \\delta\\_I^J \\, (\\textrm{trace}) \\, \\\\[3mm]\nD {\\cal H}\\_\\3^{I} = -{{\\cal H}}\\_\\2^{IJ} \\wedge \\tilde{{{\\cal H}}}\\_{\\2 J} \\hspace{3mm}, \\hspace{5mm}\nD {\\cal H}\\_{\\4}^{IJ} \\equiv 0 \\.\n\\end{array}$$ \nThe Bianchi identities ([eq:TruncBianchis]) indeed close among themselves, as we wanted to show. An equivalent way of phrasing this is that ([eq:TruncBianchis]) defines a free differential algebra (FDA) which is a sub-FDA of the FDA defined by the Bianchi identities of the full tensor hierarchy.\nWe now turn to discuss the closure of the field content ([eq:SL7fieldcontent4D]) under Hodge duality. Closure is really automatic: the magnetic two-form field strengths are dual to scalar-dependent combinations of the electric two-form field strengths; the three-form field strengths are dual to scalar dependent combinations of covariant derivatives of scalars; the four-form field strengths are dual to combinations of scalars; and all vectors and scalars have been retained in ([eq:SL7fieldcontent4D]). It is nevertheless useful to write the explicit duality relations. For the vectors and two-form potentials, these have been given in, while for the three-form potentials the duality relations have been given in. In particular, the four-form field strengths are dual to the derivative of the scalar potential (see ([VgeneralRewrite]) below) with respect to the embedding tensor.\nIn order to write the duality relations, we need to introduce two scalar-dependent symmetric matrices, ${{\\cal M}}$, real, and ${{\\cal N}}$, complex, respectively E7(7)- and SL(8)-covariant. The former is the square of the E7(7)/SU(8) coset representative, M = V V*t*, and is also related to the real and imaginary parts of the latter,\n \nNΛΣ = RΛΣ + *i* IΛΣ, \nwhere IΛΣ is invertible and negative definite. More concretely,\n \n$$\\begin{aligned}\n\\label{Mscalar}\n\\mathcal{M}\\_{\\mathbb{MN}}= 2 \\, {\\cal V\\_{( \\mathbb{M}}}^{ij} \\, {\\cal V}\\_{ \\mathbb{N}) \\, ij } \\equiv \n\\left(\n\\begin{array}{ll}\n\\mathcal{M}\\_{\\Lambda \\Sigma} & {\\mathcal{M}\\_{\\Lambda}}^{\\Sigma} \\\\[2mm]\n{\\mathcal{M}^{\\Lambda}}\\_{\\Sigma} & \\mathcal{M}^{\\Lambda \\Sigma}\n\\end{array}\n\\right)\n=\n\\left(\n\\begin{array}{ll}\n-(\\mathcal{I}+\\mathcal{R}\\mathcal{I}^{-1}\\mathcal{R})\\_{\\Lambda \\Sigma} & {(\\mathcal{R}\\mathcal{I}^{-1})\\_{\\Lambda}}^{\\Sigma} \\\\[2mm]\n{(\\mathcal{I}^{-1}\\mathcal{R})^{\\Lambda}}\\_{\\Sigma} & -(\\mathcal{I}^{-1})^{\\Lambda \\Sigma}\n\\end{array}\n\\right).\\end{aligned}$$ \nThe inverse of ${\\cal M}\\_{\\mathbb{M} \\mathbb{N}}$ is ${\\cal M}^{\\mathbb{M} \\mathbb{N}} = \\Omega^{\\mathbb{M} \\mathbb{P}} \\Omega^{\\mathbb{N} \\mathbb{Q}} {\\cal M}\\_{\\mathbb{P} \\mathbb{Q}} $, with ΩMN the Sp(56, R)-invariant matrix.\nFrom, we obtain the following duality relations for the $\\bm{56}$, $\\bm{133}$ and $\\bm{912}$ E7(7)-covariant two-, three- and four-form field strengths,\n \n$$\\begin{aligned}\n \\label{DualityH2}\n\\tilde{{{\\cal H}}}\\_{\\2 \\Lambda} &=& {\\cal R}\\_{\\Lambda \\Sigma} \\, {\\cal H}\\_\\2^\\Sigma + {\\cal I}\\_{\\Lambda \\Sigma} \\, \\*{\\cal H}\\_\\2^\\Sigma \\;, \\\\[5pt]\n\\label{DualityH3}\n{{\\cal H}}\\_{\\3 \\alpha} &=& \\tfrac{1}{12} \\, (t\\_\\alpha)\\_{\\mathbb{M}}{}^\\mathbb{P} \\, {{\\cal M}}\\_{\\mathbb{N}\\mathbb{P}} \\*D {{\\cal M}}^{\\mathbb{M} \\mathbb{N}}\\;, \\\\[5pt]\n\\label{DualityH4}\n{{{\\cal H}}\\_{\\4 \\alpha}}^\\mathbb{M} &=& - \\tfrac{1}{84} \\, (t\\_\\alpha)\\_{\\mathbb{P}}{}^{\\mathbb{R}} {X\\_{\\mathbb{NQ}}}^{\\mathbb{S}} \\mathcal{M}^{\\mathbb{MN}} \\Big( \\mathcal{M}^{\\mathbb{PQ}} \\mathcal{M}\\_{\\mathbb{RS}} + 7 \\, \\delta^\\mathbb{P}\\_\\mathbb{S} \\, \\delta^\\mathbb{Q}\\_\\mathbb{R} \\Big) \\, \\textrm{vol}\\_4 \\;.\\end{aligned}$$ \nHere, *X*MNP ≡ ΘM*α* (*t**α*)NP , see ([ET]), is the contraction of the ISO(7) embedding tensor ([Theta-compSL7]) with the generators (*t**α*)NP of E7(7) in the fundamental representation, see ([Gener63]), ([Gener70]). The duality relations for the restricted field content ([eq:SL7fieldcontent4D]) simply follow from ([DualityH2])–([DualityH4]) by branching the adjoint SL(8) index on the vectors as in ([vectors56]), and restricting the E7(7) generators to only those of $\\textrm{SL}(7) \\ltimes \\mathbb{R}^7$:\n \n$$\\begin{aligned}\n\\label{H2IJDuality}\n\\tilde{{{\\cal H}}}\\_{\\2 IJ} &=& \\tfrac{1}{2} {\\cal I}\\_{[IJ][KL]} \\, \\*{\\cal H}\\_\\2^{KL} + {\\cal I}\\_{[IJ][K8]} \\, \\*{\\cal H}\\_\\2^{K} + \\tfrac{1}{2} {\\cal R}\\_{[IJ][KL]} \\, {\\cal H}\\_\\2^{KL} + {\\cal R}\\_{[IJ][K8]} \\, {\\cal H}\\_\\2^{K} \\, \\\\[7pt]\n\\label{H2IDuality}\n\\tilde{{{\\cal H}}}\\_{\\2 I} &=& \\tfrac{1}{2} {\\cal I}\\_{[I8][KL]} \\, \\*{\\cal H}\\_\\2^{KL} + \\, {\\cal I}\\_{[I8][K8]} \\, \\*{\\cal H}\\_\\2^{K} + \\tfrac{1}{2} {\\cal R}\\_{[I8][KL]} \\, {\\cal H}\\_\\2^{KL} + {\\cal R}\\_{[I8][K8]} \\, {\\cal H}\\_\\2^{K} \\, \\\\[7pt]\n\\label{H3IJDuality}\n{{\\cal H}}\\_{\\3 I}{}^J &=& \\tfrac{1}{12} (t\\_I{}^J)\\_{\\mathbb{M}}{}^\\mathbb{P} \\, {{\\cal M}}\\_{\\mathbb{N}\\mathbb{P}} \\*D {{\\cal M}}^{\\mathbb{M} \\mathbb{N}} - \\tfrac17 \\, \\delta\\_I^J \\, (\\textrm{trace}) \\, \\\\[7pt]\n\\label{H3IDuality}\n{{\\cal H}}\\_{\\3 }{}^I &=& \\tfrac{1}{12} (t\\_8{}^I)\\_{\\mathbb{M}}{}^\\mathbb{P} \\, {{\\cal M}}\\_{\\mathbb{N}\\mathbb{P}} \\*D {{\\cal M}}^{\\mathbb{M} \\mathbb{N}} \\, \\\\[7pt]\n\\label{H4Duality}\n{{\\cal H}}\\_{\\4}^{IJ} &=& \\tfrac{1}{84} {X\\_{\\mathbb{NQ}}}^{\\mathbb{S}} \\big( (t\\_K{}^{(I|})\\_{\\mathbb{P}}{}^{\\mathbb{R}} \\mathcal{M}^{|J)K \\, \\mathbb{N}} + (t\\_8{}^{(I|})\\_{\\mathbb{P}}{}^{\\mathbb{R}} \\mathcal{M}^{|J)8 \\, \\mathbb{N}} \\big) \\big( \\mathcal{M}^{\\mathbb{PQ}} \\mathcal{M}\\_{\\mathbb{RS}} + 7 \\, \\delta^\\mathbb{P}\\_\\mathbb{S} \\, \\delta^\\mathbb{Q}\\_\\mathbb{R} \\big) \\textrm{vol}\\_4 \\. \\,\\, \\qquad \\end{aligned}$$ \nIn ([H4Duality]), only components ${{\\cal M}}^{\\Lambda \\mathbb{N}}$ in the notation of ([Mscalar]), and not ${{\\cal M}}\\_\\Lambda{}^{ \\mathbb{N}}$, are contracted with the $\\textrm{SL}(7) \\ltimes \\mathbb{R}^7$ generators. The combination of these duality relations with the Bianchi identities ([eq:TruncBianchis]) reproduces a subset of the equations of motion: see section [ScalarCriticalPoints].\nExtensions of the duality hierarchy ([eq:SL7fieldcontent4D]) may be considered that are still smaller than the full E7(7) hierarchy. A natural extension includes, besides the $\\bm{28}^\\prime$ ${{\\cal C}}^{IJ}$ three-form potentials in ([eq:SL7fieldcontent4D]) conjugate to the electric embedding tensor, also the SL(7)-singlet three-form potential $\\tilde{{{\\cal C}}}$ conjugate to the singlet magnetic component of the embedding tensor. Consistency then requires that the singlet two-form potential $\\,{{\\cal B}}\\,$ that renders ${{\\cal B}}\\_I{}^J$ traceful is also retained. The extension of the Bianchi identities ([eq:TruncBianchis]) to also include these singlets reads\n \n$$\\label{DH\\_singlets}\n\\begin{array}{lll}\nD {\\cal H}\\_{\\3} &=& {{\\cal H}}\\_\\2^{IJ} \\wedge \\tilde{{{\\cal H}}}\\_{\\2 IJ} + {{\\cal H}}\\_\\2^{I} \\wedge \\tilde{{{\\cal H}}}\\_{\\2 I} -2 g\\, \\delta\\_{IJ} \\, {{\\cal H}}\\_\\4^{IJ} - 14 \\, m \\, \\tilde{{{\\cal H}}}\\_{\\4} \\, \\\\[2mm]\nD {\\tilde{{{\\cal H}}}}\\_{\\4} & \\equiv & 0 \\,\n\\end{array}$$ \nwhile their duality relations are, from ([DualityH3]) and ([DualityH4]),\n \n$$\\label{H4Duality\\_singlet}\n\\begin{array}{lll}\n{{\\cal H}}\\_{\\3} &=& -\\tfrac{1}{12} (t\\_8{}^8)\\_{\\mathbb{M}}{}^\\mathbb{P} \\, {{\\cal M}}\\_{\\mathbb{N}\\mathbb{P}} \\*D {{\\cal M}}^{\\mathbb{M} \\mathbb{N}} \\,\\\\[2mm]\n\\tilde{{{\\cal H}}}\\_{\\4} &=& \\tfrac{1}{84} \\, {X\\_{\\mathbb{NQ}}}^{\\mathbb{S}} (t\\_8{}^{K})\\_{\\mathbb{P}}{}^{\\mathbb{R}} {\\mathcal{M}\\_{8K}}^{\\mathbb{N}} \\mathcal{M}^{\\mathbb{PQ}} \\mathcal{M}\\_{\\mathbb{RS}} \\, \\textrm{vol}\\_4 \\.\n\\end{array}$$ \nWe have used *t**I**I* = − *t*88 and Tr(*t**I**J* *t*8*K*) = Tr(*t*8*J* *t*8*K*) = 0 to simplify the results. For $\\,\\tilde{{{\\cal H}}}\\_{\\4}\\,$ in ([H4Dualitysinglet]), components ${{\\cal M}}\\_\\Lambda{}^{ \\mathbb{N}}$, and not ${{\\cal M}}^{\\Lambda \\mathbb{N}}$, in the notation of ([Mscalar]), are now contracted with the R7 generators, opposite to what happened for ${{\\cal H}}\\_{\\4}^{IJ}$ in ([H4Duality]). Although the singlet $\\,\\tilde{{{\\cal C}}}\\,$ does not play a role in the restricted duality hierarchy ([eq:SL7fieldcontent4D]), its dualised field strength $\\tilde{{{\\cal H}}}\\_{\\4}$ in ([H4Dualitysinglet]) is still crucial to recover the scalar potential, as we will show in the next subsection. The significance of this asymmetric role of $\\,\\tcC\\,$ for the massive type IIA embedding of dyonic ISO(7) supergravity will be discussed in.\nBosonic Lagrangian\n------------------\nWe will now write the Lagrangian of ${{\\cal N}}=8$ dyonically gauged ISO(7) supergravity, focusing on the bosonic terms. While it is possible to write a Lagrangian that includes higher rank fields in the E7(7) tensor hierarchy (or in the restricted hierarchy ([eq:SL7fieldcontent4D])) supplemented by duality relations, we will instead write a Lagrangian in the formulation of. The latter includes, besides the metric and scalars, only some of the vectors and two-forms in ([eq:SL7fieldcontent4D]). More concretely, the Lagrangian can be expressed in terms of the $\\bm{21}^\\prime + \\bm{7}^\\prime$ electric vectors $ {{\\cal A}}^\\Lambda = ( \\, {{\\cal A}}^{IJ}, {{\\cal A}}^I \\, ) $ and their field strengths $\\,{{\\cal H}}\\_{\\2}^{\\Lambda} = ( \\, {{\\cal H}}\\_{\\2}^{IJ} \\,, \\, {{\\cal H}}\\_{\\2}^{I}\\, )\\,$, the $\\bm{7}$ magnetic vectors $\\tilde{{{\\cal A}}}\\_I$ and their field strengths $\\tilde{{{\\cal H}}}\\_{\\2 I}$, and the $\\bm{7}^\\prime$ two-form potentials ${{\\cal B}}^I$.\nThe bosonic Lagrangian of ${{\\cal N}}=8$ dyonically gauged ISO(7) supergravity is\n \n$$\\begin{aligned}\n\\label{BosLag}\n{\\cal L} &=& R \\, \\textrm{vol}\\_4 -\\tfrac{1}{48} D {\\cal M}\\_{\\mathbb{M}\\mathbb{N}} \\wedge \\* D {\\cal M}^{\\mathbb{M}\\mathbb{N}} +\\tfrac12 \\, {\\cal I}\\_{\\Lambda \\Sigma} \\, {{\\cal H}}\\_{(2)}^\\Lambda \\wedge \\* {{\\cal H}}\\_{(2)}^\\Sigma + \\tfrac12 \\, \\mathcal{R}\\_{\\Lambda \\Sigma} \\, {{\\cal H}}\\_{(2)}^\\Lambda \\wedge {{\\cal H}}\\_{(2)}^\\Sigma \\\\[4pt]\n&& - V \\textrm{vol}\\_4 - m \\left[ {{\\cal B}}^I \\wedge \\big( \\tilde{\\cal H}\\_{\\2 I} - \\tfrac{g}{2} \\delta\\_{IJ} {{\\cal B}}^{J} \\big) \n- \\tfrac14 \\, \\tilde {{\\cal A}}\\_I \\wedge \\tilde {{\\cal A}}\\_J \\wedge \\big( d {{\\cal A}}^{IJ} + \\tfrac{g}{2} \\, \\delta\\_{KL} \\, {{\\cal A}}^{IK} \\wedge {{\\cal A}}^{JL} \\big) \\right]. \\nonumber \\end{aligned}$$ \nSee appendix [app:ConstN=8] for some details of its derivation. The second line of ([BosLag]) is entirely due to the ISO(7) gauging. It contains, on the one hand, a scalar potential,\n \n$$\\begin{array}{ccl}\n\\label{V\\_generalRewrite}\nV & = & \\dfrac{g^{2}}{168} \\, {X\\_{\\mathbb{MP}}}^{\\mathbb{R}} {X\\_{\\mathbb{NQ}}}^{\\mathbb{S}} \\mathcal{M}^{\\mathbb{MN}} \\Big( \\mathcal{M}^{\\mathbb{PQ}} \\mathcal{M}\\_{\\mathbb{RS}} + 7 \\, \\delta^\\mathbb{P}\\_\\mathbb{S} \\, \\delta^\\mathbb{Q}\\_\\mathbb{R} \\Big) \\,\n\\end{array}$$ \nwith the *X*-tensor in ([ET]) particularised for the ISO(7)*c* embedding tensor ([Theta-compSL7]). Upon using ([mcg]), this scalar potential contains pieces in *g*2, *g**m* and *m*2. On the other hand, the second line of ([BosLag]) contains some topological terms whenever *m* ≠ 0. Note, in particular, the topological mass $ gm \\, \\delta\\_{IJ} \\, {{\\cal B}}^I \\wedge {{\\cal B}}^J$, which generalises a similar term in ${{{\\cal N}}=2}$ compactifications of massive type IIA on Calabi-Yau. In the first line, the only contributions from the gauging appear in the covariant derivatives ([DISO7]) and the gauging-modified field strengths of the electric vectors given in ([eqO:2FormFieldStrengths]). In the SL(7) symplectic frame we are using, the scalar-dependent matrices ${{\\cal M}}$, R and I given in ([Mscalar]) are independent of the gauging and, in particular, of the dyonically-gauging parameter *c* = *m*/*g*.\nThe generic $\\bm{912}$ four-form field strengths ([DualityH4]) and the scalar potential ([VgeneralRewrite]) are related through the embedding tensor via\n \n$$\\begin{aligned}\n\\label{H4/Potential1}\n\\Theta\\_\\mathbb{M}{}^\\alpha \\, { {{\\cal H}}\\_{\\4 \\alpha}}^\\mathbb{M} = - 2 \\, V \\, \\textrm{vol}\\_4 \\;.\\end{aligned}$$ \nCombining ([H4Duality]) and the second equation in ([H4Dualitysinglet]), it is easy to show that this relation simplifies for the ISO(7)*c* gauging to\n \n$$\\begin{aligned}\n\\label{H4/Potential2}\n g \\, \\delta\\_{IJ} \\, {{\\cal H}}\\_{\\4}^{IJ} + m \\, \\tilde{{{\\cal H}}}\\_{\\4} = - 2 \\, V \\, \\textrm{vol}\\_4 \\;. \\; \\end{aligned}$$ \nIn particular, the dualisation of both four-forms ${{\\cal H}}\\_{\\4}^{IJ}$ and $\\tilde{{{\\cal H}}}\\_{\\4} $ contains terms linear in *g* and *m*; only when combined through ([H4/Potential2]) is the quadratic dependence of *V* on *g* and *m* reproduced.\nThe theory ([BosLag]) with ([DISO7]), ([eqO:2FormFieldStrengths]) admits three different smooth limits of the coupling constants *g* and *m*. In the limit *m* → 0, *g* ≠ 0, Hull’s purely electric ISO(7) gauging is recovered. This theory arises from consistent truncation of massless type IIA supergravity on *S*6. The limit *g* → 0, *m* ≠ 0 corresponds to a purely magnetic gauging of a nilpotent extension of U(1)6 × R with 21 non-compact central charges. This theory arises as a *T*6 truncation of massive type IIA. Finally, the *g* → 0, *m* → 0 limit yields the ungauged ${{\\cal N}}=8$ supergravity, which is well known to arise from *D* = 11 supergravity on *T*7, or massless type IIA on *T*6.\nSO(7)-covariant critical point conditions\n-----------------------------------------\nThe combination of the duality relations with the Bianchi identities of the ${{\\cal N}}=8$ tensor hierarchy gives rise to the vector equations of motion and (projections of) the scalar equations of motion. In the restricted duality hierarchy ([eq:SL7fieldcontent4D]), all vectors were retained. Accordingly, the duality conditions ([H2IJDuality]), ([H2IDuality]) reproduce all of the vector equations of motion, as derived from the Lagrangian ([BosLag]), upon substitution into (the first line of) the Bianchi identities ([eq:TruncBianchis]). In contrast, not all of the three-form field strengths of the full hierarchy were retained in ([eq:SL7fieldcontent4D]). Thus, it is interesting to enquire to which scalar equations of motion are their Bianchi identities related to when combined with the dualisation conditions. As we will now show, these are related to the equations of motion of the proper (parity even) scalars of E7(7)/SU(8). We will focus on maximally symmetric solutions for which the scalar equations of motion reduce to the extremisation conditions for the scalar potential *V*.\nFor this particular discussion, we will incorporate the singlet three-form ${{\\cal H}}\\_\\3$ in ([DHsinglets]), ([H4Dualitysinglet]) along with the three-forms ${{\\cal H}}\\_{\\3 I}{}^J$, ${{\\cal H}}\\_{\\3}^I$ of the restricted duality hierarchy ([eq:SL7fieldcontent4D]). Substituting the duality relations ([H3IJDuality])–([H4Duality]), ([H4Dualitysinglet]) into the Bianchi identities ([eq:TruncBianchis]), ([DHsinglets]), we obtain a set of $ \\bm{1} + \\bm{48} + \\bm{7}^\\prime$ equations, in representations of SL(7). From the discussion of adapted to our context, these correspond to the projections to the generators of $\\textrm{GL}(7) \\ltimes \\mathbb{R}^7 \\subset \\textrm{E}\\_{7(7)}$ of the equations of motion of the E7(7)/SU(8) scalars. Further branching into representations of SO(7) and restricting to zero tensors and constant scalars (thus, critical points of *V*), these projections become\n \n$$\\begin{aligned}\n\\label{Extreme1}\n\\bm{1} : & & \\,\\,\\, \\big( g \\, \\delta\\_{IJ} \\, {{\\cal H}}\\_{\\4}^{IJ} + 7 \\, m \\, \\tilde{{{\\cal H}}}\\_{\\4} \\, \\big)|\\_{0} = 0 \\;, \\\\[6pt]\n\\label{Extreme27}\n\\bm{27}: & & \\big( {{\\cal H}}\\_\\4^{IJ} - \\tfrac17 \\, \\delta^{IJ} \\, \\delta\\_{KL} \\, {{\\cal H}}\\_\\4^{KL} \\big) |\\_{0} = 0 \\;, \\\\[6pt]\n\\label{Extreme21}\n\\bm{21}: & & \\textrm{identically zero} \\;, \\\\[6pt]\n\\label{Extreme7}\n\\bm{7}: & & \\textrm{identically zero} \\;,\\end{aligned}$$ \nwhere ∣0 denotes evaluation at a critical point of *V*. In these equations, we have used the four-form field strengths $ {{\\cal H}}\\_{\\4}^{IJ}$ and $\\tilde{{{\\cal H}}}\\_{\\4} $ as shorthand for the scalar functions on the r.h.s. of the duality relations ([H4Duality]), ([H4Dualitysinglet]). Equations ([Extreme21]), ([Extreme7]) correspond to projections to the $\\bm{21} + \\bm{7}$ generators of the gauge group ISO(7). They turn out to be identically zero, in agreement with the scalar potential being invariant under the gauge group.\nAlthough originally obtained as projections, the SO(7)-covariant equations ([Extreme1])–([Extreme7]) are in fact in one-to-one correspondence with extremisation conditions with respect to definite scalars. The singlet equation ([Extreme1]) corresponds[3](#fn3) to the extremisation condition with respect to the SO(1, 1) dilaton that extends SL(7) into GL(7). Equation ([Extreme27]), in the symmetric traceless of SO(7), corresponds to the extremisation of the potential with respect to the 27 scalars of SL(7)/SO(7). The $\\bm{7}$ scalars of R7 ⊂ ISO(7) are Stückelberg and therefore do not enter the scalar potential, hence they do not give rise to extremisation conditions. Put together, equations ([Extreme1]), ([Extreme27]), ([Extreme7]) thus correspond to the conditions of extremisation of the potential *V* with respect to the 35 (parity even) scalars[4](#fn4) of $\\textrm{GL}(7) \\ltimes \\mathbb{R}^7 / \\textrm{SO}(7) \\subset \\textrm{E}\\_{7(7)}/\\textrm{SU}(8)$. Note, incidentally, that these equations also depend typically on the (parity odd) pseudoscalars. Finally, the $\\bm{21}$ identities ([Extreme21]) can be reinterpreted as being trivial in that the corresponding compact SO(7) scalars have been modded out from the coset E7(7)/SU(8).\nSupersymmetry transformations\n-----------------------------\nWe conclude our characterisation of ISO(7)-dyonically-gauged ${{\\cal N}}=8$ supergravity with the supersymmetry transformations. The only effects of the gauging on the supersymmetry variations of the ungauged theory occur in the fermion variations, through the gauging-modified field strengths of the vectors and new (‘shift’) scalar-dependent terms. The supersymmetry variations of the bosons are the same in gauged and in ungauged supergravity. We will nevertheless find it useful to spell out the supersymmetry variations of the SL(7)-covariant bosonic fields in ([eq:SL7fieldcontent4D]) to show that they only involve fields within the same set. See for the ${{\\cal N}}=8$ supersymmetry transformations of the fermions.\nThe ${{\\cal N}}=8$ supersymmetry transformations for the vectors *A*M are linear in the scalar coset representative[5](#fn5) $\\,{\\cal V}\\_\\mathbb{M}{}^{ij}\\,$, which also sits in the $\\bm{56}$ of E7(7). The transformations of the $\\bm{133}$ two-form potentials ${{\\cal B}}\\_\\alpha$ were worked out in, where they were shown to be quadratic in the coset representative. In order to write the variations of the $\\bm{48} + \\bm{7}^\\prime$ two-forms in ([eq:SL7fieldcontent4D]) we will only have to branch the result of accordingly and select these SL(7) representations. The ${{\\cal N}}=8$ supersymmetry variations of the $\\bm{912}$ three-forms ${{\\cal C}}\\_{\\alpha}{}^{\\mathbb{M}}$ have not appeared in the literature[6](#fn6). We conjecture these variations to be\n \n$$\\begin{aligned}\n \\label{eq:susyThreeForms}\n&& \\delta {{\\cal C}}\\_{\\mu \\nu \\rho \\, \\alpha}{}^\\mathbb{M} = (t\\_\\alpha)\\_\\mathbb{R}{}^\\mathbb{P} \\, \\Omega^{\\mathbb{R} \\mathbb{Q}} \\, \\Omega^{\\mathbb{M} \\mathbb{N}} \\Big( \\tfrac{4i}{7} {\\cal V}\\_{\\mathbb{N} \\, jl} \\, {\\cal V}\\_{\\mathbb{P}}{}^{lk} \\, {\\cal V}\\_{\\mathbb{Q} \\, ik} \\, \\bar{\\epsilon}^i \\, \\gamma\\_{[\\mu \\nu} \\, \\psi\\_{\\rho]}^j \n -i \\tfrac{\\sqrt{2}}{ 6} \\, {\\cal V}\\_{\\mathbb{N}}{}^{hi} \\, {\\cal V}\\_{\\mathbb{P} \\, [ij|} \\, {\\cal V}\\_{\\mathbb{Q} \\, | kl]} \\, \\bar{\\epsilon}\\_h \\, \\gamma\\_{\\mu \\nu \\rho} \\, \\chi^{jkl} \\nonumber \\\\[4pt]\n&& \\qquad \\qquad\\qquad\\qquad \\qquad \\qquad \\;+ \\textrm{h.c.} \\Big) +3 \\, {{\\cal B}}\\_{[\\mu \\nu| \\alpha} \\, \\delta {{\\cal A}}^\\mathbb{M}\\_{|\\rho]}\n- \\, (t\\_\\alpha)\\_\\mathbb{R}{}^\\mathbb{P} \\, \\Omega\\_{\\mathbb{P} \\mathbb{N}} \\, {{\\cal A}}^\\mathbb{M}\\_{[\\mu} \\, {{\\cal A}}^\\mathbb{R}\\_{\\nu} \\, \\delta {{\\cal A}}^\\mathbb{N}\\_{\\rho]} \\;,\\end{aligned}$$ \nup to a possible symmetrisation in *ε̄*(*i* *γ*[*μ**ν* *ψ**ρ*]*j*). As usual, *γ**μ*1…*μ**p* = *e**μ*1*α*1…*e**μ**p**α**p**γ**α*1…*α**p*. This conjecture passes several consistency checks. Being cubic in the coset representative, ([eq:susyThreeForms]) follows the pattern of the variations of the vectors and two-forms. The terms in ${{\\cal B}}\\_{[\\mu \\nu| \\alpha} \\, \\delta {{\\cal A}}^\\mathbb{M}\\_{|\\rho]}$ and ${{\\cal A}}^\\mathbb{M}\\_{[\\mu} \\, {{\\cal A}}^\\mathbb{R}\\_{\\nu} \\, \\delta {{\\cal A}}^\\mathbb{N}\\_{\\rho]}$ match the corresponding terms of the ${{\\cal N}}=1$ and ${{\\cal N}}=2$ three-form transformations. The truncation of ([eq:susyThreeForms]) to one of the singlets in the decomposition ([eq:Branchings]) of the $\\bm{912}$ under SL(7) coincides with the supersymmetry variation of the type IIA Ramond-Ramond three-form in the SO(1, 3) × SL(7)-covariant reformulation of type IIA supergravity of. Here, we will instead be interested in the $\\bm{28}^\\prime$ SL(7) components of ([eq:susyThreeForms]). As we will show in, this too can be reproduced from consistent truncation of massive type IIA on *S*6.\nWe can therefore specify the supersymmetry variations of the bosonic fields ([eq:SL7fieldcontent4D]) in our conventions as follows. The vielbein and scalar coset representatives transform as\n \n$$\\label{SUSY-Vielbeine}\n\\begin{array}{llll}\n\\delta {e\\_{\\mu}}^{\\alpha} &=& \\tfrac{1}{4} \\, \\, \\bar{\\epsilon}\\_{i} \\, \\gamma^{\\alpha} \\, {\\psi\\_{\\mu}}^{i} + \\tfrac{1}{4} \\, \\bar{\\epsilon}^{i} \\, \\gamma^{\\alpha} \\, \\psi\\_{\\mu i} &, \\\\[3mm]\n\\delta {\\mathcal{V}\\_{\\mathbb{M}}}^{ij} &=& \\tfrac{1}{ \\sqrt{2}} \\, \\mathcal{V}\\_{\\mathbb{M} \\, kl} \\, \\left( \\bar{\\epsilon}^{[i} \\, \\chi^{jkl]} \\, + \\, \\tfrac{1}{4!} \\, \\varepsilon^{ijklmnpq} \\, \\bar{\\epsilon}\\_{m} \\, \\chi\\_{npq} \\right) &, \n\\end{array}$$ \nwith the fundamental E7(7) index M on the coset representative branched out into SL(7) representations according to ([eq:Branchings]). For reference from, we do branch out the supersymmetry variations of the vectors under SL(7):\n \n$$\\label{VectorsBranchingSL7}\n\\begin{array}{llrl}\n \\delta {{\\cal A}}\\_\\mu{}^{IJ} &=& i\\, {{\\cal V}}^{IJ}{}\\_{ij} \\left( \\, \\bar{\\epsilon}^i \\psi\\_\\mu{}^j + \\tfrac{1}{2\\sqrt{2}} \\, \\bar{\\epsilon}\\_k \\gamma\\_\\mu \\chi^{ijk}\\right) + \\textrm{h.c.} &, \\\\[3mm]\n\\delta {{\\cal A}}\\_\\mu{}^{I} &=& i\\, {{\\cal V}}^{I8}{}\\_{ij} \\left( \\, \\bar{\\epsilon}^i \\psi\\_\\mu{}^j + \\tfrac{1}{2\\sqrt{2}} \\, \\bar{\\epsilon}\\_k \\gamma\\_\\mu \\chi^{ijk}\\right) + \\textrm{h.c.} &, \\\\[3mm]\n\\delta \\tilde{{{\\cal A}}}\\_{\\mu \\, IJ} &=& -i\\, \\tilde{{{\\cal V}}}\\_{IJ \\, ij} \\left( \\, \\bar{\\epsilon}^i \\psi\\_\\mu{}^j + \\tfrac{1}{2\\sqrt{2}} \\, \\bar{\\epsilon}\\_k \\gamma\\_\\mu \\chi^{ijk}\\right) + \\textrm{h.c.} &, \\\\[3mm]\n\\delta \\tilde{{{\\cal A}}}\\_{\\mu \\, I} &=& -i\\, \\tilde{{{\\cal V}}}\\_{I8 \\, ij} \\left( \\, \\bar{\\epsilon}^i \\psi\\_\\mu{}^j + \\tfrac{1}{2\\sqrt{2}} \\, \\bar{\\epsilon}\\_k \\gamma\\_\\mu \\chi^{ijk}\\right) + \\textrm{h.c.} &.\n\\end{array}$$ \nThe supersymmetry transformations of the $ \\bm{48} + \\bm{7}^\\prime $ two-forms read\n \n$$\\begin{aligned}\n \\label{susytensors4dSL7bis}\n\\delta {{\\cal B}}\\_{\\mu \\nu \\, J}{}^I & = & \\Big[ - \\tfrac{2}{3} \\big( {\\cal V}^{IK}{}\\_{jk} \\, \\tilde{{\\cal V}}\\_{JK}{}^{ik} + {\\cal V}^{I8}{}\\_{jk} \\, \\tilde{{\\cal V}}\\_{J8}{}^{ik} + \\tilde{{\\cal V}}\\_{JK \\, jk}{} \\, {\\cal V}^{IK}{}^{ik} + \\tilde{{\\cal V}}\\_{J8 \\, jk}{} \\, {\\cal V}^{I8}{}^{ik} \\big) \\bar{\\epsilon}\\_{i} \\gamma\\_{[\\mu} \\psi\\_{\\nu]}^j \\nonumber \\\\\n&& \\quad - \\tfrac{\\sqrt{2}}{3} \\, \\big( {\\cal V}^{IK}{}\\_{ij} \\, \\tilde{{\\cal V}}\\_{JK \\, kl} + {\\cal V}^{I8}{}\\_{ij} \\, \\tilde{{\\cal V}}\\_{J8 \\, kl} \\big) \\, \\bar{\\epsilon}^{[i} \\gamma\\_{\\mu \\nu} \\chi^{jkl]} +\\textrm{h.c.} \\Big] \\nonumber \\\\\n&& \\quad + \\big( {{\\cal A}}\\_{[\\mu}^{IK} \\, \\delta \\tilde{{{\\cal A}}}\\_{\\nu] JK} + {{\\cal A}}\\_{[\\mu}^{I} \\, \\delta \\tilde{{{\\cal A}}}\\_{\\nu] J} + \\tilde{{{\\cal A}}}\\_{[\\mu| \\, JK} \\, \\delta {{\\cal A}}\\_{|\\nu]}{}^{IK}+ \\tilde{{{\\cal A}}}\\_{[\\mu| \\, J} \\, \\delta {{\\cal A}}\\_{|\\nu]}{}^{I} \\big) \n-\\tfrac17 \\, \\delta\\_J^I \\, (\\textrm{trace}) \\;, \\nonumber \\\\[12pt]\n\\delta {{\\cal B}}\\_{\\mu \\nu}{}^I & = & \\Big[ \\tfrac{2}{3} \\big( {\\cal V}^{IJ}{}\\_{jk} \\, \\tilde{{\\cal V}}\\_{J8}{}^{ik} + \\tilde{{\\cal V}}\\_{J8 \\, jk}{} \\, {\\cal V}^{IJ}{}^{ik} \\big) \\, \\bar{\\epsilon}\\_{i} \\gamma\\_{[\\mu} \\psi\\_{\\nu]}^j + \\tfrac{\\sqrt{2}}{3} \\, {\\cal V}^{IJ}{}\\_{ij} \\, \\tilde{{\\cal V}}\\_{J8 \\, kl} \\, \\bar{\\epsilon}^{[i} \\gamma\\_{\\mu \\nu} \\chi^{jkl]} +\\textrm{h.c.} \\Big] \\nonumber \\\\\n&& \\quad - \\big( {{\\cal A}}\\_{[\\mu}^{IJ} \\, \\delta \\tilde{{{\\cal A}}}\\_{\\nu] J} + \\tilde{{{\\cal A}}}\\_{[\\mu| \\, J} \\, \\delta {{\\cal A}}\\_{|\\nu]}{}^{IJ} \\big) \\;. \\end{aligned}$$ \nNote the same pattern of SL(7) indices in coset and vector contributions: ${{\\cal V}}^{IK} \\, \\tilde{{{\\cal V}}}\\_{JK}$ parallels ${{\\cal A}}^{IK} \\, \\delta \\tilde {{\\cal A}}\\_{JK}$, etc. Finally, the variations that follow from ([eq:susyThreeForms]) for the $\\bm{28}^\\prime$ three-forms are\n \n$$\\begin{aligned}\n \\label{susy3forms4dSL7bis}\n\\delta {{\\cal C}}\\_{\\mu \\nu \\rho}{}^{IJ} & = & \\Big[ - \\tfrac{4i}{7} \\,\\Big( \n {{\\cal V}}^{K(I}{}\\_{jl} \\big( {{\\cal V}}^{J)L \\, lk} \\, \\tilde {{\\cal V}}\\_{KL \\, ik} + \\tilde {{\\cal V}}\\_{ KL}{}^{ lk} \\, {{\\cal V}}^{J)L}{}\\_{ ik} \\big) \\nonumber \\\\\n && \\qquad \\quad + \\, {{\\cal V}}^{K(I}{}\\_{jl} \\big( {{\\cal V}}^{J)8 \\, lk} \\, \\tilde {{\\cal V}}\\_{K8 \\, ik} + \\tilde {{\\cal V}}\\_{K8}{}^{ lk} \\, {{\\cal V}}^{J)8}{}\\_{ ik} \\big) \\nonumber \\\\\n&& \\qquad \\quad + \\, {{\\cal V}}^{(I|8}{}\\_{jl} \\big( {{\\cal V}}^{|J)K \\, lk} \\, \\tilde {{\\cal V}}\\_{K8 \\, ik} + \\tilde {{\\cal V}}\\_{K8}{}^{ lk} \\, {{\\cal V}}^{|J)K}{}\\_{ik} \\big) \\Big) \\, \\bar{\\epsilon}^{i} \\gamma\\_{[\\mu \\nu} \\psi\\_{\\rho]}^j \\nonumber \\\\\n&& + \\, i \\tfrac{\\sqrt{2}}{3} \\,\\Big( \n {{\\cal V}}^{K(I|}{}^{hi} \\, {{\\cal V}}^{|J)L}{}\\_{[ij|} \\, \\tilde {{\\cal V}}\\_{KL}{}\\_{|kl]} + \\, {{\\cal V}}^{K(I}{}^{hi} \\, {{\\cal V}}^{J)8}{}\\_{[ij|} \\, \\tilde {{\\cal V}}\\_{K8}{}\\_{|kl]} \\\\ \n&& \\qquad \\quad + \\, {{\\cal V}}^{(I|8}{}^{hi} \\, {{\\cal V}}^{|J)K}{}\\_{[ij|} \\, \\tilde {{\\cal V}}\\_{K8}{}\\_{|kl]} \\Big) \\, \\bar{\\epsilon}\\_{h} \\gamma\\_{\\mu \\nu \\rho } \\chi^{jkl} \\; + \\textrm{h.c.} \\Big] \\nonumber \\\\\n && - \\, 3 \\, \\Big( {{\\cal B}}\\_{[\\mu \\nu| K}{}^{(I} \\, \\delta {{\\cal A}}^{J)K}\\_{|\\rho]} + {{\\cal B}}\\_{[\\mu \\nu}{}^{(I} \\, \\delta {{\\cal A}}^{J)}\\_{\\rho]} \\Big) \\nonumber \\\\\n&& + \\, {{\\cal A}}^{K(I}\\_{[\\mu} \\big( {{\\cal A}}^{J)L}\\_\\nu \\, \\delta \\tilde {{\\cal A}}\\_{\\rho] KL} + \\tilde {{\\cal A}}\\_{\\nu KL} \\, \\delta {{\\cal A}}^{J)L}\\_{\\rho]} \\big) + {{\\cal A}}^{K(I}\\_{[\\mu} \\big( {{\\cal A}}^{J)}\\_\\nu \\, \\delta \\tilde {{\\cal A}}\\_{\\rho] K} + \\tilde {{\\cal A}}\\_{\\nu K} \\, \\delta {{\\cal A}}^{J)}\\_{\\rho]} \\big) \\nonumber \\\\\n&& \\qquad \\quad + \\, {{\\cal A}}^{(I}\\_{[\\mu} \\big( {{\\cal A}}^{J)K}\\_\\nu \\, \\delta \\tilde {{\\cal A}}\\_{\\rho] K} + \\tilde {{\\cal A}}\\_{\\nu K} \\, \\delta {{\\cal A}}^{J)K}\\_{\\rho]} \\big) \\;. \\nonumber\\end{aligned}$$ \nAgain, the SL(7) structure of indices in coset and vector contributions is the same.\nEquations ([SUSY-Vielbeine])–([susy3forms4dSL7bis]) show that the supersymmetry variations of the bosonic fields ([eq:SL7fieldcontent4D]) close among themselves and into the fermions *ψ**μ**i*, *χ**i**j**k*. In turn, the supersymmetry variations of the fermions close into scalars and field strengths of vectors, all of which were retained in ([eq:SL7fieldcontent4D]). This shows the consistency of the subsector ([eq:SL7fieldcontent4D]) of the duality hierarchy, plus fermions, at the level of the supersymmetry variations. Finally, the supersymmetry variations of the fields that enter the Lagrangian ([BosLag]) involve fields that appear as well in the Lagrangian. Note, however, that the fields entering the Lagrangian can still source the supersymmetry variations of fields not entering the Lagrangian, *e.g,* the terms $ {{\\cal A}}\\_{[\\mu}^{I} \\, \\delta \\tilde{{{\\cal A}}}\\_{\\nu] J} + \\tilde{{{\\cal A}}}\\_{[\\mu| \\, J} \\, \\delta {{\\cal A}}\\_{|\\nu]}{}^{I} -\\tfrac17 \\, \\delta\\_J^I \\, (\\textrm{trace})$ source the supersymmetry variation $\\delta {{\\cal B}}\\_{\\mu \\nu \\, J}{}^I$.\n[t!]\n[Table:grouptheory]\nAn N = 2 truncation: the SU(3)-invariant sector\n===============================================\nIn the remainder of the paper, we will specify the truncation of the ${{\\cal N}}=8$ theory to various interesting subsectors that preserve ${{\\cal N}}=2$ and ${{\\cal N}}=1$ supersymmetry and SU(3), G2, and an SO(4) subgroup of the ISO(7) bosonic gauge symmetry. See table [Table:grouptheory] for a summary of the field content of these subsectors, and of the SO(7) and SO(6) further subsectors of the SU(3) sector.\nWe begin by discussing the consistent truncation of the ${{\\cal N}}=8$ theory to its SU(3)-invariant sector. The analog truncation for the purely electric SO(8) gauging has been studied in and for the dyonic SO(8) gauging, in. This sector corresponds to $\\,{{\\cal N}}=2\\,$ supergravity coupled to one vector multiplet and one hypermultiplet. The corresponding 2 + 4 real scalars take values on a submanifold\n \n$$\\begin{aligned}\n\\label{ScalManN=2}\n\\frac{\\textrm{SU}(1,1)}{\\textrm{U}(1)} \\, \\times \\, \\frac{\\textrm{SU}(2,1)}{\\textrm{SU}(2) \\times \\textrm{U}(1)}\\end{aligned}$$ \nof E7(7)/SU(8) which is the product of two well known special Kähler (SK) and quaternionic Kähler (QK) manifolds. The gauging inherited in this sector from the ${{\\cal N}}=8$ ISO(7)*c* gauging is an abelian U(1) × SO(1, 1)*c* dyonic gauging in the hypersector. In section [subsec:construction] we construct the Lagrangian of this theory, including an explicit parameterisation for the scalar kinetic terms and potential, and discuss the duality hierarchy in section [subsec:SU3-hierarchy]. We then give a superpotential and the canonical ${{\\cal N}}=2$ formulations of this sector in sections [subsec:SuperPot] and [CanonicalN=2], respectively. Some further subsectors are discussed in [sec:subsecsSU3] and the vacuum structure is analysed in [subsec:CriticalPoints]. See also appendix [app:SU3sectorScalMat] for the explicit expression of the SU(3)-invariant scalar matrix ${{\\cal M}}\\_{\\mathbb{M}\\mathbb{N}}$, and appendix [app:MTheoryonSE7] for the relation of this sector to the similar model that arises from consistent truncation of M-theory on an arbitrary Sasaki-Einstein seven-manifold.\nConstruction and bosonic Lagrangian\n-----------------------------------\nThe embedding of the relevant SU(3) into SO(7) ⊂ ISO(7) can be described by the chain\n \nSO(7) ⊃ SO(6) ∼ SU(4) ⊃ SU(3), \nso that $\\,\\textbf{7} \\rightarrow \\textbf{1} \\,+\\, \\textbf{3} \\,+\\, \\bar{\\textbf{3}}\\,$. In terms of SL(8) indices, we have a splitting *A* → (*a* ⊕ 8) ⊕ (1 ⊕ *â*) with *a* = 2, 4, 6 and *â* = 3, 5, 7 , followed by a complexification of the form\n \n$$\\label{SU3\\_complexification}\nz\\_{0} = x\\_{1} + i \\, x\\_{8}\n\\hspace{3mm} \\textrm{, } \\hspace{3mm}\nz\\_{1} = x\\_{2} + i \\, x\\_{3}\n\\hspace{3mm} \\textrm{, } \\hspace{3mm}\nz\\_{2} = x\\_{4} + i \\, x\\_{5}\n\\hspace{3mm} \\textrm{, } \\hspace{3mm}\nz\\_{3} = x\\_{6} + i \\, x\\_{7} \\,$$ \nso that SU(3) is realised as a singlet (*z*0) and a triplet (*z*1, 2, 3) of complex coordinates. When restricted to this sector, the retained bosonic fields take values along the SU(3)-invariant metric[7](#fn7) *δ**i**j* , the two-form *J**i**j* , *i* = 2, …, 7 , and the complex totally antisymmetric tensor of SU(3) (or equivalently, a complex decomposable three-form Ω*i**j**k*, see appendix [app:SU3sectorScalMat]). In fact, only the scalar matrix $ {{\\cal M}}\\_{\\mathbb{MN}}$ has components along the latter. The fields in the ${{\\cal N}}=8$ duality hierarchy ([eq:SL7fieldcontent4D]) give rise to the following SU(3)-invariant fields:\n \n$$\\label{SU(3)\\_full\\_content}\n\\begin{array}{rrrrl}\n\\textrm{metric} & : & g\\_{\\mu\\nu} \\\\[3mm]\n\\textrm{scalars} & : & {{\\cal M}}\\_{\\mathbb{MN}} & \\rightarrow & (\\chi,\\varphi) \\,\\,\\textrm{ and }\\,\\, (\\phi,a,\\zeta,\\tilde{\\zeta}) \\,\\,\\,\\,\\,\\,\\,, \\,\\,\\,\\,\\,\\, \\textrm{[ see appendix~\\ref{app:SU3sectorScalMat} ]} \\\\[2mm]\n\\textrm{vectors} & : & {{\\cal A}}^{I} & \\rightarrow & {{\\cal A}}^{1} \\equiv A^{0} \\\\[1mm]\n & & {{\\cal A}}^{IJ} & \\rightarrow & {{\\cal A}}^{ij} = A^{1} \\, J^{ij} \\\\[1mm]\n & & \\tcA\\_{I} & \\rightarrow & \\tcA\\_{1} \\equiv \\tilde{A}\\_{0} \\\\[1mm]\n & & \\tcA\\_{IJ} & \\rightarrow & \\tcA\\_{ij} = \\tfrac{1}{3} \\tilde{A}\\_{1} \\, J\\_{ij} \\\\[3mm]\n\\textrm{two-forms} & : & {{\\cal B}}^{I} & \\rightarrow & {{\\cal B}}^{1} \\equiv B^{0} \\\\[1mm]\n & & {{{\\cal B}}\\_{I}}^{J} & \\rightarrow & {{{\\cal B}}\\_{1}}^{1} = \\tfrac{6}{7} \\, B\\_{1} \\,\\,\\,\\,\\,, \\,\\,\\,\\,\\, {{{\\cal B}}\\_{i}}^{j} = -\\tfrac{1}{7} \\, B\\_{1} \\, \\delta\\_{i}^{j} + \\tfrac{1}{3} B\\_{2} \\, {J\\_{i}}^{j} \\\\[3mm]\n\\textrm{three-forms} & : & {{\\cal C}}^{IJ} & \\rightarrow & {{\\cal C}}^{11} \\equiv C^{0} \\,\\,\\,\\,\\,, \\,\\,\\,\\,\\, {{\\cal C}}^{ij} = C^{1} \\, \\delta^{ij} \\;, \n\\end{array}$$ \nin agreement with the number of singlets in the last column of table [Table:grouptheory]. The real scalars (*χ*, *φ*) and (*ϕ*, *a*, *ζ*, *ζ̃*) respectively parameterise each factor of the scalar manifold ([ScalManN=2]). The superscript Λ = 0, 1 on the electric vectors *A*Λ labels them as the graviphoton and the vector in the vector multiplet, respectively, and similarly for their magnetic counterparts *Ã*Λ. The superscripts or subscripts on the two- and three-forms are just labels with no further meaning. The vectors *A*0, *Ã*0 gauge dyonically the SO(1, 1)*c* generated by *T*1 in ([linearcombt]), while *A*1 gauges electrically the U(1) generated by *T*23 + *T*45 + *T*67 in ([linearcombt]). Along with the metric and the six scalars, only *A*0, *A*1, *Ã*0, their field strengths and *B*0 enter the SU(3)-invariant bosonic Lagrangian, see ([LfullSU3]). Finally, the branching of the gravitini, in the $\\bm{\\overline{8}}$ of SU(8), under this SU(3) produces two singlets, in agreement with the N = 2 supersymmetry of this sector.\nWe can construct an explicit parameterisation of the scalar manifold ([ScalManN=2]) of this sector as follows. We first identify the generators ([Gener63]), ([Gener70]) of E7(7) that are invariant under the SU(3), ([EmbedingSO3]), ([eq:chainE7]), under consideration. These are\n \n$$\\label{Gener\\_SU3}\n\\begin{array}{cclc}\ng\\_{1} &=& {t\\_{2}}^{2} + {t\\_{4}}^{4} + {t\\_{6}}^{6} + {t\\_{3}}^{3} + {t\\_{5}}^{5} + {t\\_{7}}^{7} - 3 \\, ({t\\_{1}}^{1} + {t\\_{8}}^{8}) &, \\\\[2mm]\ng\\_{2} &=& g^{(-)}\\_{2} + g^{(+)}\\_{2} = ({t\\_{1}}^{8}) + ({t\\_{8}}^{1}) &, \\\\[2mm]\ng\\_{3} &=& {t\\_{1}}^{1} - {t\\_{8}}^{8} &, \\\\[2mm]\ng\\_{4} &=& g^{(-)}\\_{4} + g^{(+)}\\_{4} = (t\\_{4567} + t\\_{2367} + t\\_{2345} ) + (t\\_{1238} + t\\_{1458} + t\\_{1678} ) &, \\\\[2mm]\ng\\_{5} &=& g^{(-)}\\_{5} + g^{(+)}\\_{5} = ( t\\_{1246} - t\\_{1257} - t\\_{1347}- t\\_{1356} ) + ( t\\_{8357} - t\\_{8346} - t\\_{8256} - t\\_{8247}) &, \\\\[2mm]\ng\\_{6} &=& g^{(-)}\\_{6} + g^{(+)}\\_{6} = ( t\\_{3571} - t\\_{3461} - t\\_{2561} - t\\_{2471} ) + (t\\_{8246} - t\\_{8257} - t\\_{8347} - t\\_{8356} ) &,\n\\end{array}$$ \nwhere *g*1, *g*3 are Cartan generators and a subscript ( ± ) indicates a positive or negative root. The exponentiations\n \n$$\\label{coset\\_SU3}\n\\mathcal{V}\\_{\\textrm{SK}} = e^{-12 \\, \\chi \\, g^{(+)}\\_{4}} \\, e^{\\frac{1}{4} \\, \\varphi \\, g\\_{1}} \n\\hspace{10mm} \\textrm{ and } \\hspace{10mm}\n\\mathcal{V}\\_{\\textrm{QK}} = e^{a \\, g^{(+)}\\_{2} - \\, 6 \\, \\zeta \\,g^{(+)}\\_{5} - \\,6 \\, \\tilde{\\zeta} \\,g^{(+)}\\_{6} } \\, e^{\\phi \\, g\\_{3}} $$ \nlead to coset representatives for each factor in ([ScalManN=2]), and the total representative is simply the product[8](#fn8) V = VSK VQK . Finally, the scalar matrix is the quadratic combination $\\,{{{\\cal M}}=\\mathcal{V} \\, \\mathcal{V}^{t}}$. See appendix [app:SU3sectorScalMat] for its explicit expression.\nWith this scalar parameterisation, the Lagragian of the SU(3)-invariant sector can be written as\n \n$$\\label{L\\_full\\_SU3}\n\\begin{array}{lll}\n{\\cal L} &=& (R - V) \\, \\textrm{vol}\\_4 + \\tfrac{3}{2} \\left[ d\\varphi \\wedge \\* d\\varphi + e^{2 \\varphi} \\, d\\chi \\wedge \\* d\\chi \\right] \\\\[2mm]\n&+& 2 \\, d\\phi \\wedge \\* d\\phi + \\frac{1}{2} \\, e^{2 \\phi} \\, [ D\\zeta \\wedge \\* D\\zeta + D\\tilde{\\zeta} \\wedge \\* D\\tilde{\\zeta} ] \\\\[2mm] \n&+& \\tfrac{1}{2} \\, e^{4 \\phi} \\, [ Da + \\tfrac{1}{2} ( \\zeta D \\tilde{\\zeta} - \\tilde{\\zeta} D \\zeta ) ] \\wedge \\* [ Da + \\tfrac{1}{2} ( \\zeta D \\tilde{\\zeta} - \\tilde{\\zeta} D \\zeta ) ] \\\\[2mm]\n&+& \\tfrac{1}{2} \\, \\mathcal{I}\\_{\\Lambda\\Sigma} \\, H\\_{\\2}^{\\Lambda} \\wedge \\* H\\_{\\2}^{\\Sigma} + \\tfrac{1}{2} \\, \\mathcal{R}\\_{\\Lambda\\Sigma} \\, H\\_{\\2}^{\\Lambda} \\wedge H\\_{\\2}^{\\Sigma} - m \\, B^{0} \\wedge d\\tilde{A}\\_{0}\\,-\\, \\tfrac{1}{2} \\, g \\, m \\, B^{0} \\wedge B^{0} \\,\n\\end{array}$$ \nand follows by truncating ([BosLag]) according to ([SU(3)fullcontent]). Here, the covariant derivatives are\n \n$$\\label{Cov\\_aB}\nDa = da \\,+\\, g \\, A^{0} \\, - \\, m \\, \\tilde{A}\\_{0} \n\\hspace{4mm} \\textrm{, } \\hspace{4mm}\nD\\zeta = d\\zeta - 3 \\, g \\, A^{1} \\, \\tilde{\\zeta} \n\\hspace{4mm} \\textrm{, } \\hspace{4mm}\nD \\tilde{\\zeta} = d\\tilde{\\zeta} + 3 \\, g \\, A^{1}\\, \\zeta \\,\\,\\, \\,$$ \nand the electric vector field strengths\n \n$$\\label{H\\_modified\\_Elec\\_SU3}\nH\\_{\\2}^{0}=dA^{0} + m \\, B^{0}\n\\hspace{8mm} \\textrm{, } \\hspace{8mm}\nH\\_{\\2}^{1}=dA^{1} \\,\\,\\, \\,$$ \nfollow from ([eqO:2FormFieldStrengths]). The gauge kinetic matrix in ([LfullSU3]) is obtained from the scalar matrix $\\,{{\\cal M}}\\,$ through ([Mscalar]). In the scalar parameterisation that we are using here, it explicitly reads\n \n$$\\label{NMatrix}\n\\mathcal{N}\\_{\\Lambda \\Sigma} = \\mathcal{R}\\_{\\Lambda \\Sigma} + i \\, \\mathcal{I}\\_{\\Lambda \\Sigma} = \n\\frac{1}{(2\\, e^{\\varphi } \\, \\chi +i )}\n\\left(\n\\begin{array}{cc}\n -\\dfrac{e^{3 \\varphi }}{(e^{\\varphi } \\, \\chi -i )^2} & \\dfrac{3 \\, e^{2 \\varphi } \\, \\chi }{(e^{\\varphi} \\, \\chi -i )} \\\\[5mm]\n \\dfrac{3 \\, e^{2 \\varphi } \\, \\chi }{(e^{\\varphi } \\, \\chi -i)} & 3 \\, ( e^{\\varphi } \\, \\chi^2+e^{-\\varphi})\n\\end{array}\n\\right) \\.$$ \nNote that IΛΣ is negative definite so that the vector kinetic terms have the correct sign. Finally, the explicit expression of the scalar potential in ([LfullSU3]) can be derived from ([VgeneralRewrite]) to be\n \n$$\\label{VSU3}\n\\begin{array}{lll}\nV &=& \\frac{1}{2} \\, g^{2} \\left[ e^{4 \\phi -3 \\varphi } \\big(1+e^{2\\varphi} \\chi^2 \\big)^3 -12 \\, e^{2 \\phi -\\varphi } \\big(1+e^{2\\varphi} \\chi^2 \\big) -24 \\, e^{\\varphi} \\right.\n\\\\[6pt]\n&& \\left. \\qquad + \\tfrac34 \\, e^{4\\phi +\\varphi} \\big( \\zeta^2 + \\tilde \\zeta^2 \\big)^2 \\big(1+3 \\, e^{2\\varphi} \\chi^2 \\big)+ 3\\, e^{4\\phi +\\varphi} \\big( \\zeta^2 + \\tilde \\zeta^2 \\big) \\chi^2 \\big(1+e^{2\\varphi} \\chi^2 \\big) \\right.\n\\\\[6pt]\n&& \\left. \\qquad -3 \\, e^{2\\phi +\\varphi} \\big( \\zeta^2 + \\tilde \\zeta^2 \\big) \\big(1-3 \\, e^{2\\varphi} \\chi^2 \\big) \\right] - \\tfrac12 \\, g \\, m \\, \\chi \\, e^{4 \\phi+3 \\varphi } \\left( 3\\big( \\zeta^2 + \\tilde \\zeta^2 \\big) + 2\\chi ^2 \\right) \\\\[6pt]\n&& + \\, \\frac{1}{2} \\, m^2 \\, e^{4 \\phi+3 \\varphi } \\.\n\\end{array}$$ \nOut of the six real scalars in this sector, this potential effectively depends on only four. The non-compact Stückelber scalar *a* and the U(1) phase *β* of the complex combination\n \n*ζ̃* + *i* *ζ* = 2 *ρ* *e**i* *β*, \ndo not enter the potential. As we will discuss in section [subsec:CriticalPoints], this potential displays a rich structure of critical points, both supersymmetric and non-supersymmetric, when *g**m* ≠ 0 .\nDuality hierarchy\n-----------------\nThe duality hierarchy ([eq:SL7fieldcontent4D]) in this sector reduces to the field content ([SU(3)fullcontent]), which includes only singlets in the branching of SL(7) under SU(3). The formulae in section [sec:TruncTensorHierarchy] simplify accordingly. The electric vector field strengths have already been given in ([HmodifiedElecSU3]), and their magnetic counterparts are\n \n$$\\label{H2\\_SU3sector}\n\\tilde{H}\\_{\\2 0} = d\\tilde{A}\\_{0} + g B^{0} \n\\hspace{8mm} \\textrm{, } \\hspace{8mm}\n\\tilde{H}\\_{\\2 1} = d\\tilde{A}\\_{1} - 2 g B\\_{2} \\,\\,\\, \\.$$ \nThe field strengths ([eqO:3FormFieldStrengths]) for the two-form potentials (*B*0 ; *B*1, *B*2) reduce to\n \n$$\\label{H3\\_SU3sector}\n\\begin{array}{lll}\nH\\_{\\3}^{0} & = & dB^{0} \\, \\\\[1mm]\nH\\_{\\3 1} & = & DB\\_{1} + \\tfrac{1}{2} ( A^{0} \\wedge d\\tilde{A}\\_{0} + \\tilde{A}\\_{0} \\wedge dA^{0} - \\tfrac{1}{3} A^{1} \\wedge d\\tilde{A}\\_{1} - \\tfrac{1}{3} \\tilde{A}\\_{1} \\wedge d A^{1} ) + 2 g (C^{1} - C^{0}) \\, \\\\[1mm]\nH\\_{\\3 2} & = & dB\\_{2} \\, \n\\end{array}$$ \nwith *D**B*1 = *d**B*1 − *g**A*0 ∧ *B*0 + *m**Ã*0 ∧ *B*0 , and those ([eqO:4FormFieldStrengths]) for the three-form potentials (*C*0, *C*1) read\n \n$$\\label{H4\\_SU3sector}\n\\begin{array}{lll}\nH\\_{\\4}^{0} & = & dC^{0} + H\\_{\\2}^{0} \\wedge B^{0} - \\tfrac{1}{2} m B^{0} \\wedge B^{0} \\, \\\\[1mm]\nH\\_{\\4}^{1} & = & dC^{1} - \\tfrac{1}{3} H^{1}\\_{\\2} \\wedge B\\_{2} \\.\n\\end{array}$$ \nThe Bianchi identities ([eq:TruncBianchis]) simplify to\n \n$$\\label{eq:TruncBianchis\\_SU3}\n\\begin{array}{l}\nd {H}\\_\\2^{0} = m \\, H^{0}\\_{\\3} \\hspace{3mm}, \\hspace{3mm}\nd {H}\\_{\\2}^{1} = 0 \\hspace{3mm}, \\hspace{3mm}\nd {\\tilde{H}}\\_{\\2 0} = g \\, H^{0}\\_{\\3} \\hspace{3mm}, \\hspace{3mm} \nd {\\tilde{H}}\\_{\\2 1} = - 2 \\, g \\, H\\_{\\3 2} \\\\[3mm] \nd H\\_\\3^{0} = 0 \\hspace{3mm}, \\hspace{3mm} D H\\_{\\3 1} = H\\_\\2^{0} \\wedge \\tilde{H}\\_{\\2 0} - \\tfrac{1}{3} H\\_\\2^{1} \\wedge \\tilde{H}\\_{\\2 1} + 2 g \\, (H^1\\_\\4 - H^0\\_\\4 ) \\hspace{2mm}, \\hspace{2mm} d H\\_{\\3 2} = 0 \\, \\\\[3mm]\nd H\\_{\\4}^{0} \\equiv d H\\_{\\4}^{1} \\equiv 0 \\,\n\\end{array}$$ \nwhere $\\,D H\\_{\\3 1} =d H\\_{\\3 1} -g A^{0} \\wedge H\\_{\\3}^0 +m \\tilde{A}\\_{0} \\wedge H\\_{\\3}^0 \\,$. These again close among themselves, in agreement with the consistency of the SU(3)-invariant truncation.\nThe duality relations also simplify, and can be written in terms of the explicit scalar parameterisation on this sector given in section [subsec:construction] and appendix [app:SU3sectorScalMat]. The vector/vector duality relations ([H2IDuality]) and ([H2IJDuality]) reduce to\n \n$$\\label{duality:vector/vector}\n\\begin{array}{lll}\n\\tilde{H}\\_{\\2 0} &=& - \\dfrac{e^{3 \\varphi } (1+ 3 e^{2 \\varphi } \\chi ^2)}{(1+e^{2 \\varphi } \\chi ^2)^2 (1+4 e^{2 \\varphi } \\chi ^2)} \\* H\\_{\\2}^{0}\n+ \\, \\chi ^2 \\, \\dfrac{3 e^{3 \\varphi }}{1+e^{2 \\varphi } \\chi ^2 ( 5+4 e^{2 \\varphi } \\chi^2)} \\* H\\_{\\2}^{1} \n \\\\[4mm]\n&-& 2 \\, \\chi ^3 \\, \\dfrac{ e^{6 \\varphi } }{(1+e^{2 \\varphi } \\chi ^2)^2 (1+4 e^{2 \\varphi } \\chi ^2)} \\, H\\_{\\2}^{0}\n+ \\chi \\, \\dfrac{3 e^{2 \\varphi } (1+2 e^{2 \\varphi } \\chi ^2)}{1+e^{2 \\varphi } \\chi ^2 ( 5+4 e^{2 \\varphi } \\chi^2)} \\, H\\_{\\2}^{1} \\, \\\\[5mm]\n\\tilde{H}\\_{\\2 1} &=& \\chi ^2 \\, \\dfrac{3 e^{3 \\varphi }}{1+e^{2 \\varphi } \\chi ^2 ( 5+4 e^{2 \\varphi } \\chi^2)} \\, \\* H\\_{\\2}^{0} - e^{-\\varphi} \\, \\dfrac{3 (1 + e^{2 \\varphi } \\chi ^2)}{1 + 4 e^{2 \\varphi } \\chi ^2} \\, \\* H\\_{\\2}^{1} \\\\[4mm]\n&+& \\chi \\, \\dfrac{3e^{2 \\varphi } (1+2 e^{2 \\varphi } \\chi ^2)}{1+e^{2 \\varphi } \\chi ^2 ( 5+4 e^{2 \\varphi } \\chi^2)} \\, H\\_{\\2}^{0} + \\chi \\, \\dfrac{6 (1 + e^{2 \\varphi } \\chi ^2)}{1 + 4 e^{2 \\varphi } \\chi ^2} H\\_{\\2}^{1} \\,\n\\end{array}$$ \nthe duality relations ([H3IDuality]), ([H3IJDuality]) for the three-form field strengths simplify as\n \n$$\\label{duality:two-form/scalar}\n\\begin{array}{lll}\nH\\_{\\3}^{0} &=& - e^{4 \\phi} \\, \\* \\big( Da +\\tfrac{1}{2}\\, (\\zeta \\, D\\tilde{\\zeta} - \\tilde{\\zeta} \\, D\\zeta) \\big) \\, \\\\[3mm]\n{H\\_{\\3 1}} &=& \\* \\big[ 2(d \\varphi - e^{2 \\varphi} \\chi d\\chi) - 2 d\\phi + a e^{4 \\phi} \\big( Da +\\tfrac{1}{2} (\\zeta D\\tilde{\\zeta} - \\tilde{\\zeta} D\\zeta) \\big) + \\tfrac{1}{2} e^{2\\phi} ( \\zeta D\\zeta + \\tilde{\\zeta} D\\tilde{\\zeta} ) \\big], \\\\[3mm]\nH\\_{\\3 2} &=& \\tfrac{1}{2} \\* \\Big[ e^{2\\phi} (\\zeta \\, D\\tilde{\\zeta} - \\tilde{\\zeta} \\, D\\zeta) + \\tfrac{1}{2} \\, (\\zeta^2 + \\tilde{\\zeta}^2) \\, e^{4 \\phi} \\big( Da +\\tfrac{1}{2}\\, (\\zeta \\, D\\tilde{\\zeta} - \\tilde{\\zeta} \\, D\\zeta) \\big) \\Big] \\,\n\\end{array}$$ \nand the duality relations ([H4Duality]) for the four-form field strengths give rise to\n \n$$\\label{duality:three-form/ET}\n\\begin{array}{lll}\nH\\_{\\4}^{0} &=& \\Big[ \\tfrac{1}{2} \\, g \\, \\big(1+e^{2\\varphi} \\chi^2 \\big) \\Big( 12 \\, e^{2 \\phi - \\varphi } - 2 \\, e^{4 \\phi - 3\\varphi } \\big(1+e^{2\\varphi} \\chi^2 \\big)^2 - 3 e^{4 \\phi +\\varphi } \\chi^2 \\big( \\zeta^2 +\\tilde{\\zeta}^2 \\big) \\Big) \\\\[4pt]\n&& \\,\\,\\, + \\, m\\, e^{4 \\phi + 3\\varphi } \\, \\chi^3 \\Big] \\, \\textrm{vol}\\_{4} \\, \\\\[3mm]\nH\\_{\\4}^{1} &=& \\Big[ \\tfrac{1}{2} \\, g \\Big( 8 \\, e^{\\varphi } + 2 \\, e^{2 \\phi - \\varphi } \\big(1+e^{2\\varphi} \\chi^2 \\big) + e^{2\\phi +\\varphi} \\big( \\zeta^2 + \\tilde \\zeta^2 \\big) \\big(1-3 \\, e^{2\\varphi} \\chi^2 \\big) \\\\[4pt]\n&& \\qquad \\, \\, - \\, \\tfrac12 \\, e^{4\\phi +\\varphi} \\big( \\zeta^2 + \\tilde \\zeta^2 \\big) \\chi^2 \\big(1+e^{2\\varphi} \\chi^2 \\big)\n - \\tfrac14 \\, e^{4\\phi +\\varphi} \\big( \\zeta^2 + \\tilde \\zeta^2 \\big)^2 \\big(1+3 \\, e^{2\\varphi} \\chi^2 \\big) \\Big) \\\\[4pt]\n&& \\,\\,\\, + \\, \\tfrac14 \\, m \\, e^{4 \\phi + 3\\varphi } \\, \\chi \\, \\big( \\zeta^2 + \\tilde \\zeta^2 \\big) \\Big] \\, \\textrm{vol}\\_{4} \\.\n\\end{array}$$ \nFor later reference, we also give the SU(3)-invariant truncation of the duality relation ([H4Dualitysinglet]) for the four-form field strength $\\tilde{H}\\_{\\4} \\equiv \\tilde{{{\\cal H}}}\\_{\\4} $ of the singlet three-form potential $\\,\\tilde{C}\\equiv \\tcC\\,$ related to the magnetic component of the embedding tensor. It reads\n \n$$\\label{H4magneticDual}\n\\begin{array}{lll}\n\\tilde{H}\\_{\\4} &=& \\frac{1}{2} \\, g \\, \\chi \\, e^{3 \\varphi +4 \\phi } \\big(3 (\\zeta ^2+\\tilde{\\zeta}^2)+2 \\chi ^2 \\big) \\, \\textrm{vol}\\_{4} - m \\, e^{3 \\varphi +4 \\phi } \\, \\textrm{vol}\\_{4} \\.\n\\end{array}$$ \nThese duality relations manifestly show that, in the symplectic frame we are using, the magnetic vectors and the higher rank forms in the tensor hierarchy do not carry independent degrees of freedom, but rather depend on the metric, the electric vector field strengths and the scalars. Alternatively, these relations can be used to transfer independent degrees of freedom within the duality hierarchy. For example, the first relation in ([duality:two-form/scalar]) can be used to dualise the Stückelberg scalar *a* into the two-form *B*0, so that the latter can be regarded as carrying the independent degrees of freedom. This duality relation can also be obtained by varying the Lagrangian ([LfullSU3]) with respect to the magnetic graviphoton *Ã*0. Solving this duality relation and substituting into ([LfullSU3]), the following new Lagrangian is obtained:\n \n$$\\label{L\\_full\\_SU3\\_dual}\n\\begin{array}{llll}\n\\widetilde{\\mathcal{L}} &=& (R - V) \\, \\textrm{vol}\\_4 + \\tfrac{1}{2} \\, e^{-4 \\phi} \\, H\\_{\\3}^{0}\\wedge \\* H\\_{\\3}^{0} + \\tfrac{3}{2} \\left[ d\\varphi \\wedge \\* d\\varphi + e^{2 \\varphi} \\, d\\chi \\wedge \\* d\\chi \\right] \\\\[2mm] \n&+& 2 \\, d\\phi \\wedge \\* d\\phi + \\frac{1}{2} \\, e^{2 \\phi} \\left[ D\\zeta \\wedge \\* D\\zeta + D\\tilde{\\zeta} \\wedge \\* D\\tilde{\\zeta} \\right] \\\\[2mm]\n&+& \\tfrac{1}{2} \\, \\mathcal{I}\\_{\\Lambda\\Sigma} \\, H\\_{\\2}^{\\Lambda} \\wedge \\* H\\_{\\2}^{\\Sigma} + \\tfrac{1}{2} \\, \\mathcal{R}\\_{\\Lambda\\Sigma} \\, H\\_{\\2}^{\\Lambda} \\wedge H\\_{\\2}^{\\Sigma} \\\\[2mm]\n&+& H\\_{\\3}^{0} \\wedge \\, \\left[ \\, g \\, A^{0} \\, + \\, \\tfrac{1}{2} \\, ( \\zeta D \\tilde{\\zeta} - \\tilde{\\zeta} D \\zeta ) \\, \\right] \\,-\\, \\tfrac{1}{2} \\, gm \\, B^{0} \\wedge B^{0} \\.\n\\end{array}$$ \nThe kinetic terms are now expressed in terms of the field strength $H\\_{\\3}^{0}$ of *B*0 given in ([H3SU3sector]), and the magnetic vector *Ã*0 no longer appears in this Lagrangian. See *e.g.* section 4.1 of for a discussion in a similar context. In the Lagrangian ([LfullSU3dual]), *B*0 is a propagating massive two-form with conventional kinetic term $H\\_{\\3}^{0}\\wedge \\* H\\_{\\3}^{0}$ and mass term *B*0 ∧ \\* *B*0 (coming from the *m* *B*0 dependence of $H\\_{\\2}^{0}$ in ([HmodifiedElecSU3])), in addition to the topological mass term *B*0 ∧ *B*0. Lagrangians similar to ([LfullSU3dual]) but naturally written in terms of the magnetic field strengths, as in ([duality:vector/vector]), usually appear in dimensional reductions of massive IIA or M-theory to ${{\\cal N}}=2$ supergravity, see *e.g.*. See appendix [app:MTheoryonSE7] for the relation of the SU(3)-invariant sector and the theory of. See for a more general discussion of Lagragians involving higher-rank fields in the ${{\\cal N}}=8$ duality hierarchy.\nThe duality relations can be also used to relate the four-form field strengths to the potential, as discussed for the full ${{\\cal N}}=8$ theory in section [sec:TruncTensorHierarchy]. With the parameterisation of the scalars in the SU(3) sector that we gave in section [subsec:construction], the duality relation ([H4/Potential2]) can be explicitly verified in this sector. With the help of ([duality:three-form/ET]) and ([H4magneticDual]), equation ([H4/Potential2]) can be seen to reduce to\n \n$$\\label{DualitySU3H4V}\ng \\, ( \\, H^{0}\\_{\\4} + 6 \\, H^{1}\\_{\\4} \\, ) + m \\, \\tilde{H}\\_{\\4} = -2 \\, V \\, \\textrm{vol}\\_{4} \\,$$ \nwhere $H^{0}\\_{\\4}$, $H^{1}\\_{\\4}$ are the field strengths ([H4SU3sector]) of the three-form potentials *C*0, *C*1 in the truncated hierarchy ([SU(3)fullcontent]), $\\tilde{H}\\_{\\4}$ is the field strength of the three-form potential *C̃* related to the magnetic component of the embedding tensor, and *V* is the scalar potential in the SU(3) sector, given in ([VSU3]). From ([Extreme1]) and ([Extreme27]), we also find that the following relations hold at every critical point (see section [subsec:CriticalPoints]) of the scalar potential ([VSU3]):\n \n$$\\label{EOM\\_SU3}\ng \\, ( \\, H^{0}\\_{\\4}|\\_0 + 6 \\, H^{1}\\_{\\4}|\\_0 \\, ) +7 m \\, \\tilde{H}\\_{\\4}|\\_0 = 0 \\, \\qquad \nH^0\\_\\4 |\\_0 = H^1\\_\\4 |\\_0 \\.$$ \nRecall that ∣0 and *V*0 denote evaluation at a critical point. Combining ([DualitySU3H4V]) with the first equation in ([EOMSU3]) yields $\\,V\\_{0} \\,\\, \\textrm{vol}\\_{4} = 3 \\, m \\, \\tilde{H}\\_{\\4}|\\_0\\,$. This condition relates the AdS character of the critical points in this sector with a non-vanishing value of the magnetic gauge coupling *m*, provided $\\tilde{H}\\_{\\4}|\\_0 \\neq 0 $, which is indeed the case.\nSuperpotential formulation\n--------------------------\nTwo superpotentials exist (see also ) from which the scalar potential of the SU(3)-invariant sector of the electric SO(8) gauging derives. The same statement holds for the SU(3) sector of the dyonic SO(8) gauging. Here we will show that this is also true for the SU(3)-invariant sector of ISO(7)*c* supergravity. See for superpotentials in the SO(7)+ and G2 sectors of the electric ISO(7) gauging. In order to see this following a notation close to, we first introduce coordinates *t* and *u* on two copies of the upper-half of the complex plane\n \n$$\\label{mappingN1Sugra}\nt = -\\chi + i \\, e^{-\\varphi}\n\\hspace{10mm} \\textrm{ and } \\hspace{10mm}\nu = -\\rho + i \\, e^{-\\phi} \\,$$ \nwith $\\, \\rho^2=\\tfrac{1}{4}\\, (\\tilde{\\zeta}^2 + \\zeta^2)\\,$ as follows from ([polarcoord]), and then further convert into two copies of the unit-disk via\n \n$$\\label{mappingN2Chiral}\nz= \\frac{t-i}{t+i}\n\\hspace{8mm} \\textrm{ and } \\hspace{8mm}\n\\zeta\\_{12}= \\frac{u-i}{u+i} \\,$$ \nso that ∣*z*∣ < 1, ∣*ζ*12∣ < 1. In terms of the new complex fields *z* and *ζ*12 , the kinetic terms for the (*χ*, *φ*) and (*ρ*, *ϕ*) scalars in ([LfullSU3]) can be recast as\n \n$$\\label{LkinUnitDisk}\n\\tfrac{1}{2} \\, \\mathcal{L}^{\\textrm{kin}}\\_{\\textrm{scalar}} = 3 \\, \\frac{dz \\wedge \\* d\\bar{z}}{( 1- |z|^2)^2} + 4 \\, \\frac{d\\zeta\\_{12} \\wedge \\* d\\bar{\\zeta}\\_{12}}{( 1- |\\zeta\\_{12}|^2)^2} \\.$$ \nIntroducing\n \n$$\\label{WN=2-SU3}\n\\begin{array}{llll}\n\\mathcal{W} & \\equiv & (1-|z|^{2})^{-\\frac{3}{2}} \\, (1-|\\zeta\\_{12}|^{2})^{-2} \\\\[4mm]\n& \\times & \\left[ \\, g \\, \\left( \\dfrac{7}{8} \\, (1- \\zeta\\_{12})^{4} \\, (1+z)^{3} \\, + \\, 3 \\, (\\zeta\\_{12}-z) \\, (1+z) \\,(1-\\zeta\\_{12})^2 \\,(1- z \\, \\zeta\\_{12}) \\right) \\right. \\\\[4mm]\n&+& \\left. i\\, \\dfrac{m}{8} \\, (1-\\zeta\\_{12})^{4} \\, (1-z)^{3} \\, \\right] \\,\n\\end{array}$$ \nwe find that the scalar potential ([VSU3]) is reproduced through the expression\n \n$$\\label{VN=2-SU3}\n\\tfrac{1}{4} \\, V = 2 \\, \\left[ \\frac{4}{3} \\, (1-|z|^{2})^{2} \\, \\left| \\frac{\\partial W}{\\partial z} \\right|^{2} + (1-|\\zeta\\_{12}|^{2})^{2} \\, \\left| \\frac{\\partial W}{\\partial \\zeta\\_{12}} \\right|^{2} -3 \\, W^{2} \\right] \\,$$ \nwith the superpotential *W* given in terms of ([WN=2-SU3]) by either $\\,W = | {\\cal W}\\_+| \\equiv | {\\cal W} (z, \\zeta\\_{12})|\\,$ or $\\,W = | {\\cal W}\\_-| \\equiv | {\\cal W} (z, \\bar{\\zeta}\\_{12})|\\,$. We take this match as a consistency check on our calculation of the potential ([VSU3]) in section [subsec:construction] with the ${{\\cal N}}=8$ embedding tensor formalism followed by SU(3)-invariant truncation.\nAll the supersymmetric critical points of the scalar potential ([VSU3]), as given in table [Table:SU3Points] of section [subsec:CriticalPoints], are critical points of ∣W+∣ . The ${{\\cal N}}=2$ point is an extremum of both ∣W+∣ and ∣W−∣ . Under the map *ζ*12 → *ζ̄*12 (or, equivalently, *ρ* → − *ρ* in ([mappingN1Sugra]), *i.e.*, *β* → − *β* in ([polarcoord])), the ${{\\cal N}}=1$ points become extrema of ∣W−∣ , rather than ∣W+∣ . Due to the overall *c*1/3 dependence of the critical points in table [Table:SU3Points], there are two asymptotic limits: *z* = *ζ*12 = − 1 at *c* → 0 (*i.e.*, *m* → 0, *g* ≠ 0) and *z* = *ζ*12 = + 1 at *c* → ∞ (*i.e.*, *g* → 0, *m* ≠ 0). These critical points thus disappear for the purely electric and purely magnetic gaugings.\nThese superpotentials will be useful to holographically study RG flows between different Chern-Simons phases of the D2-brane field theory with at least SU(3) flavour symmetry. We leave this for future work. See for studies of RG flows with at least SU(3) invariance in the M2-brane field theory from electrically gauged SO(8) supergravity, and for similar domain wall solutions in dyonic SO(8) supergravity.\nCanonical ${{\\cal N}}=2$ formulation\n------------------------------------\nAs a further crosscheck on our calculations, we will now cast the SU(3)-invariant Lagrangian ([LfullSU3]) in ${{\\cal N}}=2$ canonical form, focusing on the special geometry quantities that enter the canonical formulation. The scalar manifold ([ScalManN=2]) is the product of two well-known special Kähler and quaternionic Kähler manifolds, corresponding to the vector multiplet and hypermultiplet scalars, respectively. The parameterisation ([cosetSU3]) leads to the familar form for the metric on this space that appears in the scalar kinetic terms in the Lagrangian ([LfullSU3]). Indices *M* = 1, …, 4 , *α* = 1, …, 8 , *u* = 1, …, 4 and *i* = *t* in this subsection correspond to Sp(4, R) vector indices, SU(2, 1) adjoint indices, SU(2, 1)/U(2) curved indices and SU(1, 1)/U(1) curved holomorphic indices – we denote by *t* the only value that *i* takes on. The index Λ = 0, 1 introduced below ([SU(3)fullcontent]) labels, as usual, ``half\" the fundamental representation of Sp(4, R) .\nLet us start by describing the special Kähler geometry of the scalars in the vector. We find the sections *X**M* = (*X*Λ, *F*Λ) , [holosections]\nlclclclc X0 = -t3 &, & X1= -t &, & F0 = 1 &, & F1= 3 t2 &,\nwhich are holomorphic in the coordinate *t* ([mappingN1Sugra]) on the upper-half plane realisation of SU(1, 1)/U(1) , to be the relevant ones for our model. In the symplectic frame in which the Lagrangian ([LfullSU3]) is written, the sections *F*Λ can be obtained from the prepotential\n \n$$\\label{Prepot\\_SU3}\n\\mathcal{F}=-2\\,\\sqrt{ X^{0} \\, (X^{1})^3} \\,$$ \nas *F*Λ = ∂F/∂*X*Λ . The Kähler potential\n \n$$\\begin{aligned}\n\\label{KahlerPot}\nK= -\\log \\big( i \\, \\bar{X}^M \\Omega\\_{MN} X^N \\big) = - 3 \\,\\log ( - i(t-\\bar{t}) ) \n\\hspace{3mm} \\textrm{ with } \\hspace{3mm}\n\\Omega\\_{MN} = \\left(\\begin{array}{cc}\n0 & \\mathbb{I}\\_2 \\\\\n"},"data_id":{"kind":"string","value":"arxiv_0184761"}}},{"rowIdx":46193,"cells":{"content":{"kind":"string","value":"**Abstract**\nThis book uses the modern theory of artificial intelligence (AI) to understand human suffering or mental pain. Both humans and sophisticated AI agents process information about the world in order to achieve goals and obtain rewards, which is why AI can be used as a model of the human brain and mind. This book intends to make the theory accessible to a relatively general audience, requiring only some relevant scientific background.\nThe book starts with the assumption that suffering is mainly caused by frustration. Frustration means the failure of an agent (whether AI or human) to achieve a goal or a reward it wanted or expected. Frustration is inevitable because of the overwhelming complexity of the world, limited computational resources, and scarcity of good data. In particular, such limitations imply that an agent acting in the real world must cope with uncontrollability, unpredictability, and uncertainty, which all lead to frustration.\nFundamental in such modelling is the idea of learning, or adaptation to the environment. While AI uses machine learning, humans and animals adapt by a combination of evolutionary mechanisms and ordinary learning. Even frustration is fundamentally an error signal that the system uses for learning. This book explores various aspects and limitations of learning algorithms and their implications regarding suffering.\nAt the end of the book, the computational theory is used to derive various interventions or training methods that will reduce suffering in humans. The amount of frustration is expressed by a simple equation which indicates how it can be reduced. The ensuing interventions are very similar to those proposed by Buddhist and Stoic philosophy, and include mindfulness meditation. Therefore, this book can be interpreted as an exposition of a computational theory justifying why such philosophies and meditation reduce human suffering.\nCopyright © Aapo Hyvärinen. All rights reserved. \nDistribution allowed as per Creative Commons Attribution-Noncommercial-NoDerivatives (CC BY-NC-ND) License.\nThis book is typeset in Utopia(R), Copyright 1989, 1991 Adobe Systems Incorporated. All rights reserved. \nUtopia is either a registered trademark or trademark of Adobe Systems Incorporated in the United States and/or other countries. \nUsed under license, as implemented in the Latex fourier package. \nPreface\n=======\nI like to write books that I would have wanted to read myself as a student. I really wish I had been able to read this book. It would probably have changed my life and my career, as I would have insisted on doing my PhD on this topic. Alas, when I was a student in the 1990s, the topic of this book was not something a reasonable PhD student would have worked on. There was hardly any literature on the topic; it would have been considered uncharted territory, if not suspicious. I hope the world has changed, and that this book may contribute to that change. With the huge increase in research on AI and computational neuroscience on the one hand, and affective neuroscience and mindfulness meditation on the other, I think the time is ripe to attempt a synthesis, which is the motivation for this book.\nWhat I should emphasize is that this book is about a scientific theory, or rather, several scientific theories. It is not a book that teaches meditation; it has little to do with self-help and certainly constitutes no clinical guidance. Nor is it really a philosophical book in the sense that the word would be used in academic circles: while there is some philosophical speculation, the main paradigm is that of the natural sciences. It may be surprising that I seem to include artificial intelligence in the natural sciences, but here it is largely used as a computational model of the brain, even if sometimes on a very abstract level. The strong neuroscience component of this book further connects it to empirical science.\nI have tried to write the book so that it is suitable for as wide an audience as possible. I believe anybody trained in computer science or neuroscience should be able to understand it. Scientific training in any discipline might be enough to understand the main ideas, and I hope that even members of the general public might find something interesting in it. Although not primarily intended as such, the book can also be used as a university-level textbook for advanced undergraduates or graduate students in computer science or cognitive science; it should also be suitable for computationally minded students in neuroscience or psychology.\nThis book was written while working in different institutions. Most of the work was done while a faculty member at the University of Helsinki (Department of Computer Science). Part of the writing was accomplished while a faculty member at University College London (Gatsby Computational Neuroscience Unit) as well as a research scientist at Université Paris-Saclay (DataIA Institute and Inria–Saclay-Ile-de-France, supported by grant ANR-17-CONV-0003). The work was further supported by a Fellowship from CIFAR (Learning in Machines & Brains Program).\nFinally, I’m very grateful to Moritz Grosse-Wentrup, Riitta Hari, Marianne Maertens, John Millar, Tiina Parviainen, Jonne Viljanen, and, especially, Michael Gutmann, for most helpful comments on the manuscript.\n*Helsinki,* Aapo Hyvärinen\nIntroduction:Understanding human suffering by AI\n================================================\nWhat is the most central question in human life? For me, it is the question of suffering. There may be questions which are more fundamental, or philosophically more fascinating, for example: Why does the world exist? Or, how is it possible that we are conscious? But those questions are rather theoretical and mainly satisfy one’s intellectual curiosity. If you found the answer to those latter questions, would that change your life, or other people’s lives, for the better?\nThe question of suffering is with us at every moment. By suffering I mean mental pain, the opposite of pleasure and happiness. In some cases, it is a result of physical pain, but usually of purely mental origin. In fact, any casual observer of human life easily comes to the conclusion that it is full of such suffering: There is frustration, anxiety, sadness, depression, and so on.\nWhy is the “human condition” so unpleasant: did somebody (or something) make a huge mistake in designing humans? And, most importantly, is there anything we can do about it: can we remove suffering, or at least reduce it? Now, this is a question that has enormous practical significance. Reducing suffering, almost by definition, makes people’s lives better.\nThe starting point of this book is the idea that we can use the theory of artificial intelligence, or AI, to understand why there is so much suffering in humans. This book will show how suffering is largely due to the inability of an intelligent system, whether an artificial intelligence or a human being, to understand its own programming and its own limitations, in particular the limitations of its computation and data.\nInvestigating intelligence by constructing it\n---------------------------------------------\nHow can I claim that the theory of AI has any relevance to understanding the human mind, let alone suffering? The answer lies in how AI can help us understand the computational design principles which are applicable to humans as well.\nWhen I asked above if somebody made a huge mistake in designing humans, that “somebody” was of course evolution, metaphorically speaking. Evolution designed the basic processes of our mental life, for good or bad. Importantly, evolution didn’t construct our brains in some random, arbitrary ways, but it designed us to be fit for certain purposes and goals. Ultimately, those evolutionary goals are about reproduction and spreading your genes, but to satisfy that ultimate goal, many more intermediate goals need to be considered. You have to get food, find sex partners, and not get killed. These, in turn, require that you know how to walk, and you are able to recognize objects as well as to plan your future actions.\nWe can learn to understand such evolutionary design goals by trying to design and construct an AI, or a robot. This is a perspective which is gaining more and more prominence in neuroscience: Trying to actually construct an intelligent system forces you to think about the computation and algorithms needed.\nOrdinary neuroscience is based on conducting experiments on humans or animals. It can establish many interesting facts about the brain; for example, where in the brain the processing necessary for vision or fear takes place. It can also tell us something about how such processing happens; for example, by explaining how the brain recognizes that the animal in front of you is a cat and not a dog, and why we would get scared if it were a tiger.\nHowever, the deepest question in neuroscience is the “why” question: Why does a certain kind of processing take place at all? What is its evolutionary purpose? Why do we, for example, have emotions like fear in the first place? Why is our mind frequently assailed by thoughts about the past and the future even when we try to concentrate on the present? And ultimately, why is there suffering?\nDesigning intelligent systems goes a long way toward answering the “why” question. If we find that an AI necessarily needs a certain kind of computation to achieve human-like intelligence, it is likely that the human brain does that same kind of computation—at least on some level of abstraction. AI can also give us a deeper understanding of “how” computations happen in the human brain, since designing it necessarily forces the scientists to figure out all the details needed in the computation.\nIs the brain a big computer?\n----------------------------\nThe prerequisite for learning about the brain by building intelligent systems is that our brain is in many ways like a computer. In fact, the modern paradigm in neuroscience and psychology considers the brain as an information-processing device. The term “cognition” is used to describe information-processing performed by the brain, while with ordinary computers we usually talk about computation.\nThe brain receives new data by seeing, hearing, or otherwise sensing things. It processes the sensory data in various ways, ultimately enabling us to recognize objects and act in the world. It can also process information retrieved from its own memory, which is necessary for what we call thinking in plain English. A system that processes information in such ways can be called, almost by definition, a computer, so it is natural to say that, actually, the brain *is* a computer.\nCertainly, the brain is very different from any ordinary computer that you can buy in a shop. For example, your PC, or your mobile phone, has a central processing unit (CPU), sometimes a couple of them. The brain has no such thing. The information-processing happens in the neural cells, or neurons. Each of them is like a tiny CPU which can only perform extremely simple processing— but there is a huge number of them, tens of billions. The crucial difference with respect to a CPU is that each neuron processes its own input independently, and all the neurons do that at the same time—this is called parallel and distributed processing.\nYet, from an abstract viewpoint, such differences can be seen as just technical details. In particular, if we are interested in the question of “why” certain computations are performed, the physical structure of the information-processing device, or even the details of the programming do not matter. What really matters for our purposes is whether the brain and the computer need to solve the same kinds of computational problems. This will be the case if humans and the AI live in the same kind of environment, have the same kind of goals for their actions, and use similar means to try to reach them. That is increasingly the case when AI develops in terms of autonomous robots, for example, and in any case, we can use our current AI theory to extrapolate what AI’s might be like in the future.\nMachine learning as analogue to evolution\n-----------------------------------------\nEven granted that humans and computers are both information-processing devices, some would argue that they process information based on very different principles. A popular claim is that a computer does exactly what it is programmed to do, and nothing else, and this is supposed to be very different from humans who do what they want themselves —so any parallels between humans and computers are impossible. I think this reasoning is fundamentally wrong, for two reasons.\nFirst, modern AI systems do not just do what they are programmed to do. That’s because their function is based on *learning*. They are programmed to learn from input data. The input may be a database determined by the programmer; it can be obtained by crawling the internet; or it can be the result of interactions with the environment, like a robot using a camera or users typing words, and so on. What the programmer really does is to provide an *algorithm* for learning. The algorithm is based on certain goals or *objective functions* that the AI is trying to optimize. An AI dedicated to searching the internet for images that resemble a given target image will learn to optimize the accuracy of its search results, for example by maximizing the number of clicks users make on each image it proposes.\nWhat this means is that anyone who programs an AI cannot really know in detail what the AI will actually do, because it is often impossible to know what kind of input the AI will receive, and it is equally difficult to understand what the AI will learn from it. Even in the simplest case where the programmer completely decides the input to the AI, the input is often so complex (say, millions of pictures downloaded from the internet) that it is impossible for a human programmer to understand what can be learned from that data.\nThe second reason why this is not a major difference between humans and AI is that just like an AI is programmed by humans, we humans are designed—one might say “programmed”—by evolution. From an evolutionary perspective, we are programmed to maximize an objective function which roughly is given by the total number of copies of our genes in the population. To satisfy such programming, we gather a lot of data—by reading things, talking to people, and simply looking around—which is not so different from an AI.\nSo, I have turned the claim about the difference between AI and humans on its head. What humans and AI have *in common* is that both are programmed *by something else* to have certain goals and needs; nobody has really decided “by themselves” to have the needs and goals they have. To accomplish those goals, both humans and AI gather data from the environment and learn from it, which leads to actions that are *very difficult to predict*. So, in the end there is little difference between AI and humans, except regarding the source of the original programming—whether it was by evolution or a human programmer.\nCan an AI actually suffer?\n--------------------------\nBy now, I hope to have convinced you that an AI is a useful model of many phenomena taking place in the human brain. But perhaps there are limits. Some would argue that we cannot talk about AI’s or robots suffering: They may *seem* to be suffering, or look like they are suffering, but in fact they are not, because they cannot *feel* anything.\nI think this argument may not be completely wrong, but it is quite irrelevant. Obviously, it depends on the exact definition of what suffering is. It is true that AI may not feel suffering in the same way as humans because that might require that it is conscious. This argument against AI’s suffering really hinges on two points: First, that an AI is not conscious, and second, that consciousness is necessary for suffering.\nHowever, conscious feeling is only one part of suffering. The situation is similar with emotions, such as fear, which are actually clever information-processing mechanisms. The conscious feeling of being afraid is only one part of a complicated process involving cognition (or information-processing), behavioural tendencies, and several other aspects. I would argue it is the same for suffering.\nSuffering is actually a signal in a complex information-processing system. The real meaning of the suffering signal is that an *error* occurred—this will be elaborated in several chapters in this book. Any information-processing system can create error signals. That’s why we can, in that specific sense, say that an AI or a robot is suffering, even if they are not conscious. All that would be missing is the conscious feeling components of suffering.\nThere is an even more important reason why it is largely irrelevant here if an AI really suffers according to some stringent definition of the word. This book does not just aim to describe the mechanisms of suffering; the primary goal here is to develop various ways of alleviating suffering. For the purpose of *reducing* suffering, it does not matter if computers actually suffer in some deep sense. If we can reduce suffering in an AI that is sufficiently human-like, then, with reasonable probability, the same methods will apply to humans, and they will reduce suffering in humans, including the conscious experience of suffering. In other words, the AI is really a simulation or a model of mechanisms that are relevant for making humans happier.\nFor those who find it impossible to think that a computer could suffer in any sense of the word, I suggest the following viewpoint that they can use while reading this book. Trying to understand human suffering by AI is one big thought experiment, where we try to understand how much the AI would suffer under various circumstances, *if* it were able to consciously experience suffering. It is like a mathematical model of atoms, or like a computer simulation of chemical processes. Everybody agrees that models and computer simulations are not the real thing, but they can help us understand the actual natural processes, and in particular, predict their behaviour. A model may tell you how a change in one quantity, say X, leads to a change in another quantity, Y. If you know that, you can perhaps choose X to maximize or minimize Y—which might be suffering.\nIntelligence is painful—overview of this book\n---------------------------------------------\nThe central claim in this book is that if we create an artificial intelligence that is really intelligent, really worthy of its name, it is necessarily going to suffer— more or less like humans. In spite of the many differences between AI’s and humans, there is a common logic in the design. In order to achieve sufficiently human-like intelligence, certain design principles have to be followed, and these lead to suffering. This book explores several interwoven ideas about such a computational basis of suffering, and the necessity of suffering as a part of intelligence.\nThe fundamental principle is that suffering is caused by error signalling, which is typically due to frustration. Frustration occurs when the system, generally called an “agent”, fails to achieve a goal. Such errors are inevitable in a complex world, where things are uncertain and unpredictable, and we have limited control over them. Error signalling is necessary for any sufficiently intelligent system, since such error signals are used by learning algorithms. Our brain produces error signals automatically, and we simply cannot shut off the error-signalling system.\nIn fact, the complexity of the world is overwhelming for any known intelligent system, whether the very best supercomputer in the world, or the most intelligent human brain. The computations available to them cannot handle all the different possibilities, for example, in choosing action sequences to reach a given goal.\nModern AI uses *learning* to cope with such complexity. Learning from complex data enables particularly sophisticated information processing. However, for such learning to be really successful, huge data sets are required. Obtaining data sets which completely capture the complexity of the world is rarely possible in practice.\nThese two factors, lack of computational resources together with scarcity of data, mean that the intelligent agent cannot work optimally. Its intelligence, and its control over the world, are limited. Thus, there will be errors: Its actions do not always lead to the desired outcome. This is the fundamental reason why such errors, and suffering, are ubiquitous.\nSuffering is greatly enhanced by several information-processing principles inherent in the design of human-like intelligent systems. One is the phenomenon of experience replay, where memories related to past errors are recalled and repeated in the system in order to optimize learning about past experiences. Likewise, plans for future actions are constantly computed, which means the agent simulates or “imagines” them in its mind, together with the ensuing errors. Such replay and planning multiply any suffering arising from real events. Errors are signalled as if those bad, imagined events happened for real. Replay and planning even lead to conscious suffering in humans. Somehow, some part of the brain does not understand that recalling or imagining an event is not the same as actually living it. That is why we suffer from mishaps which only happen in our imagination.\nModern AI has found systems based on parallel and distributed information processing to be useful for programming intelligent systems, which makes it understandable that our brain uses similar principles. However, parallel and distributed systems are difficult to control by any central “executive”. Instead, different computational modules are competing for control, making, for example, any sustained attention or concentration difficult. Any internal control of the agent’s computations is further reduced by emotions such as fear, which work as evolutionarily conditioned “interrupts” of ongoing processing. Thus, the agent has little control of even its own internal processing, let alone the external world.\nYet another problem is the difficulty of understanding how uncertain most perceptions and inferences are. Perceptions are often highly subjective and contextual interpretations, sometimes little more than guesses. Yet, humans often mistakenly think that our perceptual systems are able to discover some underlying objective reality. To simplify the overwhelming complexity of the world, an agent may further divide it into categories. However, the categories may be arbitrary and assigning objects to categories difficult. Our inability to appreciate such uncertainty and even arbitrariness leads to more suffering.\nFinally, the goals and desires that have been programmed in us by evolution are ultimately counterproductive and make us unhappy. Evolution never had our happiness as its goal anyway. In fact, it forces us to do things which are clearly bad for our happiness, something I call “evolutionary obsessions”. Evolution makes us worry about our survival and our evolutionary performance, creating a sense of self. In fact, evolution does *not* want us to reduce suffering because the error-signalling system is necessary for learning and optimal behaviour. Evolution *does* want us to learn to act in more and more efficient ways, but the goals towards which this intelligence is used are those set by evolution, not us. Even worse, both AI and humans are usually trying to satisfy their drives and desires endlessly, without any limits; at no point do they become satiated and think that they have achieved enough.\nHowever, there is hope. At the very end of the book, I sketch methods that can be used to decrease suffering, based on the theories outlined in this book. What is needed is a reprogramming of the brain. The key method is to retrain the brain by inputting new data into the learning system. The new data will change the computations in such a way that error signals, and in particular frustration, are reduced. This is difficult and takes a lot of time, but various forms of philosophical contemplation and meditation attempt to do it. These methods are rather logical consequences of the theory, while at the same time, they have mostly been proposed earlier in Buddhist, and to some extent Stoic, philosophy. Thus this book can be seen as an attempt to construct a scientific, computational theory on the underpinnings of such philosophies.\nDefining suffering\n==================\nIn this chapter, I try to define the word “suffering”. This is not an easy task, as we will quickly see. Defining the term properly requires, to some extent, elucidating the underlying mechanisms creating suffering.\nOne fundamental point here is that I exclude physical pain from the definition of suffering; I use the word suffering synonymously with mental pain. Nevertheless, I will start the search for a definition of suffering by considering the closely related concept of pain, taken here in the medical sense of physical pain.\nThe central conclusion of this chapter is that the main definitions of suffering consider it based on either *frustration* or a *threat to the intactness* of the person. These two definitions, and especially the definition based on frustration, are the basis of the developments of the rest of this book. From a more abstract viewpoint, I will argue that such suffering can be seen as *error signalling*, similarly to physical pain.\nMedical definitions of pain\n---------------------------\nLet us start by defining pain. Pain has been given a widely accepted consensus definition by the International Association for the Study of Pain (IASP) as\n> \n> [iasp] Pain is an unpleasant sensory and emotional experience associated with actual or potential tissue damage or described in terms of such damage.\n> \n> \n> \nSurprisingly, while this definition was originally adopted in 1979, it is still used with minimal modifications. It posits damage to any tissue of the person, or any threat of such damage, as the origin of pain. Pain is then defined as an ensuing unpleasant experience. While this definition has been found to be quite useful in a clinical context, deeper theoretical analyses have found various problems.[1](#fn1)\nOne important controversy is whether one should define pain as a subjective experience, or as something that has a more objective existence. The definition above talks about an “experience” which is here interpreted as a conscious, subjective experience: something that only I am aware of, and which you cannot measure in any objective way. As we will discuss in more detail in Chapters [emotions.ch] and [consciousness.ch], this problem of subjective conscious experience vs. objectively observed phenomena is ubiquitous in neuroscience and psychology.\nThe problem with talking about such subjective experience in a scientific context is that objective, reproducible measurement is the basis of science. Fortunately, subjective experience can be measured in various indirect ways, such as verbal report. That is, we can ask the patient if there is pain. Yet we will never know for sure what the patient actually feels. In particular, we cannot tell how her experience of pain compares with other people’s experience: Does she feel more pain than some other patient?\nThis problem in the definition is to some extent alleviated by the reference to tissue damage, which is objectively measurable and reasonably well-defined. Yet, as this definition clearly points out, actual tissue damage is not *necessary* for pain—since it can be just “potential”—and thus it does not provide a basis for measuring pain or for objectively defining it. (In fact, the definition does not actually say that pain is in any sense proportional to the amount of damage— it is well-known that tissue damage that creates a lot of pain in one person may create little pain in another—which complicates any measurement even more.) Another related problem with the IASP definition above is that it relies heavily on the word “unpleasant”, which is not a very well-defined term, and, again, quite subjective.\nOne approach to solve these problems is to take an evolutionary approach. To begin with, we could replace “unpleasant experience” in the definition by “experience that has evolved to motivate behaviour, which avoids or minimises tissue damage, or promotes recovery”.[2](#fn2) Here we go towards defining pain using its evolutionary, functional role, while still acknowledging the subjective nature of pain by talking about an “experience”. The downside of such an approach is that it works on a very abstract level, and provides no details on what might cause pain, in contrast to the IASP definition which explicitly points at tissue damage (even if only potential). This definition, in a sense, shifts the burden to understanding the evolutionary goals of certain experiences, which is not easy either. However, one obvious candidate for such an evolutionary goal is minimizing tissue damage and recovering from it, which links this evolutionary approach with the IASP definition. In more general terms, the evolutionary goal could be the maintenance of “homeostasis”, that is, an optimal balance in the physiological condition of the body.[3](#fn3) Such evolutionary logic can be applied on suffering as well, and we will see related argumentation throughout this book.\nMedical and psychological definitions suffering\n-----------------------------------------------\nIn contrast to pain, suffering is a rather neglected term in science, and there is nothing like a consensus definition. Intuitively, most people would think suffering also contains an *unpleasant feeling or experience* as an integral part, while being more abstract and general than physical pain, in particular including more psychological and emotional aspects. A typical dictionary definition is “Feeling of pain or strong stress, either physical or emotional”.[4](#fn4) Like pain, suffering is often considered a subjective experience which cannot be objectively measured.[5](#fn5)\nOne simple and concrete approach to define suffering is to give examples of phenomena related to suffering and possibly producing suffering. A typical list would contain grief, sadness, discomfort, distress, anguish, fear—which is just a random sample, and many different lists can be produced. While this is a good starting point, it does not lead to a solid scientific theory.\nTerms such as psychological pain or mental pain are often preferred in neuroscience, and some attempts at definitions of those terms have been made.[6](#fn6) In this line of thinking, suffering is really a generalization of pain. This may not solve the problem of defining suffering, since the burden is then simply shifted to defining pain, but then we can leverage the large literature on pain, in particular the IASP definition just given, as well as any of its critique and improvements.\nOne approach distinguishes three kinds of pain: physical pain, social pain, and psychological pain.[7](#fn7) An interesting emphasis in this line of research is that all these different kinds of pain are neurally very similar in the sense that the brain areas responsible are the same.[8](#fn8) Here, physical pain is primarily due to physical damage to the body, but it can also be felt when there is a strong anticipation of such physical damage (think about going to a dentist), reminiscent of the IASP definition. In contrast, social pain is an unpleasant feeling due to social exclusion or rejection. Psychological pain is largely the same as what I call mental pain or suffering, and attempt to define here.\nImportantly for our purposes, in such an approach, mental or psychological pain is often assumed to be due to *reward loss*, defined as follows[9](#fn9)\n> \n> [Reward loss is] a negative discrepancy between expected and obtained rewards.\n> \n> \n> \nIn other words, reward loss happens when you expect a reward but don’t get it, and it leads to mental pain. Reward loss can also be called *frustration*, although sometimes this term is reserved for the actual suffering caused by reward loss. This provides one important computational viewpoint: reward loss is a function of computations involving expectations, observations of the obtained reward, and their difference.\nAn alternative approach emphasizes how suffering is related to our person, or self. Psychological or mental pain has been characterized as an aversive state of high self-awareness of inadequacy,[10](#fn10) or a negative appraisal of an inability or deficiency of the self.[11](#fn11) This is analogue to IASP definition of physical pain in the sense that there is “damage”, but on a more mental level, to our image of ourselves as a psychological and social entity.[12](#fn12)\nA particularly potent and influential idea in this vein is that suffering necessarily involves a threat to, or a loss of, the *intactness of the person*, as proposed by Cassell:[13](#fn13)\n> \n> [casselldef] Suffering is a state of severe distress induced by the loss of the intactness of person, or by a threat that the person believes will result in the loss of his or her intactness.\n> \n> \n> \nThis is a natural generalization and abstraction of the IASP definition of pain as related to “tissue damage”. Damage to the intactness of the person includes tissue damage but is something much more general, in particular including damage to one’s self-image. It is of course crucial here to understand what “intactness” means more precisely; Cassell emphasizes the generality of this notion, saying that “suffering may occur in relation to any aspect of personhood”.[14](#fn14)\nA unifying theory, which combines several different kinds of suffering and pain in a single framework by linking them through the concept of frustration, has been developed by van Hooft.[15](#fn15)[hooft] He starts from an Aristotelian conception of the human person as having four “parts of the soul”. They range from the lowest level of biological functioning to the emotional/desiring functions and the rational functions, finally reaching the sense of the meaning of existence. In his theory, each of these parts has its own goals, its own form of “fulfillment”, which is again an Aristotelian idea. Suffering is then nothing else than frustration, namely “frustration of the tendency towards fulfillment” of one of the different parts of the soul. In this theory, the lowest level of biological functioning is even below ordinary pain and pleasure, and simply about staying healthy and alive. Ordinary physical pain is the frustration on the emotional/desiring level, where the goal of the organism is to gain pleasure and avoid pain. Frustration of rational (intellectual) function refers to suffering which happens when it is not possible to reach long-term goals that one would usually expect to reach and plan for. Frustration on the highest, “spiritual” level happens when it is impossible to understand why it is me that is sick—in the medical context where van Hooft writes—or life seems meaningless due to the despair and fear which a malady brings with it. This last kind of suffering brings us close to the kind of suffering considered in existential philosophy.[16](#fn16)\nClosely related definitions can be found in the literature on *stress*: Lazarus and collaborators define “psychological stress” as “a particular relationship between the person and the environment that is appraised by the person as taxing or exceeding his or her resources and endangering his or her well-being”.[17](#fn17) So, we have to consider the possibility that stress is another kind of suffering, or a mechanism for suffering. However, I don’t take such a view in this book because the classic definition by Hans Selye, “the father of stress”, proposes that “stress is the non-specific response of the body to *any* demand” (my italics). This is a very general definition, and Selye has explicitly emphasized that positive, happy events can induce stress just as well as negative, threatening ones; think about an athlete engaged in a competition. Based on this definition, it does not seem possible to simply consider stress as one kind of suffering, unless we focus on the negative kind of stress, termed “distress” by Selye.[18](#fn18) The distinction between distress and “pleasant” stress is, unfortunately, not very clear; it has been proposed that it is the *unpredictability and uncontrollability* of a situation which distinguish the unpleasant distress from other kinds of stress.[19](#fn19) Their connection to suffering will be considered from different viewpoints in this book.\nAncient philosophical approaches to suffering\n---------------------------------------------\nCenturies before any such modern developments, some ancient philosophers already made great progress in understanding suffering. The best expert on the topic may have been the Buddha, and in fact the whole of Buddhist philosophy can be seen as a theory of suffering—especially when considering the original version proposed by the Buddha himself. He gave the following description of suffering:[20](#fn20)\n> \n> [fournoble1] Union with what is displeasing is suffering; separation from what is pleasing is suffering; not to get what one wants is suffering.\n> \n> \n> \nThis is actually not so much a definition of what suffering is, but rather an attempt to describe what the main causes of suffering are.\nStoic philosophers in ancient Greece and Rome had very similar ideas. Epictetus, one of the most famous Stoics, describes mechanisms that lead to suffering as follows:[21](#fn21)\n> \n> [epictetospromise] [D]esire promises the attainment of that of which you are desirous; and aversion promises the avoiding that to which you are averse. However, he who fails to obtain the object of his desire is disappointed, and he who incurs the object of his aversion wretched.\n> \n> \n> \nThese are essentially a reformulation of the points given by the Buddha above. We can summarize these philosophical ideas as the following two causes for suffering, each with two variants:[22](#fn22)\n1. Not getting what one wants (Buddha, Epictetus)\n2. Something pleasant, which one would like to be present, is absent (Buddha)[23](#fn23)\n3. Not being able to avoid what one is averse to, i.e., wants to avoid (Epictetus)\n4. Something unpleasant is present (Buddha)\nThen, the definitions by the Buddha and Epictetus can be interpreted in terms of wanting (point 1) and aversion (point 2) only. Point 1 in particular defines the typical case of frustration, related to reward loss considered above. Using the term somewhat liberally, we can also call the suffering in point 2 frustration, since the desire to avoid something is frustrated. Thus, we see that both the ideas of both the Buddha and Epictetus can be simply summarized as saying that suffering comes from frustration.\nTwo main kinds of suffering\n---------------------------\nNow I shall try to recapitulate the ideas above, both ancient and modern, as succinctly as possible. I think we only need to talk about two kinds of suffering, or rather two mechanisms producing suffering, namely:\n1. Frustration (e.g., Buddha, Epictetus, several neuroscientists[24](#fn24))\n2. Threat to the intactness of the person, including their self-image (e.g., IASP, Cassell)\nFurthermore, van Hooft’s theory could be seen as combining these two aspects.\nBased on this dichotomy, this book will develop two computational definitions of suffering, one each for these two aspects. Frustration will be specifically dealt with in Chapters [planning.ch] and [rpe.ch], and the threat to the person (or “self”) in Chapter [self.ch]. Chapter [self.ch] will also propose how the two aspects can be seen in a unified framework by seeing threats to the person as frustration of certain long-term goals, a bit in the same sense as van Hooft’s theory.\nThe emphasis in the following chapters is, obviously, on information processing. As already argued in the introduction, my main justification for talking about information-processing is that the framework of information-processing is a *practically useful* way of describing suffering in the precise sense that it can tell us something about how to *reduce* suffering. Information-processing is something that we can influence, something we can intervene on, so from a practical viewpoint, it is a very important aspect of suffering to investigate. Focusing on information-processing is also perfectly in line with the current emphasis on cognition in neuroscience and psychology; I see cognition as synonymous with information-processing.\nUsing the pain system for broadcasting errors\n---------------------------------------------\nTo conclude this chapter, I discuss some computational principles that explain why pain and suffering are so closely related. First, I propose that on a more abstract computational level, both pain and suffering are essentially *error signals*, messages that something is going wrong from the viewpoint of the goals and rewards of the system. Clearly, frustration signals that something went wrong in terms of not getting what one wants, and a similar case will be made for the threat to the person in Chapter [self.ch]. Such error signals are in fact ubiquitous in artificial intelligence, where, in particular, they can be used for learning to choose actions better in view of maximizing rewards. We will see several kinds of error signals in the following chapters, and see how some of them can be interpreted in terms of suffering.\nPain is thus an evolutionarily primitive form of an error signal. Its unique feature is that pain signals are *broadcast widely* in the information-processing system. This is important in an agent whose computation is distributed into different modules (whether processors or brain regions). For such an agent, it is necessary that any really important signal uses a special pathway that allows it to be broadcast to all, or most of, the modules. The pain signal is indeed broadcast widely to different neural systems, and the signal can change the behaviour of the whole organism in terms of making it stop whatever it is doing and pay close attention to the pain. Furthermore, when an error signal drives the learning of the system, as we will consider in later chapters, it often needs to be observed by several of the modules, and such broadcasting is essential.[25](#fn25)\nSuffering is largely using the neural systems originally developed for physical pain, as already mentioned. This makes evolutionary sense if we think that computationally more sophisticated forms of error signalling, such as frustration, simply started using the evolutionarily older pain signalling pathway, adapting it for their own purposes. That was practical because the pain system already existed, and served well the purpose of broadcasting error signals to many brain regions. Using the physical pain system for signalling mental pain is thus a useful computational shortcut.[26](#fn26)\nYet, merely talking about information-processing, as in a computer, may seem a rather incomplete description of suffering. Why does suffering *hurt*, if it is merely a signal in an information-processing system? This is in fact exactly the same problem that we encountered with the IASP definition of pain above: Is it a subjective experience, or something more objective and measurable?\nThe evolutionary rationale just described explains why suffering “hurts” in the same way as physical pain: the physical pain system is hijacked for the purposes of suffering or mental pain. (Perhaps this explains why we talk about mental “pain” in the first place.) The very dichotomy of experience vs. objective measurements is thus exactly the same for pain and suffering, since it is a question of similar experiences and neural pathways. Nevertheless, explaining why physical pain actually subjectively feels like it does in the first place, is an extremely difficult question; it is intimately related to the question of consciousness, which we defer to Chapter [consciousness.ch]. We shall rather continue, in the next chapter, by elucidating the computational underpinnings of a particular form of error signal: frustration.\nFrustration due to failed plan\n==============================\nIn this chapter, I propose a first model where frustration is a fundamental mechanism for suffering. It is assumed that an agent, whether a human or an AI, engages in *planning* of action sequences in order to get to a desired *goal state*. A state is here an abstraction of the properties such as location, context, and possessions of the agent. Frustration happens when the goal state is not reached in spite of the agent executing the planned sequence of actions.\nI start by emphasizing the great computational difficulty of such planning of action; it is one reason why frustration happens. Another central concept here is wanting or *desire*, which is a complex phenomenon we will return to several times in this book. As an initial definition, I consider desire as a computational process that suggests goals for the planning system. Finally, I discuss the importance of committing to a single plan, even in the presence of conflicting desires, based on Bratman’s concept of *intention*. This chapter lays out the framework in simple, largely intuitive terms; the main terms and concepts will be greatly refined in later chapters.\nAgents, states, and goals\n-------------------------\nOne may be tempted to think of an artificial intelligence as a system which just takes input, and processes information. However, information-processing in itself will actually be rather pointless unless it leads to some kind of visible output or action regarding the external world. In the very simplest case, action can just mean printing some text on a computer screen, so this is not necessarily a big leap.\nIn AI, the basic unit of analysis is often what is called an intelligent *agent*, i.e. a system which not only processes information but also takes actions. In fact, the word “agent” literally means “one that acts”. An intelligent agent can be artificial, such as a robot or an AI program, but the term also encompasses biological agents, that is, animals. In one extreme, an artificial agent could be just a program inside a computer, working in a virtual world with no physical body; actions would essentially consist of sending messages inside an information network. In the other extreme of human-like artificial agents, it could be a robot having a body with arms and legs; actions would include walking and grasping objects. In this book, we will see examples of both extremes—in addition to agents that actually are animals or humans.\nSuch an agent needs at least two things: perception and action selection. Perception is actually a tremendously difficult task but we defer its discussion to Chapter [ml.ch] and especially Chapter [perception.ch]. To begin with, we assume perception is somehow satisfactorily performed, and consider the question of how the agent is to choose its actions.[27](#fn27)\nPerhaps the simplest and most intuitive approach to action selection is to think in terms of *goals*. This is an introspectively compelling approach: We usually think of ourselves as acting because there are certain goals that we try to reach. That may be why it has also been a dominant approach in the history of AI, starting from around 1960. For example, a very simple thermostat has the goal of keeping room temperature constant; a cleaning robot has the goal of removing dust and dirt from the room.\nThe goals of humans are fundamentally determined by evolution, complemented by societal and cultural influences. In this book, I simply use the word “evolution” for brevity to describe the joint effect of biological evolution, culture and society. The assumption here is that the latter two are ultimately derived from biological evolution, although this is of course a controversial point. Fortunately, for the purposes of this book, the exact relationship between biology and culture is irrelevant: What matters is that the goals that humans strive for are largely, even if sometimes very indirectly, determined by some outside forces. Humans can set some intermediate goals, such as getting a job, but those are usually in the service of final biological or societal goals, such as being nourished or raising one’s social status. In the case of AI, in contrast, the goals are usually supplied by its human designers. This may seem to be a fundamental difference between AI agents and humans, but we will see in later chapters that it may not matter very much; the human designer plays a role similar to evolution in terms of being an outside force. In any case, regardless of where the goals come from, the way they are translated into action may still be rather similar in both cases.\n### Modelling the world as states\nIn order to choose its actions, the agent should have some kind of a model of how the world works, where the agent itself is seen as a part of the “world” modelled. The model expresses the agent’s beliefs of what the world is typically like, and how the world changes from one moment to another, in particular as a function of the actions the agent takes.\nAI research uses a very abstract kind of a world model based on the concept of a *state*, where each possible configuration of the world is one state. For example, if a cleaning robot is in the corner of a room, facing south, and there is only a single speck of dust in the room, at exactly two meters east from the robot, that is one state, we can call it state #1. If the agent finds itself 10 cm further to the west, it is in another state, say state #2; likewise, if another speck of dust appears in the room, that means the agent is in state #3 (and if the speck of dust appears *and* the agent is 10 cm further to the west, that is yet another state). In the simplest case, such a world model has states which are categorical, or discrete; in other words, there is a finite number of possible states. This is a very classical AI approach, but we will see alternatives in later chapters.\nAny effects of the agent’s actions can now be described in terms of moving from one state to another, called state transitions. Indeed, in addition to knowing what the states of the world are like, the agent should know something about the transitions between the states caused by its actions. If it finds itself in state #1 and decides to move forward, does it find itself in state #2, or #47, or something else? If the agent’s world model can predict the effects of its actions in terms of transitions from one state to another, it is ready to start taking actions.\nUsing this formalism of states, the basic approach to action selection is that one of the states is designated as the goal state, by some mechanisms to be specified. The agent then uses its capabilities to reach the goal state, starting from its current state. That is surprisingly difficult, usually requiring complicated computation called *planning*, which is a central concept we consider next.[28](#fn28)\nPlanning action sequences, and its great difficulty\n---------------------------------------------------\nThe fundamental problem in action selection is that you must actually select *sequences* of short actions. For example, if the agent in question is you, and you decide to get something from the fridge in the kitchen, you need to take a step with your left foot, take a step with your right foot, repeatedly, until you finally can open the fridge door—which consists of several actions such as: raise your arm, grab the handle, pull it down, pull the door, and so on. In some cases, you may easily know how to choose the right sequence, but it is not easy at all in many cases. For programming AI, this has turned out to be quite a challenge.\nThis is known as the problem of planning in AI. Using the formalism of world states, action sequences can be represented graphically as what is called a tree (Figure [searchtree.fig]). The “root” of the tree represents the state you’re in at the moment. Any action leads to a branching of the tree, and depending on the action, you will find yourself on any of the new branches. (In this figure, taking an action means moving down in the tree, and we assume for simplicity that there are just two actions you can take at any time point). At the end of a given number of actions, or levels in the tree, you find yourselves at one of those states which are depicted at the outer “leaves” of the tree. Of course, the tree continues almost forever since you can take new actions all the time, but to keep things manageable, we consider a tree of a limited depth.\n[searchtree.fig]\nLet’s now assume that the agent has been given a goal state by the programmer. It would be one of the states at the lowest level of the tree. The central concept here is *tree search*; many classical AI theories see intelligence as a search for paths, or action sequences, among a huge number of possible paths in the action tree. In particular, the planning system finds a path which leads from the current state to the goal state. Such search may look simple, but the problem is that with such paths or action sequences, the number of possibilities grows exponentially. If you have, at any one time point, just two different actions to choose from, then after 30 such time points you have more than a billion (more exactly 2 to the power of 30) possible action sequences to choose from. What’s worse is that typically an AI would have many more than just two possible courses of action at any one point. The computations involved easily go beyond the capacity of even the biggest computers or brains. So, it may be impossible to “look ahead” more than a couple of steps in time.\nThe difficulty of such planning may be difficult for humans to understand since evolution has provided various tricks and algorithms that solve the problem quite well, as we will see below. We may only be able to grasp the difficulty of planning in some slightly artificial examples such as the search tree above. One of the more realistic examples would be planning a route between two points. Say you find yourself in a random location in Paris and want to go to the Eiffel Tower using public transportation. Even if you remembered every detail of the metro map as well as the geography of Paris itself, you would still need quite a lot of computation, in the form we would usually call “thinking”. Which metro station should I walk to, or should I perhaps use the bus? What is the best itinerary once inside the metro station? It is not surprising that people tend to use mobile phone apps to solve this problem.[29](#fn29)\nBoard games are an extreme example of the difficulty of planning. Humans playing chess have great difficulties in thinking more than one or two moves ahead. The search tree has a lot of branches at every move because there are so many moves you can take. Even worse, your opponent can do many different things. (The uncertainty regarding what your opponent will do further adds to the complexity, but that is another story.)\nA lot of the activity we would casually call *thinking* is actually some kind of planning. If you are thinking about where to go shopping for a new electronic gizmo, or how to reply to a difficult message from your friend, you are considering different courses of action. Basically, you’re going through some of the paths in the search tree. Often, such thinking or planning happens quite involuntarily, even when you’re supposed to be doing something else, a topic to which we will return in Chapter [replay.ch].\nFrustration as not reaching planned goal\n----------------------------------------\nEquipped with this basic framework for action selection, we are ready to define frustration in its most basic form. We start by considering one part of the Buddha’s definition of suffering mentioned above: “not to get what one wants”: This is in fact a typical dictionary definition of frustration. To achieve a deeper computational understanding of the phenomenon, we integrate this with the framework of planning.\nJust like AI, complex organisms such as humans engage in planning: Based on their perception of the current environment, they try to achieve various goals by some kind of tree seach. For such organisms, it is vital to know if a plan failed, so that they can re-plan their behaviour, and even learn to plan better in the future, as will be considered in detail in the following chapters.\nWe thus formulate the basic case of frustration as *not reaching a goal that one had planned for*, and the ensuing error signal. Sometimes frustration rather refers to the resulting unpleasant mental state; that is, frustration refers to the actual suffering instead of the cause for suffering. I use frustration in both meanings in this book.[30](#fn30)\nThis initial definition will be refined and generalized in later chapters. In later chapters, we will see how central error signals are to any kind of learning. For example, a neural network that learns to classify inputs, or predict the future, is essentially minimizing an objective function which gives the error in such classification or prediction. Frustration can be seen as a special case of such error signalling: It signals that an action plan failed. In complex organisms like humans, which are constantly engaged in planning, frustration is an extremely important learning signal, and the basis of a large part of the suffering.\nDefining desire as a goal-suggesting mechanism\n----------------------------------------------\nHowever, there is a slight inconsistency here: Frustration was actually defined as not getting what one *wants* in Chapter [suffering.ch]. How is this related to our computational formulation based on planning above? In other words, what exactly is wanting, or desire, in a computational framework like ours?\nIn everyday intuitive thinking, action selection is indeed supposed to be based on wanting, or desires: An agent takes an action because it *wants* something, and it thinks it is reasonably likely to achieve or obtain it by that action. I choose to go to the fridge because I want orange juice. However, the account earlier in this chapter made no reference to the concepts of desire or wanting. In AI, the term “desire”, which I consider synonymous with “wanting”, can actually be used in a couple of different meanings.\nIn the very simplest definition, if the agent has a goal to plan for, one could simply say the agent “wants” to reach the goal state; desires would essentially be the same as goals. In such a meaning, desire is a kind of purely rational, “cold” evaluation of states and objects. However, the word has many more connotations in everyday language. Desire also has an affective aspect we could call “hot”, in which we are “burning with desire”, unable to resist it.\nA definition that is a bit more in the direction of “hot” can be obtained by considering desire as a specific computational process inside the agent. To begin with, desire has been defined as a “psychological state of motivation for a specific stimulus or experience that is anticipated to be rewarding”.[31](#fn31) While a “psychological state” may mean different things, here we consider a state as something where a particular kind of information processing is performed—another meaning would be related to conscious experience which we treat in Chapter [consciousness.ch]. In practical terms, desire is often triggered by the perception of something that is rewarding to possess.[32](#fn32) Such perception of an object often means that the agent should be able to get the object after a rather short and uncomplicated action sequence: If you see something, it is likely to be within reach.\nFrom the viewpoint of information-processing, we thus define desire as: *A computational process suggesting as the goal a state that is anticipated to be rewarding and seems sufficiently easily attainable from the current state.*[desiredef1] I want to emphasize that I am considering desire as a particular form of information-processing: Desire is not simply about preferring chocolate to beetroot, nor is it merely an abstract explanation of the behaviour where I grab a chocolate bar. It is sophisticated *computation* that is one step in the highly complex process that translates preferences into planning and, finally, into action.\nThe starting point for that processing is that your perceptual system, together with further computations, estimates that from the current state, you can relatively easily get into a state of high reward—the exact formalism for “rewards” will be introduced in Chapter [rpe.ch]. This realization will trigger, if you are properly programmed, further computational processes that will try to get you in that desired state by suggesting it as the goal for your planning system. When all this happens, you want the new state, or have a desire for that new state, according to the definition just given. For example, if chocolate appears in your visual field, your brain will compute that the state where you possess the chocolate is relatively easy to reach, and produces high reward; so it will choose the chocolate-possessing state as a possible goal and input that to the planning system.\nTherefore, the definition of desire just given shows how the intuitive definition of frustration as not getting what one wants and the computational definition of not reaching the goal are essentially the same thing. This definition also solves a question which many readers must have asked while reading this chapter: Where do the goals for planning come from? In a very simple AI, there might be just a single goal, or a small set, defined by the programmer. But for a sophisticated agent, that is certainly not the case: The number of possible goals for a human agent is almost infinite. Here, we define desire as a computational process that suggests new goals to the planning system, and thus this is where the goals come from. (More details on how the desire system could actually choose goals will be given in Chapter [dual.ch], which also considers a different aspect of desire related to its interrupting and irresistible quality.)\nA closely related concept is *aversion*, which is in a sense the opposite of desire. However, from a mathematical viewpoint, aversion is very similar to desire: The agent wants to *avoid* a certain state (or states) and wants to be in some other state. For example, the agent wants to be in a state in which some unpleasant object is not present. Thus, it is really a case of wanting and desire, just framed in a more negative way. I do not use the term aversion very much in this book since it is mathematically contained in the concept of desire. Whenever I use the word “desire”, aversion is understood to be included.[33](#fn33)\nIntention as commitment to a goal\n---------------------------------\nWe have seen that a desire is something that suggests the goal of the agent. Note that I’m not saying that desire *sets* the goal, but it *suggests* a goal to the planning system. This difference is important because there might be conflicting goals; you don’t grab the chocolate every time you have desire for it. The agent needs to choose between different possible objects of desire. This is particularly important because attaining the desired goal state often takes time: The whole plan has to be executed from beginning till end, and new temptations—activations of the desire system which suggests new states as possible goals—may arise meanwhile. Some method of arbitrating between different desires is necessary.\nSuppose the desired state for a monkey is one where the monkey has eaten a banana. The banana is currently high up the tree which is in front of the monkey, so the monkey needs to perform a series of actions to reach that desired state: it must climb up the tree, take the banana, peel it, and eat it. The monkey must thus figure out the right sequence of actions to reach the desired state— this is just the planning problem discussed above—and launch its execution.\nBut, suppose the monkey suddenly notices another banana in another tree near-by. Its desire system may suggest that the new banana looks like an interesting goal. The monkey now faces a new problem: Continue with the current banana plan, or set the new banana as a new goal? It may be common sense that after the monkey has launched the first banana plan, the monkey should, in most cases, persist with that plan until the end. The monkey should not start pondering, halfway up the first tree, whether it actually prefers to get the other banana in the other tree, even if it looks a bit sweeter. The key idea here is *commitment* to the current plan, and thus to a specific goal.\nThe reason why commitment is important comes fundamentally from computational considerations. Since computing a plan for a given goal takes a lot of computational resources, it would be wasteful to abandon it too easily in favour of a new goal. It might be wasteful even to just consider alternative goals seriously, because that would entail computing possible plans to achieve all those goals. The agent must settle on one goal and one plan and execute it without spending energy thinking about competing goals.[34](#fn34)\nAnother utility of commitment is that the agent has a better idea of what will happen in the future, and it can start planning further actions, e.g. a plan on what to do after reaching the goal of the current plan. So, while the monkey is climbing up the tree, it *is* a good idea to start thinking about the best way of getting the banana in the other tree *after* having grabbed the first banana. That would be planning the long-term future after completing the execution of the current plan; perhaps the monkey can directly jump to the other tree from the location of the first banana. Such long-term plans would obviously collapse if the monkey didn’t first get the first banana due to lack of commitment, being distracted by yet another thing.\nCommitment to a goal is also called *intention* in AI, and leads to an influential AI framework called belief-desire-intention (BDI) theory. Belief refers to the results of perception, which give rise to desires. BDI theory argues for the importance of intentions, as commitment to specific goals, on top of beliefs and desires.[35](#fn35) Of course, there must be some limits to such commitment: If something unexpected happens, the goal may need to be changed. If a tiger appears, the monkey cannot persist with the goal of just eating a banana. Chapter [emotions.ch] will consider the importance of emotions such as the fear aroused by the tiger as one computational solution.[36](#fn36)\nThe concept of intentions has important implications for suffering, as will be discussed in detail in later chapters. To put it simply, I will propose that frustration and suffering are stronger if an intention is frustrated, as opposed to frustration of a simple desire as in the basic definition.\nHeuristics can help in planning\n-------------------------------\nStill, we have not yet solved the central problem regarding the planning system. We saw above that because of the huge number of possible paths in planning, a complete tree search is quite impossible in most cases. Is planning then impossible? Fortunately, there are a couple of tricks and approximations that can be used to find reasonable solutions to the planning problem. Here we first consider what is called *heuristics*, while a more sophisticated solution is given in the next two chapters. These solutions also have important implications for the definition of frustration, and understanding suffering.\nA heuristic means some kind of method for evaluating each state in the search tree, usually by giving a number that approximately quantifies how good it is, i.e. how close to the goal it is. The point is that a heuristic does not need to be exact—if it were, we would have already solved the problem. It just gives a useful estimate, or at least an educated guess, of how “good” a state is.[37](#fn37)\nSometimes, it is quite simple to program some heuristics in an AI agent. Consider a robot whose goal is to get some orange juice from the fridge and deliver it to its human master. Clearly, when the robot has orange juice in its hand, it is rather close to the goal; we could express that by a numerical value of, say, 8. If it is, in addition, close to its master, it is very close to its goal, say a value of 9. The most important thing is, however, to assist the robot at the beginning of the search, and that is where the heuristic is the most powerful. So, we could say that when the robot is close to the fridge, the heuristic gives a value of 2. When it has opened the fridge, the value is 3, and so on.\nWith such heuristics, the search task would not require that much computation. The robot just has to figure out how to get to some easily reachable state with a higher heuristic than the current state. Assuming the robot starts at an initial state with heuristic value 0, it would quickly compute that what it can achieve rather easily is a state of heuristic value of 2, by going to the fridge. The length of the tree to be searched for is thus much shorter, i.e. much fewer actions steps need to be taken in that subproblem. Once there, it only has to figure out how to open the door to get to the state with heuristic value of 3. Thus, the heuristic essentially divides a long complex search task into smaller parts. Each of these parts is quite short, so the exponential growth of the number of branches is much less severe.[38](#fn38)\nThere is one famous success of AI where such tree search with heuristics was hugely successful: The Deep Blue chess-playing machine,[39](#fn39) which beat the chess world champion, for the first time, in 1997. Its main strength was the huge number of sequences of moves (i.e. paths in a search tree) it was able to consider, largely because it was based on purpose-built, highly parallel hardware that was particularly good in such search computations on the chessboard. But its success was also due to clever heuristics, the main one being called “piece placement”, computed as the sum of the predetermined piece values with adjustments for location, telling how good a certain position is. (In chess, the state is the configuration of all the pieces on the board, and called a “position” in their jargon.)\nEvolution has also programmed a multitude of heuristics in animals. Think about the smell of cheese for a rat. The stronger the smell, the closer the rat is to the cheese. The rat just needs to maximize the smell, as it were, and it will find the cheese. No complex planning is needed—unless there are obstacles in the way.[40](#fn40)\nHowever, the crucial problem is how to find such heuristics for a given planning problem. In fact, this is a very difficult problem, and there is no general method for designing them. Nevertheless, there is a general principle which has been found tremendously useful in modern AI, and can be used here as well: *learning*. Modern AI is very much about using learning from data as an approach to solving the problem of programming intelligence. In the case of planning, it turns out that a general approach for solving the planning problem is to learn to rate the states, i.e. learn to associate some kind of heuristic to each world state. This is why in the next two chapters, we delve into the theory of machine learning. Its specific application to solving the planning problem will be considered in Chapter [rpe.ch], where we also consider a different approach to defining frustration.\nMachine learning as minimization of errors\n==========================================\nIn this chapter, we will go through some of the basics of the backbone of modern AI: machine learning. Such AI crucially relies on learning from incoming data, which is also true of the brain. Machine learning is most often used in conjunction with neural networks, which are powerful function approximators, loosely mimicking how computations happen in the brain. We will also consider an alternative, older approach to intelligence based on symbols, logic, and language, which is now called “good old-fashioned AI”. (The preceding chapter with its discrete, finite states, was an example of this latter approach.)\nA central message in this chapter is that learning is often based on some measure of error. Minimizing such errors means optimizing the performance of the system. The fundamental importance of computing and signalling such errors is important in future chapters where such errors are directly linked to suffering, generalizing the concept of frustration. I conclude this chapter by claiming that any kind of learning from complex data can lead to quite unexpected results, something that the programmer could not anticipate.\nNeurons and neural networks\n---------------------------\nModern AI is based on the observation that the human brain is the only “device” we know to be intelligent for sure and without any controversy. It is actually not easy to define what “intelligence” means, and I will not attempt to do that in this book.[41](#fn41) Yet, nobody denies that the brain is intelligent—or, to put it another way, it enables us to behave in an intelligent way. The brain is intelligent as if by definition; it is the very standard-bearer of intelligence. If you want to build an intelligent machine, it makes sense to try to mimick the processing taking place in the brain.\n### Neurons as tiny processors\nThe computation in the brain is done by specialized cells called neural cells or neurons. A schematic picture of a neuron is in Fig. [neuron.fig]. A neuron receives input from other neurons, processes that input, and outputs the results of its computations to many other neurons. There are tens of billions of neurons in the human brain. Each single neuron can be seen as a simple information-processing unit, or a processor.[42](#fn42)\n[neuron.fig]\nAll these tiny processors do their computations simultaneously, which is called *parallel* processing in technical jargon. The opposite of parallel processing is serial processing, where a single processor does various computational operations one after another—this is how ordinary CPU’s in computers work. Another major difference between the brain and ordinary computers is that processing in neurons is also *distributed*. This means that each neuron processes information quite separately from the others: It gets its own input and sends its own output to other neurons, without sharing any memory or similar resources. Compared to an ordinary PC, the brain is thus a *massively parallel and distributed* computer. Instead of a couple of highly sophisticated and powerful processors as found in a PC, the brain has a massive amount—billions—of very simple processors. (Parallel and distributed processing is discussed in detail in Chapter [self.ch].)\nWhile the actual neurons are surprisingly complex, in AI, a highly simplified model of a real neuron is used. Sometimes, such a model is called an artificial neuron to distinguish it from the real thing, but for simplicity, we call them just neurons. Like a real neuron, an artificial neuron gets input signals from other neurons, but each such input signal is very simple, just a single number; we can think of it as being between zero and one, like a percentage. Based on those inputs, the neuron computes its output which is, again, a single number. This output is, in its turn, input to many other neurons.\nIn such a simple model, the essential thing is to devise a simple mathematical formula for computing the output of the cell as a function of the inputs. In typical models, the output is computed, essentially, as a *weighted sum* of the inputs. The weights used in that sum are interpreted, in the biological analogy, as the strengths of the connections between neurons, or the incoming “wires” on the left-hand-side of Fig. [neuron.fig]. These weights can get either positive or negative values: The weight is defined as zero for those neurons from which no input is received. The weighted sum is usually further thresholded (i.e. passed through a nonlinear function) so that the output is forced to be between zero and one. In the brain, the connections are implemented through small communication channels called synapses, which is why the weights can also be called “synaptic”.\nImportantly, these weights can be interpreted as a pattern, or a template, which the neuron is sensitive to. Thus, a neuron can be seen as a very simple *pattern-matching unit*. The neuron gives a large output if the pattern of all the input signals matches the pattern stored in the vector of weights or connection strengths.\nAs an illustration, consider a neuron which has a weight with the numerical value +1 for inputs from another neuron, let’s call it neuron A, as well as a zero weight from neuron B, and a weight of -1 neuron C. This neuron will give a maximal output (it is maximally “activated”) when the input to the neuron is similar to the pattern of those weights: It will output strongly when neuron A gives a large output and the neuron C has a small output, while it does not care what the output of neuron B might be. In other words, the neuron computes how well the pattern stored in its synaptic weights matches with the pattern of incoming input, and its output is simply a measure of that match.\nSuch pattern-matching is obviously most useful in processing sensory input, such as images. Consider a neuron whose inputs come from single pixels in an image. That is, the input consists of the numerical values of each pixel, telling how bright it is, i.e. whether it is white, black, or some sort of grey. Then, we can plot the synaptic weights as an image, so that the grey-scale value in each pixel in this plot is given by the corresponding synaptic weights. If they are -1 or +1 as in the previous example, we can plot those values as black and white, respectively. A neuron could have synaptic weights as in Fig [digitneuron.fig]. Clearly, this neuron is specialized for detecting a digit, in particular number two.\n[digitneuron.fig]\nOf course, in reality, to recognize digits (or anything else) in real images, things are much more complicated. For one thing, the pattern to be recognized could be in a different location. If the digit is moved just one pixel to the right or to the left, the pattern matching does not work anymore, and the neuron will not recognize the digit. Likewise, if the digit were white on a black background instead of black on a white background, the same pattern-matching would not work. To solve these problems, we need something more sophisticated.\n### Networks based on successive pattern matching\nBuilding a *neural network* greatly enhances the capabilities of such an AI, and solves the problems just mentioned. A neural network is literally a network consisting of many neurons. Networks can take many different forms, but the most typical one is a hierarchical one, where neurons are organized into layers, each of which contains several cells, actually quite a few sometimes. The incoming input first goes to the cells in the first layer which compute their outputs and send them to neurons in the second layer, and so on. This is illustrated in Figure [neuralnetwork.fig].\nFrom the viewpoint of pattern-matching, such a network performs successive and parallel pattern matching. The input is first matched to all the patterns stored in the first-layer neurons, and those neurons then output the degrees to which the input matched their stored patterns or templates. These outputs are sent to the next layer, whose neurons then compare the pattern of first-layer activities to their templates. So, the second-layer patterns are not patterns of original input (such as the pixels of an image) but patterns of the first-layer activities, which form a description of the input on a slightly more abstract level. This goes on layer by layer, so that each neuron in each layer is “looking for” a particular kind of pattern in the activities of the neurons in the previous layer. The patterns are always stored in the synaptic weights of the neurons.\nThe utility of such a network structure is that it enables much more powerful computation. For example, consider the problem of a digit which could be in slightly different locations in the image, as mentioned above. The problem of different locations can be fixed by having several neurons in the first layer, each of which matches the digit in one possible location. All we need in the second layer is a neuron that adds the inputs of all first-layer neurons, and thus computes if any of them finds a match. With such a scheme, the second-layer neuron is able to see if there is a digit “2” at any location in the image.\n[neuralnetwork.fig]\nFinding the right function by learning\n--------------------------------------\nNow, a crucial question is how the synaptic connection weights can be set to useful values. In modern AI, the synaptic weights between neurons are *learned from data*, hence the term *machine learning*. Learning is really the core principle in most modern AI. Especially in the case of neural networks, it is actually difficult to imagine any alternative. How could a human programmer possibly understand what kind of strengths are needed between the different neurons? In some cases, it might be possible: in image processing, the first one or two layers do have rather simple intuitive interpretations, as we have alluded to above. However, with many layers—and neural networks can have thousands of them— the task seems quite impossible, and hardly anybody has seriously tried to design such neural networks by fixing the weights manually, based either on some theory or intuition.\nIn the brain, the situation is quite similar. There is simply not enough information in the genome—which is somewhat analogous to the programmer here—to specify what the synaptic connection strengths should be for all the neurons. It would hardly be optimal anyway to let the genes completely determine the synaptic connections, since animals live in environments that may change from one generation to another, and some individual adaptation to circumstances is clearly useful. What happens instead is that the synaptic weights change as a function of the input and the output of the neuron, or as a function of perceptions and actions of the organism. The capability of the brain to undergo such changes is called “plasticity”, and those changes are the biophysical mechanism underlying most of learning in humans or animals.\nHow such changes precisely happen in the brain is an immensely complex issue, and we understand only some basic mechanisms. Nevertheless, in AI, a number of relatively simple and very useful learning algorithms have been developed. Neural networks using them learn to perform basic “intelligent” tasks such as recognizing patterns (is it a cat or a dog?) or predicting the future (if I turn left at the next intersection, what will I see?). Learning in a neural network in such a case is based on learning a *mapping*, or function, from input data to output data. Let’s first consider a single neuron. It can basically learn to solve simple classification problems, as illustrated in Figure [twoclusters.fig]. If the classes are nicely separated in the input space, a single neuron can learn, as its synaptic weights, the pattern that precisely describes the difference between the classes.\n[twoclusters.fig]\nHowever, a network with many neurons can learn to represent much more complex functions from input to output. The input data could be photographs and the output data could be a word describing the main content of the photograph (“cat”, “dog”, or “unicorn”). The learning of the input-output mapping then consists of changing the synaptic weights of all the neurons in all the layers. In successive layers, the network performs increasingly sophisticated and abstract computations, consisting of matching the inputs successively to the templates given by the weight vectors in each layer. After learning the right mapping, you can input a photograph to the system, and the output of the neural network will give its estimate of what the photograph depicts.\nThere is an infinite number of different ways you can use a neural network by just defining the inputs and outputs in different ways. If you want to learn to predict future stock prices, the input data would be the past prices and the output, current stock prices. You can create a recommendation system that recommends new products to people in online shopping by defining the inputs to be some personal information of the customers, and the output whether the customer bought a certain item or not.[43](#fn43) In a rather unsavoury application, the inputs would be what a social media user likes, and the output some sensitive personal information (say, sexual orientation), and then you can predict that sensitive information for anybody. Whether the prediction is accurate is another question, of course.\nHere, we see one main limitation of machine learning: The availability of data. Where do you get the sensitive personal information of social media users in the first place, i.e. where do you get the data to train your network? Maybe nobody wants to give you such sensitive data. In other cases, the data may be very expensive to collect; for example, in a medical application, useful measurements and their analyses may cost a lot of money. Finding suitable data is a major limiting factor in neural network training; this is a theme we will come back to many times. Learning needs data, obviously; but it also needs the right kind of data, and enough of it.\nLearning as minimization of errors\n----------------------------------\nAfter we have got the data, we need to define how to actually perform the learning. Most often, the learning is based on formulating some kind of error, and the network then tries to minimize it by an algorithm. The error is a function of the data, i.e. something that can be computed based on the data at our disposal, and tells us something about how well the system is performing.\nSuppose the data we have consists of a large number of photographs and the associated categories (cat/dog etc.). To recognize patterns in the images, the network could learn by minimizing the percentage of input images classified incorrectly, called classification error. Alternatively, suppose we want to learn to predict how an agent’s actions change the world—say, how activation of an artificial muscle changes the position of the arm of a robot. In that case, what is minimized is prediction error: The magnitude of the difference between the predicted result of the action and the true result of the action (which can be observed after the action). Such errors don’t usually go to zero, i.e. there will always be some error left even after a lot of learning. This is due to the uncertain and uncontrollable nature of the world and an agent’s actions; that’s another theme we will discuss in detail in later chapters.\nWe also need to develop an algorithm to minimize the errors, but there are standard solutions that usually are satisfactory. What most such algorithms have in common is they learn by making tiny changes in the weights of the networks. This is because optimizing an error function, such as classification error, is actually a very difficult computational task: There is usually no formula available to compute the best values for the weight vectors. In contrast, what is usually possible is to obtain a mathematical formula that gives the *direction* in which the weights should be changed to make the error function decrease the fastest. That direction is given by what is called the *gradient*, which is a generalization of the derivative in basic calculus.\nSo, you can optimize the error function step by step as follows. You start by assigning some random values to the weight vectors. Given those values, you can compute the gradient, and then take a small step in that direction (i.e. move the weight vector a bit in that direction), which should reduce the error function, such as prediction error. But you have to repeat that many times, often thousands or even millions, always computing the gradient for the new weight values obtained at the previous step. (The direction of the gradient is different at every step unless the error function is extremely simple.) Such an algorithm is called *iterative* (repeating) because it is based on repeating the same kind of operations many times, always feeding in the results of the previous computation step to the next step. Using an iterative algorithm may not sound like a very efficient way of learning, but usually an iterative gradient algorithm is the only thing we are able to design and program.\nSuch an algorithm is a bit like somebody giving you instructions when you’re parking your car and cannot see the right spot precisely enough. They will only give instructions which are valid for a small displacement. When they say “Back”, that means you need to back the car *a little bit*, and then follow some new instructions. This is essentially an iterative algorithm, where you get instructions for the direction of a small displacement, and they are different at every time step.\nNeural networks use an even more strongly iterative method, based on computing the gradient for just a few data points at a time. A data point is one instance of the input-output relation data, for example, a single photograph and its category. In principle, a proper gradient method would look at all the data at its disposal, and push the weights a small step in the direction that improves the error function, say the classification error, for the data set as a whole. However, if we have a really big data set, say millions of images, it may be too slow to compute the error function for all of them. What the algorithms usually do is to take a small number of data points and compute the gradient only for those. That is, you just take a hundred photographs, say, and compute the gradient, i.e. in which direction the weights should be moved to make the classification accuracy better, for those particular images. Importantly, at every step you randomly select a new set of a hundred images, and do the same thing for those images.\nThe point is that you are still *on average* moving the weights in the right direction, so this is not much worse than computing the real gradient. But crucially, you can take steps much more quickly, since the computation of the gradient is much faster for the small sample. It turns out that in practice, the benefit of taking more steps often overwhelms the slight disadvantage of having just an approximation of the gradient.[44](#fn44)\nPutting these two ideas together, we get what is called the *stochastic gradient descent* algorithm. Here, “descent” refers to the fact that we want to minimize an error. “Stochastic” means “random”, and refers to the fact that you are computing the gradient for randomly chosen data points, so you are going in the right direction only on average.\nSuppose you’re in an unfamiliar city and you need to get to the railway station. Your “error function” is the distance from the station. You can ask a passer-by which direction the station is, and you get something analogous to the gradient for one data point. Now, of course, that direction given by the passer-by is not certain, she could very well be mistaken; maybe she even said she is not quite sure about the direction. But you probably prefer to walk a bit in that direction, and then ask another passer-by. This is like stochastic gradient descent, where you follow an approximation of the gradient, given by each single data point. The opposite would be that instead of following each person’s advice one after the other, you first ask everybody you see on the street where they think the station is, and move in the average of the directions they are giving. Sure, you would get a very precise idea of what the right direction is, but you would advance very slowly—this is analogous to using the full, non-stochastic gradient.\nGradient optimization vs. evolution\n-----------------------------------\nThere are many more ways of optimizing an error function, and many systems that can be conceptualized as the optimization of a function. In particular, evolution is a process where the error function called *fitness* is optimized. Fitness is basically the same as reproductive success, which can be quantified as the expected number of offspring of an organism.[45](#fn45)\nFitness is, of course, maximized, while errors in AI are minimized. However, this difference is completely insignificant on the level of the optimization algorithms, since maximization of a function is the same as minimizing the negative of that function. Thus, evolution can equally well be seen as minimization of the negative fitness.\nIn general, such a function to the optimized—whether minimized or maximized—is called an “objective function”. The objective function does not necessarily have to be any kind of a measure of an error, although in AI, it often is. Note that the objective function is different from the function from the input to the output that the neural network is computing, as described above. The objective function is what enables the system to learn the best possible input-output function, so it works on a completely different level.\nEvolution works in a very different way than stochastic gradient descent. But it is actually possible to mimick evolution in AI and use what is called evolution strategies, evolutionary algorithms, or genetic algorithms. These are iterative algorithms which are sometimes quite competitive with gradient methods. They can optimize any function, which does not need to have anything to do with biological fitness. For example, we can learn the weights in a neural network by such methods. The idea is to optimize the given error function by having a “population” of points in the weight space, which is like a population of individual organisms in evolution.\nLike real evolution, such algorithms are based on two steps. First, new “offspring” is generated for each existing “organism”. In the simplest case, you randomly choose some new weight values close to the current weight values of each organism, which is a bit like asexual reproduction in bacteria, with some mutations to create variability. Then, you evaluate each of those new organisms by computing the value of the error (such as classification error) for their values for weights. Finally, you consider the value of the error as an analogue of fitness in biological evolution, albeit with the opposite sign because fitness is to be maximized while an error function is to be minimized. What this means is that you let those organisms (or weight values) with the smallest values of the error “survive”, i.e. you keep those weight values in memory and discard those weight values which have larger (that is, worse) values of the error function. You also discard the organisms of the previous iteration or “generation”, since those individual organisms die in the biological analogy.\nSuch an evolutionary algorithm will find new weight values which are increasingly better because only the organisms with the best weight values survive in each iteration. Thus, it is an iterative algorithm that optimizes the error function. It is a randomized algorithm, like stochastic gradient descent, in the sense that it randomly probes new points in the weight space. In fact, an evolutionary algorithm is much more random than stochastic gradient descent, since gradient methods use information about the shape of the error function to find the best direction to move to, while evolutionary methods have no such information. This is a disadvantage of evolutionary algorithms, but on the other hand, one step in an evolutionary algorithm can be much faster to compute since you don’t need to compute the gradient, just random variations of existing weights.[46](#fn46)\nSo, we see that *both evolution and machine learning are optimizing objective functions.* The optimization algorithms are often quite different, but they need not be. One important difference is that in machine learning, the programmer knows the error function, and explicitly tells the agent to minimize it. In real biological evolution, fitness is an extremely complicated function of the environment; it cannot be computed by anybody, nor can its gradient. Real biological fitness can only be observed afterwards, by looking at who survived in the real environment, and even then you only get a rough idea of the values of fitness of the individual organisms concerned: Those who die *probably* had a low fitness, but it is all quite random—even more than stochastic gradient descent. (If an organism were actually able to compute the gradient of its fitness, that would give it a huge evolutionary advantage.) Another important difference is that in biology, evolution works on a very long time scale, over generations, while in AI, the learning in the neural networks happens typically inside an individual’s life span. The evolutionary algorithms in AI typically learn within an individual agent’s lifespan as well, only simulating “offspring” of a neural network in its processors.\nLearning associations by Hebbian rule\n-------------------------------------\nSo far, we have seen learning as based on finding a good mapping from input to output. Such learning is called *supervised* because there is, metaphorically speaking, a “supervisor” that tells the network what the right output is for each input. Yet, sometimes it is not known what the output of a neural network should be, or whether there is any point at all in talking about separate input and output—especially if we are talking about the brain. In such a case, learning needs to be based on completely different principles, called *unsupervised learning*. In unsupervised learning, the learning system does not know anything about any desired output (such as the category of an input photo). Instead, it will try to learn some regularities in the input data.\nThe most basic form of unsupervised learning is learning associations between different input items. In a neural network, they are represented as connections between the neurons representing those two items. For example, if you have one neuron representing “dog” and another neuron representing “barking”, it is reasonable that there should be a strong association between them.\nOne theory of how such basic unsupervised learning happens in the brain is called Hebbian learning.[hebb] Donald Hebb proposed in 1949 that when neuron A repeatedly and persistently takes part in activating neuron B, some growth process takes place in one or both neurons such that A’s efficiency in activating B is increased.[47](#fn47) Such Hebbian learning is fundamentally about learning associations between objects or events. A very simple expression of the Hebbian idea is that “cells that fire together, wire together”, where “firing” is a neurobiological expression for activation of a neuron. In this formulation, Hebbian learning is essentially analysing statistical correlations between the activities of different neurons.[48](#fn48)\nOne thing which clearly has to be added to the original Hebbian mechanism is some kind of forgetting mechanism. It would be rather implausible that learning would only increase the connections between neurons. Surely, to compensate, there must be a mechanism for decreasing the strengths of some connections as well. Usually, it is assumed that if two cells are not activated together for some time, their connection is weakened, as a kind of negative version of Hebb’s idea.[49](#fn49)\nHebbian learning has been widely used in AI, and it has turned out to be a highly versatile tool. You can build many different kinds of Hebbian learning, depending on how the inputs are presented to the system and on the mathematical details of how much the synaptic strengths are changed as a function of the firing rates. You can also derive Hebbian learning as a stochastic gradient descent for some specially crafted error functions.[50](#fn50)\nLogic and symbols as an alternative approach\n--------------------------------------------\nThe inspiration for neural networks is that they imitate the computations in the brain. Since the brain is capable of amazing things, that sounds like a good idea. But historically, before neural networks, the initial approach to AI was quite different. It was actually more like the world of planning we saw in Chapter [planning.ch], where the world states are discrete, and there are few if any continuous-valued numerical quantities.\nIn early AI, it was thought that logic is the very highest form of intelligence, and therefore, AI should be based on logic. Also, the principles of logic are well-known and clearly defined, based on hundreds of years of mathematics and philosophy, so they should provide, it was thought, a solid basis on which to build AI. In modern AI, such logic-based AI is not very widely used, but it is making a come-back: It is increasingly appreciated that intelligence is, at its best, a combination of neural networks and logic-based AI—now called “good old-fashioned” AI, or GOFAI for short. Such logic provides a form of intelligence that is in many ways completely different from neural network computations, as we will see next.\n### Binary logic vs continuous values\nMathematical logic is based on manipulating statements which are connected by operators such as AND and OR. For example, a robot might be given information in the form of a statement that “the juice has orange colour AND the juice is in the fridge”. Any statement can also be made negative by the NOT operator. An important assumption is such systems, in their classical form, is that any statement is either true or false; no other alternatives are allowed. This goes back to Aristotle and is often called the law of the “excluded third”. That is, truth values are binary (have two values).\nSuch logic is perfectly in line with the basic architecture of a typical computer. Current computers operate on just zeros and ones, and those zeros and ones can be interpreted as truth values: Zero is false and one is true. Such computers are also called “digital”, meaning that they process only a limited number of values, in this case just two. Our basic planning system in the preceding chapter, with its finite number of states of the world, was an example of building AI with such a discrete approach, and planning is fundamentally based on logical operations.\nThe brain, in contrast, computes with quantities which are in “analog” form, which means the continuous-valued numbers, which take a potentially infinite number of possible values. Artificial neural networks do exactly the same, as they are trying to mimick even this aspect of the brain. It is rather unnatural for the brain to manipulate binary data or to perform logical operations. It is possible only due to some very complex processes which we do not completely understand at the moment.[digitalvsanalog]\nThis distinction between digital and analog information-processing is another important difference between ordinary computers on the one hand, and the real brain or its imitation by neural networks on the other. (Earlier we saw the distinction between parallel and distributed processing in the brain versus the serial processing in an ordinary computer.) The digital nature of ordinary computers implies that any data that you input has to be converted to zeros and ones. This is actually a bit of a problem because a lot of data in the real world does not really consist of zeros and ones. For example, images are really intensities of light at different wavelengths, measured in a physical unit called “lux”. One pixel in an image might have an intensity of 1,536 lux and another 5,846 lux. It is, again, rather unnatural to represent such numbers using bits, which is why processing non-binary data such as images is relatively slow in modern computers, compared to binary operations.[51](#fn51)\n### Categories and symbols\nSaying that things are either true or false is related to thinking in terms of *categories*. Human thinking is largely based on using categories: We divide all the perceptual input—things that we see, hear, etc.—into classes with little overlap. Say, you divide all the animals in your world into categories such as cats, dogs, tigers, elephants, and so on, so that each animal belongs to one category—and usually just one. Then, you can start talking about the animals in terms of true and false. You can make a statement such as “Scooby is a dog”, and that is either true or false based on whether you included that particular animal in the dog category; any other (third) option is excluded.\nCategories are usually referred to by *symbols*, which in AI are the equivalent of words in a human language. For example, we have a category referred to by the word “cat”, which includes certain “animals” (that’s another category, actually, but on a different level). Ideally, we have a single word that precisely corresponds to each single category, like the words “cat” and “animal” above. Such symbols are obviously quite arbitrary since in different languages the words are quite different for the same category. An AI system might actually just use a number to denote each category.\nWe see that logic-based processing goes hand-in-hand with using categories, which in its turn leads to what is sometimes called symbolic AI. These are all different aspects of GOFAI.\n### From hand-coded logic to learning\nHistorically, one promise of GOFAI was to help in medical diagnosis, where the programs were often called “expert systems”. This sounds like a case where categories must be useful since medical science uses various categories referring to symptoms (“cough”, “lower back pain”) as well as diagnoses (“flu”, “slipped disk”).\nThe basic approach was that a programmer asks a medical expert how a medical diagnosis is made, and then simply writes a program that performs the same diagnosis, or makes the same “decisions” in the technical jargon. For example, one decision-making process by the human expert might be translated into a formula such as\n> \n> IF cough AND nasal congestion AND NOT high fever THEN diagnosis is common cold\n> \n> \n> \nHowever, this research line soon ran into major trouble. The main problem was that medical doctors, and indeed most human experts in any domain, are not able to verbally express the rules they use for decision-making with enough precision. This is rather surprising since we are operating with human language and well-known categories. The situation is different from neural networks where it is intuitively clear that no expert can directly tell what the synaptic weights should be, because their workings are so complex and counterintuitive. Yet, it turned out that even medical diagnoses are often based on intuitive recognition of patterns in the data, which is a form of *tacit knowledge*. Tacit knowledge means knowledge, or skills, which cannot be verbally expressed and communicated to others.\nA major advance in such early AI was to understand that expert systems should actually learn the decision rules based on data. Again, learning provides a route to intelligence that is more feasible than trying to directly program an intelligent system. Given a database with symptoms of patients together with their diagnoses given by human experts, a machine learning system can learn to make diagnoses. Such learning is not so fundamentally different from learning by neural networks. What is different is that the data is categorical (“cough”, “no cough”), and the functions are computed in a different way, for example by combining logical operations such as AND, OR, and NOT.\n### Categorization and neural networks\nBy definition, such logic-based AI can only learn to deal with data which is given as well-defined categories. Yet, real data is often given as numbers instead of categories; even medical input variables often include numerical data in the form of lab test results. In this medical diagnosis, we have indeed a category called “high fever”. The system is effectively dividing the set of possible body temperatures into at least two categories, one of which is “high fever”. How are such categories to be defined? What is fever? What is low fever and what is high fever? Here we see a deep problem concerning how categories should be defined based on numerical data, such as sensory inputs.\nAgain, some progress can be made by learning, this time learning the categories themselves from data. The AI can consider a huge number of possible categorizations of body temperatures: It could try setting the threshold for high fever at any possible value. If there is enough data on previous diagnoses by human doctors, the system could use that to learn the best threshold. In fact, the right threshold could be found as the one that minimizes classification error.\nHowever, while it is possible to learn categories in such very simple numerical data, GOFAI has great difficulties in processing complex numerical data. It is virtually impossible to use it to process high-dimensional sensory input, such as images consisting of millions of pixels. This was a rather big surprise for GOFAI researchers in the 1970s and 80s. After all, categorization of visual input is done so effortlessly by the human brain that it may seem to be easy. Yet, AI researchers working in the GOFAI paradigm found it to be next to impossible. The early research on GOFAI was fundamentally over-ambitious, grossly underestimating the complexity of the world, as well as the complexity of the brain processes we use to perceive and make decisions.\nOne reason for the current popularity of neural networks is that processing high-dimensional sensory data is precisely what they are good at. In fact, it is clear by now that neural networks operate in a completely different regime from such logic-based expert systems. There are no categories, and no symbols, in the inner workings of neural networks: What they typically operate on is numerical, sensory input such as images, or some transformations of such sensory input. Raw grey-scale values of pixels are kind of the opposite of neat, well-defined categories.\nAs such, neural networks and logic-based systems can complement each other in many ways. Usually the categories used by a logic-based system need to be recognized from sensory input, so a neural network can tell the logic-based system whether the input is a cat or a dog. In particular, a neural network can take sensory data as input, and its output can *identify the states* used in action selection; to begin with, it can tell the planning system what the current state of the agent is. In Chapter [dual.ch] we will consider in more detail this fundamental distinction between two different modes of intelligent information-processing, which are found both in AI and human neuroscience.\nEmergence of unexpected behaviour\n---------------------------------\nFinally, let me mention a phenomenon that is typical of any learning system. “Emergence” means that a new kind of phenomenon appears in some system due to complex interactions between its parts. It is a special case of the old idea, going back to Aristotle at least, that “the whole is more than the sum of its parts”. For example, systems of atoms have properties that atoms themselves do not—consider the fact that a brain can process information while single atoms hardly can. Likewise, evolution is based on emergence. Its objective function is given by evolutionary fitness, which sounds like a very simple objective function. Nevertheless, it has given rise to enormous complexity in the biological world, as well as human society. What is typical of such emergence is that its result is extremely difficult to predict based on knowledge of the laws governing the system. If the objective function given by fitness had been described to some super-intelligent alien race a few billion years ago, they would hardly have been able to predict what the world looks like these days.\nMachine learning is really all about the emergence of artificial intelligence. We build a simple learning algorithm and give it a lot of data, and hope that intelligence emerges. It is the interaction between the algorithm and the data that gives rise to intelligence. This seems to work, if the algorithm is well designed, there is enough data of good quality, and sufficient computational power is available. Such emergence in machine learning is actually a bit different from emergence in other scientific disciplines. In physics, very simple natural laws *by themselves* can give rise to highly complex behaviour. In machine learning, the complexity of the behaviour of the system is, to a large extent, a function of the complexity of the *data*. In some sense, one could even say the complex behaviour learned by an AI does not emerge but is extracted, or “distilled”, from input data. The complexity of the input data is due to the complexity of the real world, which is obvious when inputting a million photographs into a neural network.\nThe emergent nature of the behaviour learned by an AI implies that, just like in evolution, there is often something unexpected in the resulting system. The complexity of the input data usually exceeds the intellectual capacities of the programmer. So, the programmer of an AI cannot really know what kind of behaviour will emerge: Often the system will end up doing something surprising.\nIn this book, we will encounter several forms of emergent properties in learning systems which are related to suffering. While some kind of suffering may be necessary as a signal that things are going wrong, we will also see how an intelligent, learning system will actually undergo *much more* suffering than one might have expected. To put it bluntly, a particularly intelligent system will find many more errors in its actions and its information-processing. In fact, finding such errors was necessary to make it so intelligent in the first place. Therefore, a learning system may learn to suffer much of the time, even though that is not what the programmer intended.\nFrustration due to reward prediction error\n==========================================\nNow, armed with modern machine learning theory, we revisit the problem of action selection and the concept of frustration. In planning treated in Chapter [planning.ch], the main computational problem is looking several steps ahead, which can lead to quite impossible demands of computational capacities. Another constraint is that it requires a model of how your actions affect the world, i.e. where do you go in the search tree when you perform a given action in a given state. As such, planning is not really a good method for action selection if computational resources are very limited, as in a simple computer, or a very simple animal such as an insect.\nIn this chapter, we consider an alternative way to action selection, based on learning. A paradigm called reinforcement learning enables learning intelligent actions without any explicit planning, thus avoiding many of its problems. It also generalizes the framework of a single goal to maximization of rewards obtained at different states. While it can be performed even in very simple animals and computers, it is also used by humans; it is similar to how habits work.\nWe then consider how frustration can be defined in such a case; it can no longer be simply defined as not reaching the goal—since there is no explicit goal. We define more general error signals called reward loss and reward prediction error, which have been linked to signals of certain neurons in the mammalian brain. Thus, we expand the view where frustration is related to error signalling by linking it to errors in prediction.\nRepeated frustration is thus something necessary for learning algorithms to work, and intelligence may not be possible without some frustration. We further see how the very construction of an agent based on reward maximization means that it is insatiable, never satisfied with the amount of reward obtained. Moreover, it can be directed towards intermediate goals which are not valuable in themselves, but simply predictive of future reward. Evolutionary rewards, in particular, can lead to behaviour which resemble obsessions.\nMaximizing rewards instead of reaching goals\n--------------------------------------------\nIn modern AI, action selection is most often not based on planning, but a framework where the obtained *rewards*, or reinforcement, is maximized. This is useful because often an AI does not have just a single goal to accomplish, but many things it should take care of. Defining behaviour as maximization of rewards as opposed to reaching goals is also often thought to be more appropriate for modelling behaviour in simple animals, which are thought to be incapable of the sophisticated computations needed in planning (more on this in Chapter [replay.ch]).\nFor example, if a cleaning robot disposes of some dust in the dustbin, it could be given a reward signal. Since there are many rooms and many dustbins in the building, it makes sense to give a reward whenever the robot disposes of some of the dust. In principle, we could decide to give it a single reward when all the rooms are completely clean; however, it is common sense to rather give it a reward every time it removes some dirt or dust from any of the rooms. After all, the robot has done something useful every time it reduces the amount of dust in the room; telling this to the robot is highly useful information, and it would simply complicate the learning if the reward were postponed until the robot has completed some larger part of the task.\nIn fact, giving a single reward at the end would mean the robot has to engage in long-term planning, which is difficult. A “piece-wise” training by giving rewards for small accomplishments is not very different from how you would teach a child to perform a rather demanding, long task, say tying shoelaces: Divide the task into successive parts and give the child a small encouragement when it completes each small part. This is computationally advantageous since it eliminates the need for long-term planning, a bit like the heuristics we saw above.[52](#fn52)\nReinforcement or reward can also be negative; if the robot tries to put household items in the dustbin, it can be given some. Negative reinforcement is really what we usually call a punishment—but the word is interpreted without any moral connotations here.\nThus, we actually ground action selection in the optimization of an objective function, i.e. a quantity to be optimized. Earlier, we saw that minimization of an error function, such as the number of images incorrectly classified, is the way an AI can learn to recognize objects in images. Here we define a different kind of objective function which is the basis of action selection: It is equal to the *sum of all future rewards*. It is a function of the action selection parameters of the agent, and more precisely, it expresses how much reward the agent can obtain by behaving according to its current action selection system.\nSuch a learning process based on maximization of future rewards by learning a value function is called *reinforcement learning*. Reinforcement learning can be seen as a third major type of learning in AI, in addition to supervised and unsupervised learning.\nIn a sense, this future reward is the ultimate objective function of an agent. Its maximization, by tuning the action-selection system, is the very meaning of life of the agent. The objective functions we saw earlier, used to learn things like pattern recognition by minimization of errors, are there merely to help in maximizing this reward-based objective function.\nIn such a reward-based objective function, more weight is often put on the rewards in the near future as opposed to rewards in the far-away future, which is called discounting. The justification for this is complicated, but suffice it to say that such discounting is often evident in human behaviour: Humans prefer to have their reward right now, and value it less if they have to wait. To keep the discussion simple, I sometimes ignore discounting in what follows, but it could be used in almost every case considered in this chapter.[53](#fn53)\nLearning to plan using state-values and action-values\n-----------------------------------------------------\nAs such, the sum of future rewards gives a more general framework than having a single goal as in Chapter [planning.ch], since trying to reach a single goal can be accommodated in the reward framework by simply giving a reward when the agent reaches the goal, and no reward otherwise. In such a case, discounting further means the agent receives more reward if it reaches the goal more quickly, which is intuitively reasonable.\nIt turns out that we can use this reformulation of planning as reward maximization to our advantage, since the algorithms developed for maximizing future rewards give a particularly attractive way of solving the problem of planning. In Chapter [planning.ch], we saw how difficult planning is due to the exponential explosion in the number of possible plans to choose from. While heuristics were proposed as a practical trick to make the computations more manageable, there is no universal way of designing good heuristics.\nLike in other branches of AI, it has been found that *learning* solves many of these problems. Intuitively, if the agent encounters the same planning problem again and again, it can store information about the previous solutions (or attempts) in memory. For example, a cleaning robot will probably clean the same building many, many times, and a delivery robot will deliver the parcels to the same addresses quite a few times. So, such agents should be able to learn something about planning in their respective worlds. This would be a clear improvement compared to heuristics which need to be explicitly programmed in the system by programmers, as in our examples above, and it is often not at all clear how to do that.[54](#fn54)\nIn reinforcement learning, there is a sophisticated mathematical theory that tells how to learn a particularly good substitute of a heuristic, called the *state-value function*. It is a clever way of learning to deal with the complexity of the search in a planning tree. The basic principle is simple: Using the previous planning results in its memory, the agent can compute something like a heuristic based on how well it performed starting from each possible state. If it found the goal quickly starting from a certain state, that state gets a large state-value.\nIn the case where we have a single goal, the state-value function basically tells you how far from the goal you are, thanks to discounting which takes account of the time needed to reach the goal. A delivery robot that frequently delivers stuff to the same building (say town hall) would easily learn the distance from any other building to the town hall. In the beginning, when it had a delivery to the town hall, it had to spend a lot of time and effort in planning the path there. But little by little, it gained information by storing any results of executed plans in its memory, and learned the distance from any other building to the town hall. Such distances now give the state-value function for that goal (the state-value is actually a decreasing function of that distance). When the robot next needs to go to the town hall, it recalls the distances—to the town hall—from those buildings that are close to its current location, and simply decides to move in the direction of the near-by building which has the smallest distance to the town hall. Thus, it has learned a kind of a heuristic that avoids planning action sequences altogether.\nThis works even in a very general setting when there is no particular goal. In general, the value of a state is defined as the sum of all future rewards the agent can obtain *starting* from that state.[55](#fn55) After successful learning of the state-values, the solution to the problem of action selection is that at each step, whatever state the agent may be in, the agent just selects the single action which leads to the state with the highest state-value (e.g. closest to the town hall above). There is no need to compute several steps ahead, or make a search within the huge search tree anymore. This reduces the complexity of the computations radically: instead of planning all its actions up to reaching the goal, the state-values now provide kind of intermediate goals, one step ahead, that the agent can very easily reach. However, a lot of time and computation still needs to be spent on first learning the state values.[56](#fn56)\n### Completely reactive action selection by action-values\nWhile we have thus solved the problem of the computational explosion of planning, there is still the problem that the agent needs to have a model of how the world works. Even using the state-values, it needs to understand which action takes it into a state with higher state-value. Now, consider an extreme case where the agent has no model of how the world works in the sense that it has no idea what about the effects of its actions. Then, it is not enough to assign values to different states since the agent does not know how to get from state A to state B. (Still, we assume the agent knows in which state it is, at any given moment, so it does have some minimal model of the world.)\nThe trick to learning to act even with such a minimal model of the world is to learn what is called the *action-value* function. When the agent is in a given state, the action-value function tells the value of each of its actions, in terms of how much the total future reward is if the agent performs that action.[57](#fn57) This makes action selection really easy and extremely fast: Just compare the values of different actions and choose the one which gives the maximum. In fact, all the relevant information about the effects of the agent’s actions are implicitly included in the action-value function. The agent still has to learn the action-values, but that is not really more difficult than learning the state-values.[58](#fn58)\nAt the end of the 19th century, Edward Thorndike put cats in a box where they would need to press a lever to get out of the box and receive some fish to eat. He observed that in successive trials, the cats were pressing the lever more and more often. Such learning is called *instrumental conditioning* (as opposed to classical conditioning as in the famous Pavlov’s dogs, to be considered below).[instrucond] This shows how learning to choose actions is possible by simply associating what we call a state in AI theory (here, being in the box) with an action.[59](#fn59)\nSo, using reinforcement learning, an AI or an animal can actually learn to act without doing any real planning and having almost no model about the world. If it learns the action-value function, it only needs to look at the single actions immediately available, and then take the action which has the largest action-value—at the state where it happens to find itself. Since the action is here triggered immediately without any deliberation, like a *habit* or a knee-jerk reaction, the resulting behaviour is often called habit-based, or reactive.[60](#fn60)\nReinforcement learning has recently become popular as a model of human behaviour in neuroscience, where humans may not be considered too different from experimental animals such as cats or rats. Current thinking is that the same reinforcement learning algorithms can be used to model at least one part of the action selection system in most animals, including humans. Nevertheless, there is little doubt that some animals, probably most mammals, engage in planning as well.[61](#fn61)\nIn fact, reinforcement learning using value functions is not a magic trick that will obliterate the complexity of the action selection: It simply shifts the computational burden from search in the tree to learning a value function. Sometimes, this is a good idea, but not always. We will discuss the pros and cons of reinforcement learning vs. planning in Chapter [dual.ch]. Let me just mention here the main disadvantages of habit-based behaviour: such learning often needs a lot of time and data, and leads to inflexible behaviour. This is quite in line with the common-sense idea we have about habits.\nFrustration as reward loss and prediction error\n-----------------------------------------------\nWe have thus divided action selection into planning and habits, where habits refer to more automated action selection mechanisms.[62](#fn62) Now we consider how to define frustration in the case of habits, where there are no goals but rather rewards obtained here and there, so we cannot talk about frustration in the sense of not reaching the goal.\nWhat defines frustration in this case is an error signal called reward loss[63](#fn63) which we already saw briefly in Chapter [suffering.ch]. It is computed by the following simple formula:\n> \n> [frustdef] reward loss = expected reward - obtained reward\n> \n> \n> \nwhich is set to zero in case the difference is negative. That is, a reward loss is incurred when an agent expects to get some reward but actually gets less reward than expected. Maybe a cleaning robot expected to find a lot of dust in a room, but in fact there was much less. If it happens that the obtained reward is actually larger than expected, there is obviously no reward loss, so the reward loss is defined as zero if the difference in the expression above is negative. Reward loss can also occur if the expected reward is negative, and the obtained reward is negative while even larger (in absolute value): the agent did expect something bad to happen, but it turned out to be even worse.[64](#fn64)\nExpectation of reward here refers to the mathematical expectation as defined in probability theory. It is obtained by weighting the possible values by their probabilities: if the probability of obtaining a reward is 50% and the reward is 10 pieces of chocolate, the expected reward is 5 pieces of chocolate.[65](#fn65) Expectations of the future are often called predictions, which are in fact ubiquitous in the brain: it is likely that the brain makes a prediction of almost any important quantity in the environment.[66](#fn66)\nIn contrast to our basic definition of frustration in Chapter [planning.ch], which works only on the level of plans, such a reward loss can be computed after every single action and at every single time point. This definition of reward loss is, in fact, quite flexible since the time interval in which the reward is computed can be specified to be anything from seconds to days. Therefore, it provides a general framework encompassing both planning and habit-based action. Reward loss coming from planning and reward loss coming from single actions are similar except that they work on very different time scales. We shall consider this point in more detail at the end of Chapter [dual.ch].[67](#fn67)\nReward loss, in its turn, is related to what is called the reward prediction error (RPE), a most fundamental quantity in machine learning theory. RPE means any error made in the prediction of the reward. This definition is very general because the expected reward can be greater or less than the obtained one, and thus RPE can be positive or negative. If the obtained reward was larger than expected, that is the opposite of reward loss and suffering, and related to pleasure.[68](#fn68)\nAs the very expression “reward prediction *error*” indicates, the theory of RPE also shows how suffering is related to learning by minimization of errors, which is a fundamental approach in machine learning. If the agent can predict the rewards obtained by different actions in different states, it will be able to act so as to maximize the obtained rewards (at least if it has a good model of other aspects of the world as well). To learn and improve such predictions, it is necessary to compute the errors incurred by the current system to predict the reward. This is how minimizing RPE is related to maximization of rewards. In fact, it is possible to devise reinforcement learning algorithms that work simply by minimizing reward prediction error.[69](#fn69)\nThe exact mathematical definition of RPE is quite involved and relegated to a footnote.[RPE][70](#fn70) Let me just point out that RPE is a more general concept than reward loss also in the sense that it enables an error signal even when the agent is far away from any actual or expected rewards, but it receives “bad news” about future reward. This is in contrast to reward loss which does not make any sense except when a reward is actually expected to be obtained right now (the exact meaning of “now” depends on the time scale). In particular, if the expectation of total *future* reward decreases, RPE signals frustration, while the reward loss has no meaning if no reward was expected to be obtained at this moment. Suppose a cleaning robot is on its way to a room where there is a lot of dust (yummy!), and thus its expected (predicted) reward is high, but this reward will in any case not be obtained for quite a while. Then, it finds that the door to the room is locked and it cannot enter; that is bad news it didn’t expect. Thus the robot finds itself in a new state that has a much lower expected total future reward since the dust in that room cannot be reached. It is this difference between the earlier prediction and the new prediction that creates RPE and suffering. This is not an ordinary reward loss because no actual reward was expected to appear at this time point anyway: the robot has not yet even entered the room, and the dust is still far away. However, RPE can create suffering merely based on predictions: if information arrives that makes the agent reduce its prediction of future reward, frustration is created. This is intuitively appealing since a lot of our frustration is actually about such negative news and the lowering of expectations they create. Suppose I’m planning to attend an event that I expect to enjoy, and then, well in advance, I hear the event has been cancelled. I will suffer, although I didn’t expect to obtain anything enjoyable yet, and I may not have taken any action either; it was all just predictions in my head.[71](#fn71)\nExpectations or predictions are crucial for frustration\n-------------------------------------------------------\nReward loss and RPE highlight the importance of expectations and predictions. Clearly, there must be some expectation or prediction in order for them to occur. If the cleaning robot were so primitive that it had no expectations or predictions at all, it might be just enjoying every single speck of dust it finds. Making it more intelligent so that it can predict the future thus deprives it of its “innocence”, and enables frustration to occur. Likewise, Cassell says that “to suffer, there must be a source of thoughts about possible futures”, even though his approach to suffering is quite different.[72](#fn72)[cassellprediction]\nThe importance of predictions is well appreciated in neuroscience. It has been observed that in the brain, RPE is coded by certain neurons using a neurotransmitter called dopamine. More precisely, it is coded by quick changes in the level of dopamine (called “phasic dopamine signal”), typically originating in evolutionarily old areas such as the midbrain, which is literally in the very center of the brain.[73](#fn73) In case the obtained reward is higher than expected, there is a temporary peak in the amount of dopamine in the signalling pathways, which is called by some a “dopamine surge”. That’s why many drugs of abuse target the dopamine pathways in the brain. For example, cocaine blocks the removal of dopamine in the synapse so that its signal is amplified.[74](#fn74) Such drugs are fooling the reward-processing system in the brain, thus leading to a strong desire for such drugs, in addition to a pleasurable feeling. This has led some to think that dopamine is the neurotransmitter responsible for the feeling of pleasure itself. Such a viewpoint is probably incorrect, and the actual feeling of pleasure is mainly mediated by other transmitters, namely those in the opioid family, while dopamine is more related to “cold” action selection and learning.[75](#fn75)\n### Classical conditioning\nTo emphasize the importance of predictions in the brain, let’s consider an extremely famous kind of prediction learning in the animal realm: classical conditioning. Ivan Pavlov, doing physiological experiments on dogs around the year 1900, observed that the dogs began to salivate when they saw the staff person who was responsible for feeding them, even before receiving any food. Pavlov was intrigued and tried to see if the dogs would be able to associate any arbitrary stimuli to food. He succeeded in making the dogs associate food with many different kinds of stimuli, including the sound of a bell or a metronome, provided that these stimuli were consistently presented just before food was given.\nWhat the animal is clearly doing here is predicting the future: After the bell, food is likely to arrive. Such predictions are ubiquitous in the brain; the brain is constantly trying to predict what happens, using many different systems. Predicting the results of any action you might take is important if you want to choose good actions, as we already saw in the case of instrumental conditioning above (page ). Predicting where the rabbit will be a second or two later is necessary if you want to catch it. Note the crucial distinction between classical conditioning and instrumental conditioning: In classical conditioning, the agent does not yet learn to *choose* actions, but merely to predict future states, independently of any rewards.[76](#fn76)\nIt would be natural to assume that such classical conditioning could be easily performed by Hebbian learning. It is just the kind of association of two stimuli—bell and food— that Hebbian learning seems to be good at. That is to some extent true, although this is a bit tricky; the most successful models actually use supervised learning, with the bell as input and the food as the output. Such learning, again, proceeds by minimizing prediction error.[77](#fn77)\n### Does a low level of rewards produce suffering?\nIntuitively, however, it might seem that talking about frustration based on expectations and predictions is unnecessarily complicated. If the agent is in a state with low state-value (in its own estimation), would that not in itself imply suffering? Being in a state of low value means that the agent believes it will not obtain much net reward in the future, which sounds like a good reason for mental pain. Or, even more fundamentally, why not just say that lack of rewards, presumably during recent history, is suffering?\nOne fundamental problem with such an approach would be that it is not obvious how to define a suitable baseline or comparison: What level of state-value is actually low, and how small should recent reward actually be to create suffering? The reward loss or prediction error actually solves this problem by using the expected reward as the baseline. Thus, the obtained level of rewards is compared with the expected level, and if it is “low” in this particular sense, suffering occurs.[78](#fn78)\nUnexpected implications of state-value computation\n--------------------------------------------------\nIn the rest of this chapter, I will consider some practical implications of the theory presented here. First, let us consider how the computation of state-values, as proposed in basic reinforcement learning theory, fundamentally changes the behaviour of human agents. Originally, of course, evolutionary forces demand that an action is pursued by a biological organism if it helps it reproduce and spread our genes, and an action is avoided if it hampers this effort. So, evolution “tells” us that kicking a stone is bad because it can cause damage to our foot, and the damage decreases our potential for reproduction—thus giving us negative reward for such an action. Having sex is very good, and rewarded by basic evolutionarily mechanisms, because then we are fulfilling our deepest evolutionary calling and spreading our genes.\nThe computation of state-values changes the situation: The organism will not only try to reach states directly giving reward—such as having sex—but also states that have higher state-values. This is a mechanism for looking forward in time: instead of immediate reward, the organism will try to maximize the total reward in the future, and just that is given by the state-value.\nSeemingly valueless states are now valued by the agent since they *predict* that more actual reinforcement can be found sooner. Such states provide intermediate goals in the pursuit of the actual reward, similarly to heuristics in tree search. If you train a robot to get orange juice from the fridge, it must of course first go to the fridge, and open it. So, the state where the robot is standing next to the fridge acquires a positive state-value and we could even say that the robot “likes” being next to a fridge, even more so if it is open.\nThe situation is even more complex due to the existence of human civilization and society. Culture plays an important role in determining the state-value function, and it is often difficult to separate the influences of biology and culture. In neuroscience, this is called the “nature vs. nurture” question. There can be extremely complex chains of value computation which transform the original evolutionary goals to behaviour based on intermediate goals. For example, humans have evolved to strive for high social status. From an evolutionary perspective, this is because it helps humans get more sexual partners and increases the number and the survival probability of their offspring. This then implies that we want to increase our status: for example, winning a gold medal in the Olympics is a good behavioural goal. Clearly, a gold medal in Olympics has no evolutionary value in itself: it does not satisfy your hunger, thirst, or sexual appetite in itself. It is just an arbitrary piece of metal. There is no logical connection between such a piece of metal and sex. It is only due to a complex interplay of value function calculation *and* cultural meanings that the original evolutionary reward of sex has been subtly transformed into a goal such as excelling in sports—or science, or politics.\nSuch slightly weird desires are another manifestation of the phenomenon discussed earlier: emergence of unexpected phenomena due to the interaction between the learning agent and a complex environment. If we program sufficiently complex AI, the same thing is likely to happen as with human evolution. The AI will pursue goals that were not intended by the programmers, but which still happen to produce a high state-value.[79](#fn79)\nEvolutionary rewards as obsessions\n----------------------------------\nNow, if we admit that our desires are based on evolution, even if quite indirectly, is that a good thing or a bad thing? Should we just follow our desires, or think twice, or even try to follow some completely different goals? There are actually people who try to justify certain kinds of behaviour by saying they are evolutionarily conditioned, i.e. “evolution made me do it”. In popular science magazines and web sites, such logic is not very uncommon. Fortunately, it is rejected by many as an example of sloppy thinking.[80](#fn80) In the following, I argue the very opposite: following evolutionary desires is often a bad idea and even morally wrong.\nIn fact, even the evolutionarily conditioned *rewards* themselves can go wrong, sometimes quite catastrophically. One reason is that evolutionarily, we may be adapted to the environment where our evolutionary ancestors lived, often assumed to be the “African savannah”. However, the modern world is different and, therefore, our evolutionary programming may not be very suitable.[81](#fn81) With humans, a well-known example is the addictive quality of sugary food. The sweet taste of sugar must have signalled the high nutritious quality of food in the environment where our ancestors lived.[82](#fn82) But these days it tends to signal added refined sugar which is bad for your health; evolutionarily speaking, sweet taste should rather be punishing in the modern context, not rewarding.[83](#fn83) Yet, the state of having a sweet taste in your mouth is rewarding, and humans tend to try to reach such a “sweet” states.[84](#fn84)\nWhat is even more serious is that evolution makes us want particularly questionable goals, especially from a societal viewpoint. Evolution is fundamentally based on selfish, merciless competition between different organisms (or strictly speaking, between their genes). Many behavioural tendencies evolution has imposed on us should be seen as instruments for such egoistic competition. Evolution is all about maximally spreading our genes. It makes us hoard finite resources such as food to ourselves in order to spread our genes. It makes us violent; it even makes us go to war, again for the sole purpose of spreading genes. This is in stark contrast to most ethical systems in the world which see such selfishness as evil, and recommend quite opposite courses of action.[85](#fn85)\nEven more fundamentally, the rewards defined by evolution never had the goal of making us happy in any meaningful sense. They are a force that drives us to do exactly those things which are good for our evolutionary fitness. Even if you come to the conclusion that the evolutionary reward system makes you suffer, it cannot be switched off or modified. You cannot decide to be rewarded by something you consider more meaningful and good for society.\nI think what evolution offers us is something I would call [evobs] *evolutionary obsessions*.[86](#fn86) That is, the evolutionary rewards, together with the learned state-values, make us desire, even crave for many things which we would actually prefer *not* to desire if we could rationally decide what we desire. If you could just consciously, rationally, “switch off” your desire for, say, sugary food—would you not do that? Chapter [freedom.ch] explains how Buddhist and Stoic thinking are based on the rather extreme tenet that switching off *all* desires would actually be very good for you. Whether one agrees with that extreme viewpoint or not, surely, most people have certain desires that they would rather not have. I call them obsessions because they are automatically created, they often override any conscious deliberation, and they may even feel unwanted and intrusive. (We will look at the computational mechanisms for this in Chapters [dual.ch] and [emotions.ch].)\nReward maximization is insatiable\n---------------------------------\nFinally, let me mention another dark side to this reinforcement learning theory.[insatiability] One crucial property of the algorithms based on reward prediction error is that they drive the system to get more and more reward, and there is never any long-term satisfaction. This is because any prediction of the future is learned by the agent, and *constantly updated by learning*. Thus, in the reward loss, the level of expected reward is updated based on what the agent has obtained recently.\nSuppose that an agent gets an exceptional amount of rewards for a while, maybe because a cleaning robot finds itself in a building with lots of nice dust to clean, and it is rewarded for every speck of dust it sucks away. Now, the agent’s prediction system is updated so that an equally large amount of rewards is predicted in the future as well. An environment that produced an unexpectedly large amount of reward for a while becomes the new baseline. That level of reward is not unexpected anymore and, therefore, does not produce any particular “pleasure” anymore either.[87](#fn87) What’s worse is that when things get back to normal, the agent will get less rewards than what it has now learned to expect, since the prediction was updated to reflect the particularly nice environment that lasted for a while. Therefore, the agent suffers enormously when it has to go back to a normal room with a modest amount of dirt.\nSimilar computations take place in our brain, since our brain also computes the reward prediction error and updates its expected level of reward. No wonder that Wolfram Schultz, one of the leading neuroscientists on dopamine, calls the dopamine neurons “little devils”.[88](#fn88)[treadmill] In fact, this is a logical consequence of the guiding principle of AI agent design: the agent should maximize obtained reward. The reward prediction system has no other goal than helping in maximization of rewards. If you program an agent to maximize reward, then by definition, nothing can possibly be enough; the system will be *insatiable*. The agent will relentlessly try to get more and more reward, and it is precisely the frustration signal that will force the agent to try harder and harder.[89](#fn89)\nA merciful programmer might program some stopping criterion to limit the greed of the agent: Once you have obtained X units of reward, you can stop. Unfortunately, evolution knows no mercy, and humans don’t seem to have any such stopping criterion programmed in them. We need more money, more power, more sex (and better sex), and better food (and more food). If we follow our evolutionary “obsessions”, as I called them, nothing is enough.\nSuppose you program a robot called Pat to clean a building. You would like the building to be superclean, and the building is quite large with dozens of rooms. So, you would be very tempted to program Pat so that it will spend all its time cleaning the building. You probably want to program a couple of other functions in Pat as well, such as a routine for charging its batteries, some basic maintenance procedures, as well as safety systems to prevent it from hurting people or breaking things. But you would probably program Pat to spend all the rest of the time in tirelessly cleaning the rooms, with no breaks in between. This is what most programmers would do. Here, you have implemented a kind of a “cleaning drive” which is without mercy. Pat will spend all its time and energy just making the rooms spotlessly clean. This may seem completely natural, given that it is “just” a robot.\nNow, suppose your colleague, responsible for the visual design of the robot, decides to make Pat look really cute, giving it the shape of a little kitten. It even says “Meow” using its loudspeakers. Many people may suddenly start feeling sympathy for this poor little kitten. “Does it really have to be working all the time? Can’t it ever play, or take a rest?” they would ask. What would you reply?\nSuffering due to self-needs\n===========================\nIn addition to frustration, we identified another cause of suffering in Chapter [suffering.ch]: loosely speaking, suffering related to self. Self is a concept with a bewildering array of meanings. Psychology, philosophy, and neuroscience offer a multitude of definitions, and I can make no claim to treat the concept comprehensively.\nI focus here on two meanings of “self” directly related to suffering. First, self as the target of evaluation of some kind of long-term success of the agent. The human brain, in particular, has a system that constantly evaluates the agent, checking whether the goals set were reached or rewards obtained, and seeks to improve its general performance. Second, we have self as the target of self-preservation, or survival instinct: all animals have behavioural tendencies to avoid death or organic damage. (A third meaning of self, related to control, will be treated in Chapter [control.ch], and the concept of self-awareness, in Chapter [consciousness.ch].)\nSuch self-evaluation and self-preservation are computational mechanisms which are constantly operating in animals, and it is easy to justify their computational utility for any intelligent agent. Although at first sight, these aspects of self may seem to provide a mechanism for suffering which is completely different from frustration, I will also show how they are related to frustration of internal, higher-level goals and rewards. As the title of this chapter indicates, these aspects of self can thus be seen as needs, or desires, and they can be frustrated.\nSelf as long-term performance evaluation\n----------------------------------------\nLet us start with self as something whose performance is being constantly evaluated at different levels. As we saw earlier, in reinforcement learning, every single action is always evaluated to improve future actions. The reward prediction error is computed even in the simplest algorithms. If the reward is incorrectly predicted, the error is used by the learning algorithm to improve the prediction—if the prediction was too high, set it lower in the future, for example. While such computations are crucial for learning to act optimally, the errors also trigger the suffering signal according to the theory of the preceding chapter.\nHowever, the situation is complicated by the fact that the learning algorithms themselves contain many parameters describing how the algorithm itself works. One fundamental parameter is how quickly the system should learn: If it learns too quickly, the new information will tend to override the old one, thus leading to forgetting. Below, we will see another parameter which is how much of the time the agent should spend on relatively random exploration of the environment. There are many such parameters in a sophisticated learning system.\nTherefore, sophisticated AI should be able to adjust such internal parameters by itself. This is called *learning to learn*.[90](#fn90) Such learning to learn requires constant monitoring of the performance of the basic learning algorithms. If the current internal-parameter settings do not lead to good learning, adjustments have to be made. This requires an internal signalling system, not unlike the suffering signal, but typically working on a longer time scale, since it takes a long time to see if a learning system learns well.[91](#fn91)\n### Self-esteem and depression\nIn humans, mood is a signalling system working on a longer time scale. Mood is defined as an emotional state which is more long-lasting than single emotional episodes (such as being angry or feeling afraid). A low mood may take days, if not weeks or months, to change. A psychological concept which works on an even longer time scale is self-esteem: an overall view of the self as worthy or unworthy.[92](#fn92)\nDepression may in fact be an extreme case of the performance signalling made by the self-evaluation system. One theory proposes that depression occurs when goals are not reached, and moreover, constant attempts to improve performance fail.[93](#fn93) That is, the agent has to admit that whatever it tries, nothing works. In such a case, there is still one last strategy that may help: wait and see. The environment may eventually change by itself, even if you do nothing. Perhaps, after a while, with some luck, the circumstances will be more favourable. Such a “wait and do nothing” program may explain some depressive symptoms, such as passivity and lack of interest in any activities.[94](#fn94)\nIt would clearly make sense to program such a “depressive” mechanism in an AI. If the current algorithms are simply not working at all, it would be better for the agent to just wait and see if the world changes for the better. Such waiting will save energy, and perhaps will also enable the AI to perform some further computations to improve its performance in the meantime.\n### Self-destructing systems\nWhat if an AI comes to the conclusion that it is not able to fulfill its task at all? Perhaps something went very much wrong in the design of the learning algorithm, or the task is completely impossible, and the circumstances do not seem to change for the better. The most extreme solution would then be for the AI to “destroy” itself.\nSuppose you launch many AI agents, or programs, that work more or less independently inside some computing system. If one of the agents is not achieving anything, it would be natural that you terminate its execution. This would free up computational resources for other agents—assuming all the agents are running on the same shared processors—and other agents might be more successful.\nTo make this possible, there has to be a system for evaluating each AI agent’s performance as a whole. Importantly, the evaluation does not have to be done by an external mechanism; it could be part of the agent itself, which could then decide to self-destruct. There is nothing paradoxical or impossible in such a self-destruction system. It can be explicitly programmed in the agent by a human programmer—while it may indeed be quite impossible for the agent itself to learn such self-destruction behaviour.\nIt is possible that in some cases, even biological organisms may engage in such self-destruction sequences. Such an idea is quite speculative because it is not obvious why evolution would favour such behaviour. It is clearly possible that the designer of an AI system can explicitly create the self-evaluation and destruction systems, but in biological evolution, there is no such explicit designer. It may actually sound completely nonsensical to think that evolution could lead to self-destruction mechanisms, since an organism which destroys itself cannot spread its genes anymore.\nHowever, evolution is a bit more complicated than just the survival of the fittest individual. It is widely appreciated that in evolutionary arguments, we should take into account not only the survival and reproduction of an individual, but also the survival and reproduction of the closest relatives. This leads to the concept of “inclusive fitness”, where the fitness of an individual takes into account the fitnesses of the relatives weighted by the proportion that they share genes. Close relatives of an individual spread partly the same genes anyway, so their survival is evolutionarily useful for that individual.\nAccording to one suggestion, if a person is seriously ill, and finds himself a great burden to his relatives, it might actually be evolutionarily advantageous for that person to commit suicide. If this helps the relatives with whom he shares a large proportion of genes, the suicide might actually help in spreading those genes, thus increasing the inclusive fitness.[95](#fn95)\nThus, self-destruction programs may be useful not only to maximize the utility of AI agents, but also from an evolutionary perspective. This may sound abhorrent from a moral perspective, but that is often the case with evolution which has no reason to be nice or good from a human perspective—as already argued in the preceding chapter, where I compared evolutionary desires to obsessions.\nSelf as self-preservation and survival\n--------------------------------------\nAnother rather obvious reason why some kind of concept of self should be programmed in an AI is that the AI may need to protect itself against anything that might destroy it. A robot must take care not to be run over by a car: This is the concept of self-preservation. There is no doubt we can, and probably want to, program some kind of self-preservation mechanism in an AI agent.\nEven the simplest biological organisms have behavioural programs that are activated when their existence is threatened; we talk about self-preservation, or survival instinct. We already encountered related ideas in considering definitions of pain and suffering. The widely-used IASP definition related pain to “tissue damage” (page ), while Cassell’s definition of suffering talked about the “intactness of the person” (page ). However, what we are talking about here is threats to the very existence of the agent, not just damage.\nWhile it seems relatively straightforward to program self-preservation behaviours in an AI, an open question is whether an AI can somehow develop a survival instinct by itself. In other words, can self-preservation emerge without being explicitly programmed; can the agent learn to perform certain actions for the main purpose of avoiding its own destruction? This is one of the deepest questions in AI, extremely relevant from the viewpoint of developing safe AI systems, and the subject of intense debate.[96](#fn96) We have seen earlier that learning in AI can have various side-effects and unintended consequences; this would be one of the most extreme ones.\nOn the one hand, there are those who point out that biological organisms have developed their survival instinct via evolutionary mechanisms. They have been subject to natural selection, which has ruthlessly weened out those organisms which do not fight for their survival. In contrast—this line of argumentation goes—AI is not subject to natural selection; it has no evolutionary pressures. So, it will not learn a survival instinct, unless perhaps we explicitly decide to program it to learn one.\nOther experts disagree and point out that some kind of survival instinct may be automatically created as an unintended side-effect of creating sufficiently intelligent machines. If a robot is given any mundane task, say fetching a bottle of milk from a near-by shop, a super-intelligent robot would understand that in order to perform that task, it has to stay alive. If the robot were damaged or destroyed in a collision with a car, for example, its task cannot be performed. Thus, the robot might decide to destroy the car somehow (let’s assume the robot is really big) to get the milk safely delivered. If everybody in the car gets killed, that is irrelevant, if the programmer didn’t tell the robot to avoid human casualties. The idea here is that there is no need to explicitly program a survival instinct, or any reward related to that: the general goal of maximizing future rewards will direct the robot’s behaviour towards avoiding destruction. In fact, this line of thinking means that almost any sufficiently intelligent AI will *by logical necessity* strive to survive. If it is intelligent enough, it will understand what death is, and how death makes it impossible to obtain any further rewards or accomplish goals. This is the opposite of what has happened in biological evolution, where even the very simplest organisms have a survival instinct, and sophisticated intelligence develops later. In AI, intelligence is programmed first, and later, possibly by chance, the AI might obtain a tendency for self-preservation behaviour and related information processing, which might then be called a survival “instinct”.\nClearly, these two views are based on very different assumptions about the AI. The argument where the robot understands that a car on a crash course has to be destroyed assumes a very, very intelligent robot. The robot must have a sophisticated model of the world, infer that it risks being overrun by the car, and understand that being overrun by the car will prevent it from delivering the milk. Most current robots would be nowhere near the intelligence required—but we don’t know if they will be in the future. We are even further away from an AI which could intellectually infer, on an abstract level, that there is such a thing as death, and that various measures should be taken to avoid it.\nNevertheless, if an AI is learning using evolutionary algorithms instead of the conventional gradient-based algorithms, it might be perfectly possible for an AI to obtain a survival instinct, even at the current level of AI development. As reviewed earlier (page ), optimization procedures mimicking evolution are sometimes actually used in AI. Large-scale application of evolutionary methods definitely has the potential of creating a survival instinct in AI agents. It is a necessary logical consequence of fundamental evolutionary pressures: To spread its artificial “genes”, an agent has to survive long enough to produce offspring if the evolutionary optimization method is similar enough to biological evolution.\nSelf as desires based on internal rewards\n-----------------------------------------\nGoing back to our main topic, suffering, it is clear that both self-preservation and self-evaluation are important sources of suffering.[97](#fn97) First, it is well-known that depression and low self-esteem create suffering— and they are largely produced by the self-evaluation system. It"},"data_id":{"kind":"string","value":"arxiv_0184763"}}},{"rowIdx":46194,"cells":{"content":{"kind":"string","value":"Synchrotron emission from virial shocks around stacked OVRO-LWA galaxy clusters\n===============================================================================\n### Accepted —. Received —; in original —\n[firstpage]\n01 0.2in\nGalaxy clusters accrete mass through large scale, strong, structure-formation shocks. Such a virial shock is thought to deposit fractions *ξ**e* and *ξ**B* of the thermal energy in cosmic-ray electrons (CREs) and magnetic fields, respectively, thus generating a leptonic virial ring. However, the expected synchrotron signal was not convincingly established until now. We stack low-frequency radio data from the OVRO-LWA around the 44 most massive, high latitude, extended MCXC clusters, enhancing the ring sensitivity by rescaling clusters to their characteristic, *R*500 radii. Both high (73 MHz) and co-added low (36–68MHz) frequency channels separately indicate a significant (4–5*σ*) excess peaked at (2.4--2.6)*R*500, coincident with a previously stacked *Fermi* *γ*-ray signal interpreted as inverse-Compton emission from virial-shock CREs. The stacked radio signal is well fit (TS-test: 4–6*σ* at high frequency, 4–8*σ* at low frequencies, and 8–10*σ* joint) by virial-shock synchrotron emission from the more massive clusters, with $\\dot{m}\\xi\\_e\\xi\\_B\\simeq (1\\mbox{--}4)\\times 10^{-4}$, where $\\dot{m}\\equiv \\dot{M}/(MH)$ is the dimensionless accretion rate for a cluster of mass *M* and a Hubble constant *H*. The inferred CRE spectral index is flat, *p* ≃ 2.0 ± 0.2, consistent with acceleration in a strong shock. Assuming equipartition or using $\\dot{m}\\xi\\_e\\sim0.6\\%$ inferred from the *Fermi* signal yields *ξ**B* ≃ (2--9)%, corresponding to *B* ≃ (0.1–0.3) *μ*G magnetic fields downstream of typical virial shocks. Preliminary evidence suggests non-spherical shocks, with factor 2–3 elongations.\ngalaxies: clusters: general - galaxies: clusters: intracluster medium - intergalactic medium - magnetic fields - radio continuum: general\nIntroduction\n============\nGalaxy clusters and filaments are the largest gravitational bound objects in the Universe, with clusters located at the nodes of the cosmic web of large-scale structure (LSS). A cluster of mass *M* is thought to grow by accreting surrounding matter at a rate $\\dot{M}=\\dot{m}MH$, where *H* is the Hubble parameter and $\\dot{m}$ is a dimensionless parameter of order unity. As the accreted gas is violently decelerated, strong, collisionless, so-called virial or structure formation shocks form near the virial radius of the cluster. These virial shocks define the edges of clusters, so observing them can provide useful information about the formation of LSS, in particular the local accretion rate. As the accreted gas is pristine and weakly magnetized, virial shocks also provide a useful laboratory for studying collisionless shock physics. A virial shock is distinct from the weak shocks found in its heated downstream, known as the intracluster medium (ICM); such weak shocks, usually observed at $r\\lesssim 1\\Mpc$ radii following a merger, are more difficult to model as they propagate into a pre-heated, magnetised plasma, already enriched by metals and cosmic-rays.\nVirial shocks are thought to accelerate charged particles to highly relativistic, $\\gtrsim$ 10 TeV energies, in resemblance of strong supernova remnant shocks that share similar, $\\sim10^3\\km\\se^{-1}$ velocities. The cosmic ray (CR) electrons (CREs) and ions (CRIs) accelerated by such strong shocks are thought to develop a nearly flat, *d**N*/*d**E* ∝ *E*− *p*, energy spectrum with *p* ≃ 2 (equal energy per logarithmic CR energy bin), as expected in diffusive shock acceleration theory when scattering is not too anisotropic. A fraction *ξ**e* of the downstream thermal energy, deposited in CREs, is subsequently radiated away, forming a distinctive non-thermal signature, which should surface at the extreme ends of the electromagnetic spectrum.\nThe high energy CREs accelerated by the virial shock cool rapidly, primarily by inverse-Compton scattering off cosmic microwave background (CMB) photons, resulting in a thin shell of energetic CREs. These CREs produce a nonthermal, detectable leptonic ring around the cluster, as predicted analytically and calibrated using cosmological simulations. The rings are expected to be fairly spherical, but somewhat elongated toward the main filament feeding the cluster.\nEvidence for the virial inverse-Compton signal first surfaced in the Coma cluster: an ideal virial-shock target, thanks to its high mass, proximity (redshift *z* ≃ 0.023), and location in a low-foreground region near the north Galactic pole. Consistent signals were reported in $\\sim220\\GeV$ data from VERITAS, $\\sim\\GeV$ data from *Fermi*-LAT, and $\\sim0.1\\keV$ data from *ROSAT*. The inferred virial ring is elongated toward the large-scale filament between Coma and the nearby cluster A1367, with a semi-minor axis ∼ 2.1*R*500 coinciding with the virial radius, and is thin; searches for a thick or spherical LAT signal did not recover the signal. Here, *R**δ* is a characteristic cluster radius, where subscript *δ* designates an enclosed mass density *δ* times above the critical mass density of the universe. The three aforementioned signals are all consistent with a normalisation $\\dot{m}\\xi\\_e\\simeq 0.3\\%$ (up to an uncertainty factor ∼ 3) and with a $2.0\\lesssim p\\lesssim2.2$ spectrum.\nGalaxy cluster signals can often be amplified by stacking data around multiple clusters, utilising the self-similarity of the latter. By co-adding *Fermi*-LAT data around 112 massive, high latitude, extended clusters, rescaled by their *R*500 radii to a dimensionless, *τ* ≡ *r*/*R*500 scale, binning the outcome radially, and utilising the anticipated flat spectrum, a high significance ( > 5*σ*) stacked -ray ring was identified. The stacked signal is best fit by a normalised shock (subscript *s*) radius *τ**s* ≡ *r**s*/*R*500 = 2.3 ± 0.1, a CRE injection rate $\\dot{m}\\xi\\_e=(0.6\\pm0.1)\\%$, and a spectral index *p* = 2.1 ± 0.2, all consistent with predictions and with the Coma signal. While stacking increases the sensitivity, it carries an elevated systematic uncertainty (here admitting best-fitting peak radii in the range $2.2\\lesssim\\tau\\_s\\lesssim2.5$, and an additional uncertainty factor ∼ 2 in $\\dot{m}\\xi\\_e$) and washes away non-spherical components and variations among clusters, due to the co-addition and radial binning. Moreover, the projected shock signal of an individual cluster should not be precisely circular, so stacking effectively biases $\\dot{m}$ towards lower values.\nA direct tracer of virial shocks, independent of any particle acceleration, is their anticipated imprint on the thermal Sunyaev-Zel’dovich signal. A spatial correlation was identified between the -ray ring in Coma and an outward drop in the *y*-parameter inferred from *WMAP* data, supporting the association of both signals with the virial shock. A highly significant, 8.6*σ* drop in *y*-parameter was later identified in the cluster A2319, which has the highest signal-to-noise detection in the *Planck* SZ catalogueues, indicating a strong shock around *τ* ≃ 3 with a Mach number *Υ* > 3.25. A joint search for SZ and *Fermi* virial signals in the massive clusters Coma, A2319, and A2142 showed a significant drop in *y* coincident with a weak -ray excess in all three clusters, consistent with $0.2\\%\\lesssim\\dot{m}\\xi\\_e\\lesssim0.7\\%$. Combining the SZ signal with galaxy counts gives a measure of the accretion rate, allowing independent measurements of *ξ**e* ∼ 0.5% and $\\dot{m}\\simeq 1.1$ in A2319. More recently, stacking the *Planck* *y*-parameter around 10 galaxy groups indicated a significant drop around $2.0\\lesssim\\tau\\lesssim2.6$, and stacking the *y*-parameter among 500 clusters using South Pole Telescope data indicated a projected drop starting at *τ* ≃ 2, both consistent with a virial shock at the stacked *γ*-ray radius of. Interestingly, stacking the 500 clusters also indicated a second drop starting at *τ* ≃ 6; if verified, this would suggest elongated virial shocks with a ∼ 2.5 ellipticity ratio, as indicated by the virial ring found in Coma by.\nIn addition to the inverse-Compton signal, the CREs accelerated by the virial shock should also emit a synchrotron radio signal, as they gyrate in the post-shock magnetic fields. Assuming that the shock deposits a fraction *ξ**B* ≃ 1% of the thermal energy in magnetic fields, the cumulative signal from all clusters should constitute a considerable fraction ( ∼ 30% for *ξ**e**ξ**B* = 10− 4) of the diffuse extragalactic radio background below 500 MHz, where it dominates fluctuations on $1\\arcmin\\lesssim\\theta\\lesssim1{^{\\circ}}$ scales. The signal, similar in morphology but somewhat more patchy than its inverse-Compton counterpart, has proven difficult to detect, although a correlation found between the *WMAP* synchrotron signal and the VERITAS -ray ring in Coma corresponds to *ξ**B* ∼ 1%.\nThe flat, *α* ≃ *p*/2 ≃ 1 radio spectral index of the anticipated virial shock signal from cooled CREs is somewhat softer than that of the Galactic foreground, giving low-frequency interferometers an advantage in searching for this elusive signal. The Owens Valley Radio Observatory Long Wavelength Array (OVRO-LWA; hereafter LWA) is particularly suited for such a search, providing clean sky maps of the northern sky in the (36–73) MHz frequency range. Searching the low-frequency sky for the virial shock signature is complicated by the weakness of the signal, an abundance of unrelated point and diffuse sources, including radio halos and relics associated with the galaxy clusters themselves, and instrumental difficulties associated in particular with beam sidelobes of sources in the centres of clusters. Such studies therefore require careful modelling and extensive sensitivity tests.\nWe select the 44 most massive, nearby, extended clusters in the northern sky from the Meta-catalogue of X-ray Clusters, after avoiding regions of strong Galactic contamination or a suboptimal point spread function (PSF). LWA data are excised around these clusters, cleaned from point sources and from smooth background and foreground signals, rescaled to the *R*500 extent of each cluster, co-added over all clusters, and radially binned, to test if a residual virial-shock signal can be identified. A robust (in particular, independent of PSF residuals) and highly-significant signal, found around the virial radius and coincident with the previously stacked -ray signal, is then modelled as the cumulative synchrotron emission from cluster virial shocks.\nThe paper is organised as follows. We outline the LWA data and the cluster sample in §[sec:Data]. The stacking analysis, including background modelling and Monte-Carlo simulations of control cluster samples, is described in §[sec:analysis]. In §[sec:result], we outline the results of the stacking, in particular a strong central signal and a significant virial excess. Simple models for the central emission are presented and fit to the data in §[sec:centralemission]. Models for virial-shock emission are fit to the peripheral signal, alone and jointly with the central signal, in §[sec:virialemission]. Finally, the results are summarised and discussed in §[sec:summary].\nThe Supplementary Material provides sky maps (§[append:skymaps]), technical details concerning different aspects of the analysis, including cluster co-addition (§[append:intensity]), background and foreground removal (§[append:backgroundremove]), control sample analyses (§[append:describecontrolsample]), noise and correlation corrections (§[append:controlsample]), modelling the central emission from clusters (§[append:CentralResults]), additional tests constraining residual sidelobe artefacts (§[append:psf]), sensitivity and consistency tests (§[append:sensi]), the theory of synchrotron emission from virial shocks (§[append:model]), cross contamination by central and virial modelling residuals (§[append:mutualeffects]), methods of inferring the peripheral magnetic fields (§[append:decouplexib]), the list of clusters in our sample (§[append:MCXCclusterslist]), and extended 2D significance images (§[append:2Dsigmaps]).\nWe adopt a flat ΛCDM cosmological model with a Hubble constant $H\\_0=70\\km\\se^{-1}\\Mpc^{-1}$, a mass fraction Ω*m* = 0.3, and a cosmic baryon fraction *f**b* = 0.17. Assuming a 76% hydrogen mass fraction, we use a mean particle mass *m̄* ≃ 0.59*m**p*, where *m**p* is the proton mass. Error bars designate 68% containment projected for a single parameter.\nData and data preparation\n=========================\ncccccccccc *ν* (Δ*ν*) & *T*0 & FWHM & Restored & & \n[MHz] & [K] & (*δ* = 45∘) & FWHM & *η**s* & *η̄* & *ψ* & *η**s* & *η̄* & *ψ* \n(1) & (2) & (3) & (4) & (5) & (6) & (7) & (8) & (9) & (10) \n36.528 (0.024) & 595 & $20\\arcmin.2 \\times 16\\arcmin.9$ & $16\\arcmin.1$ & 1.24 (1.20) & 1.17 (1.14) & 0.70 (0.71) & 1.91 (1.86) & 1.90 (1.83) & 0.85 (0.85) \n41.760 (0.024) & 541 & $18\\arcmin.5 \\times 16\\arcmin.0$ & $14\\arcmin.5$ & 0.99 (0.95) & 0.93 (0.90) & & 1.54 (1.49) & 1.51 (1.45) & \n46.992 (0.024) & 417 & $17\\arcmin.4 \\times 15\\arcmin.2$ & $14\\arcmin.0$ & 1.06 (1.03) & 1.02 (0.99) & & 1.63 (1.57) & 1.61 (1.55) & \n52.224 (0.024) & 418 & $16\\arcmin.2 \\times 15\\arcmin.0$ & $13\\arcmin.1$ & 0.89 (0.86) & 0.87 (0.84) & & 1.33 (1.28) & 1.30 (1.25) & \n57.456 (0.024) & 354 & $15\\arcmin.9 \\times 15\\arcmin.0$ & $12\\arcmin.9$ & 0.84 (0.80) & 0.81 (0.78) & & 1.18 (1.13) & 1.17 (1.13) & \n62.688 (0.024) & 309 & $15\\arcmin.8 \\times 14\\arcmin.9$ & $12\\arcmin.7$ & 0.77 (0.74) & 0.75 (0.72) & & 1.05 (1.02) & 1.05 (1.01) & \n67.920 (0.024) & 281 & $15\\arcmin.9 \\times 14\\arcmin.7$ & $12\\arcmin.6$ & 0.71 (0.68) & 0.69 (0.67) & & 0.94 (0.91) & 0.93 (0.90) & \n73.152 (0.024) & 154 † & $12\\arcmin.2 \\times 10\\arcmin.0\\,\\, ^{\\dagger}$ & $9\\arcmin.3$ & 0.97 (0.94) & 0.94 (0.91) & — & 1.26 (1.22) & 1.24 (1.20) & — \n[tab:mapssum]\n**Columns:** (1) Central frequency (and bandwidth) (MHz) of each channel; (2) Noise temperature ; (3) FWHM major axis × minor axis of the synthesized beam at a typical declination *δ* ∼ 45∘; (4) Effective FWHM of the restored data in the 20∘ < *δ* < 60∘ range (see §[sec:CentralModels]); (5) Correction factor *η**s* ≡ *η*(2.25 ≤ *τ* < 2.5) for the product *T*0 ⋅ FWHM in the virial shock region, inferred from control samples for the beam- (cluster-)stacked cleared data; (6) Correction factor *η̄* ≡ *η*(0 < *τ* < 10) uniformly averaged over the full control range for the cleared data; (7) Correlation parameter of low-frequency channels uniformly averaged over the range 0 < *τ* < 10, for the cleared data; (8) *η**s* for the restored data; (9) *η̄* for the restored data; (10) *ψ* for the restored data. \n† — In the high-frequency channel we use a dedicated high-resolution data set, provided by the OVRO-LWA collaboration, rather than the nominal-resolution $16\\arcmin.8 \\times 14\\arcmin.6$ data set of. The *T*0 = 154 K of the nominal-resolution map is shown to approximately hold at high-resolution. The *δ* = 45∘ FWHM provided here is extracted from the tabulated high-resolution PSF.\nLWA sky maps\n------------\nThe LWA is a 288-antenna interferometer with a maximum baseline of ∼ 1.5 km. We use data produced from 28 hours of observation during 2017–2018. The data, in the form of eight low-frequency maps of the sky north of declination *δ* = − 30∘, were constructed with Tikhonov-regularised *m*-mode analysis imaging. The maps were taken at frequencies 36.53, 41.76, 46.99, 52.22, 57.46, 62.69, 67.92, and 73.15 MHz, with a 24 kHz bandwidth, a 15ʹ–20ʹ FWHM angular resolution, and ∼ 800 mJy/beam thermal noise. More precise parameters, and the thermal noise level *T*0(*ν*) in each frequency *ν*, are discussed in and summarised in Table [tab:mapssum].\nFor the high-frequency, $\\nu\\_h=73.152\\MHz$ channel, the sky map was prepared with a substantially better, ∼ 11ʹ FWHM resolution. Therefore, we separate the study of data in this channel (henceforth referred to as the high-frequency channel) from the analysis of the other, lower-frequency data sets (referred to as the low-frequency channels).\nPoint sources were removed from the maps using a version of the CLEAN algorithm, adapted to deconvolve the PSF for *m*-mode analysis imaging by drift-scanning telescopes like the OVRO-LWA. To better understand the point-source contribution and the performance of CLEAN, we utilise two sets of maps in each channel, denoted ’cleared’ and ’restored’, provided by the OVRO-LWA collaboration. The cleared maps are the usual output of CLEAN, after the flux associated with each source was removed by convolving a point source with the PSF. In the restored maps, the flux of each source thus removed was subsequently restored using a simpler, Gaussian beam. Figure [fig:allsky] shows the resulting sky map of the cleared data in the high-frequency channel.\n [fig:allsky]\nOur nominal analyses are based on the cleared data. To further avoid contamination from bright source residuals, we mask all HEALPix pixels within a radius $0{{\\overset{^\\circ}{.}}}3$ of $F\\_{\\rm 1.4GHz} > 1$ Jy sources in the NRAO VLA Sky Survey. Our results are not sensitive to small variations in the masking radius or in the flux cut, as illustrated in §[append:sensi].\nThe PSF of the LWA varies with declination. The PSF sidelobes become increasingly less circularly-symmetric toward lower declination, in particular for *δ* < 20∘. This gradual distortion of the PSF renders sidelobe artefacts increasingly difficult to test and correct. Therefore, we impose a cut on the cluster sample, and include only clusters with *δ* > 20∘ in our sample, thus selecting for a well-behaved, approximately circular PSF. The basic information and properties of the maps are summarised in Table [tab:mapssum].\nThe central regions of galaxy clusters are often radio-bright, due to both point sources and diffuse emission. Searching for an annular signal around such a cluster is challenged by the possible presence of sidelobes that were not fully accounted for. This difficulty is addressed and tested throughout the text, in particular in §[subsec:sidelobe], §[subsec:LowFrequencyVirial], §[append:centrerestored], and §[append:psf], as summarised in §[sec:summary]. Another difficulty is that if the virial ring is sufficiently clumpy, it may in part be misidentified by CLEAN as a collection of point sources and thus removed; we show in §[subsub:sigprof] that this is not the case.\nThe sky maps are discretised using a HEALPix scheme of order 11 (nside = 2048), involving 3 × 224 ≃ 5 × 107 pixels on the sky, so each pixel spans a solid angle\n \n$$\\delta \\Omega\\simeq 2.5\\times 10^{-7}\\sr\\fin$$ \nThe mean, $\\sim 0{{\\overset{^\\circ}{.}}}03$ separation between pixels is sufficiently small with respect to the angular resolution of the LWA to avoid discretisation errors. Our modelling utilises tabulated PSF maps, provided for the full declination range by the OVRO-LWA collaboration.\nCluster sample\n--------------\nWe select groups and clusters of galaxies (henceforth, clusters) for radio stacking from the MCXC catalogue. This catalogue, based on the *ROSAT* all sky-survey, was homogenised to an overdensity of 500, thus providing the characteristic length and mass scales, required for our analysis, for 1743 clusters. In addition to the position of each cluster on the sky and the cluster mass *M*500 enclosed within $\\myR$, the catalogue specifies the redshift *z* and radius *R*500 of each cluster, so the corresponding angular radius *θ*500 can be computed.\nTo construct an optimal sample of clusters for the virial shock search, we select only clusters that satisfy all of the following criteria.\n1. Massive clusters: the anticipated synchrotron signal arises from massive clusters, so we select clusters with a mass *M*500 > 1013*M*⊙. Note that the mass dependence of synchrotron emission is stronger than its inverse-Compton counterpart, due to expected scaling of the magnetic field.\n2. Resolvable *R*500 core: an angular radius $\\theta\\_{500} > 0{{\\overset{^\\circ}{.}}}2$, chosen such that *R*500 is marginally resolved at the $\\sim 12\\arcmin$ PSF scale of our nominal, high frequency channel.\n3. Avoiding excessively extended, bright clusters of $\\theta\\_{500}>0{{\\overset{^\\circ}{.}}}4$, which are too bright and extended for our analysis; in particular, such clusters complicate the background removal procedure.\n4. Avoiding contamination from the Galactic plane by excluding clusters located at latitudes ∣*b*∣ < 20∘.\n5. Avoiding possible contamination from Loop-I, by excluding a circle on the sky centred on Galactic coordinates {*l*, *b*} = {337.2, 24.6} with a 55∘ radius. This region is based on LWA data, and is largely consistent with previous findings.\n6. Avoiding possible contamination from the Fermi bubbles, by excluding the regions defined as ∣*b*∣ < 55∘ and either *l* < 45∘ or *l* > 315∘,.\n7. Optimising for PSF: as in the cut on the data, we select only clusters with *δ* > 20∘ to avoid the PSF deterioration at lower declination.\nOut of the 1743 clusters in the MCXC catalogue, the above cuts leave us with a sample consisting of *N**c* = 44 high-quality clusters. These are nearby clusters, found at redshifts $z \\lesssim 0.07$. The clusters in our sample and their basic properties are listed in the supplementary material §[append:MCXCclusterslist]. Their median values are *z* ≃ 0.03, $\\theta\\_{500} \\simeq 0{{\\overset{^\\circ}{.}}}24 $, and *M*500 ≃ 0.56 × 1014*M*⊙. Variations in the above cuts are discussed in §[append:sensi]; our results are shown to be robust to reasonable changes in all cluster selection criteria.\nStacking method\n===============\nAlong with the stacking of multiple rescaled maps around clusters, detecting the virial shock signal requires careful removal of backgrounds and foregrounds (henceforth background, for brevity), angular data-folding or radial binning, and Monte-Carlo simulations to calibrate the noise and correlation parameters and validate the relevant sky statistics.\nThe procedures of scaling, co-adding, and binning the data are outlined in §[sec:StackingBinning]. The stacked quantities and their computation are detailed in §[sec:StackedQuantities]. Different background removal methods are discussed in §[sec:T0]. Finally, we describe the Monte-Carlo simulations of control cluster samples in §[subsec:controlsamples]. These control samples, consisting of random position in the sky analysed as if they contain clusters analogous to the real sample, are used for calibration purposes throughout §[sec:StackingBinning]–§[sec:T0], where we refer to them as ’control clusters’; in later sections, where we inject mock signals into these random sky locations for modelling purposes, we refer to them as ’mock clusters’.\nStacking and binning procedure\n------------------------------\nIn terms of (proper) spatial scales, virial shocks span a wide range of radii, due to the diversity in cluster parameters. The dispersion in terms of angular scales is even larger, due to the $0.017 \\lesssim z \\lesssim 0.072$ range of cluster redshifts in our sample. Therefore, before stacking cluster maps, we normalise the sky coordinates around each cluster as {*τ**x*, *τ**y*} ≡ {*θ**x*, *θ**y*}/*θ*500, and define the normalised radius *τ* ≡ *θ*/*θ*500 or equivalently $r/\\myR$. Here, angles $\\bm{\\theta}=\\{\\theta\\_x,\\theta\\_y\\}$ are measured with respect to the X-ray peak of the cluster. The data around each cluster are rotated randomly in the {*τ**x*, *τ**y*} plane to minimize background effects, although this is inconsequential for most parts of the analysis, where radial binning is employed. With a background model, the excess flux and brightness above the background can now be co-added over clusters, and the significance of the excess can be estimated directly from the sky statistics or from Monte-Carlo simulations.\nThe virial shock models and detected signals reviewed in §[sec:Intro] indicate that the anticipated signal, after stacking the scaled maps, should show a ring of enhanced brightness peaked near the $2.2\\lesssim\\tau\\lesssim2.5$ radius of its similarly stacked, -ray counterpart. The radio ring might be somewhat offset from this radius by the different weighting of clusters — which are not precisely spherical — in the present sample, and by the effect of magnetic fields on the synchrotron signature of virial shocks. The anticipated signal is weak, so it is beneficial to bin it radially, although we show in §[subsec:virialexcess] that the signal can be identified even without such binning. The corresponding scaled, stacked, binned, radial plot — based on bins in the form of scaled concentric rings about the centres of the clusters in the sample — should then show a peak in, or close to, the above *τ* range. As projected virial shock signals are not expected to be circular, some information and flux are lost when randomly rotated clusters are stacked or when the data are radially binned; see discussion in §[sec:summary].\nWe choose Δ*τ* = 0.25 as our nominal radial bin size, and 2.25 ≤ *τ* < 2.5 as the a-priori most-likely virial shock bin, based (only) on the -ray stacking of. Although such a resolution may seem finer than the beam FWHM, it exceeds the limiting resolution near the virial radius after integrating over the 2*π**τ* circumference of the radial bin. Both larger and smaller choices of bin size are examined throughout the text; the robustness of the results to resolution is demonstrated in §[append:sensi]. While co-adding the flux is straightforward, there is more than one way to co-add the brightness and determine the significance of the excess, as we next discuss.\nStacked quantities\n------------------\nIn each frequency channel *ν*, each cluster c, and each HEALPix pixel h, denote the measured brightness temperature as *T*(*ν*, c, h), and the brightness temperature of the background as *T**b*(*ν*, c, h). Estimates of the background *T**b* are discussed in §[sec:T0] and in §[append:backgroundremove]. We may now define the local brightness temperature excess as\n \n$$\\Delta T({{\\nu}},{{\\mathsf{c}}},{{\\mathsf{h}}}) \\equiv T-{{T\\_b}}\\fin$$ \nWe anticipate a spectrally flat synchrotron signal from virial-shocks, with a specific flux *F**ν* ∝ *ν*− 1, with a photon index *α**v**i**r* ≃ *p*/2 ≃ 1 corresponding to rapidly cooled CREs injected with an initially flat, *p* ≃ 2 spectrum. Therefore, it is useful to estimate the excess flux over the background per e-fold in photon energy (flux excess; henceforth),\n \n$$\\label{eq:FjDef}\n\\Delta F({{\\nu}},{{\\mathsf{c}}},{{\\mathsf{h}}}) \\equiv \\nu \\,\\Delta F\\_{\\nu} \\simeq 2\\frac{{{\\nu}}^3}{c^2}k\\_B \\Delta T\\,\\delta\\Omega \\coma$$ \napproximately independent of frequency for the expected signal. Here, *k**B* is the Boltzmann constant and *c* is the speed of light. We use Δ*F* ∼ *ν**F**ν* instead of *F**ν* for convenience; this of course does not bias the measurements of the spectrum, carried out in §[sec:centralemission] and §[sec:virialemission].\nWe may now stack the flux excess around the clusters in our sample to obtain an image of the mean flux excess per cluster at a given channel *ν*,\n \n$$\\Delta F\\left({{\\nu}}, \\bm{\\tau}\\equiv\\{ \\tau\\_x,\\tau\\_y \\}\\right) = \\frac{1}{N\\_c} \\sum\\_{{{\\mathsf{c}}}=1}^{N\\_c} \\sum\\_{{{\\mathsf{h}}}=1}^{N\\_p (\\bm{\\tau},{{\\mathsf{c}}})} \\Delta F({{\\nu}},{{\\mathsf{c}}},{{\\mathsf{h}}}) \\coma\n\\label{eq:excess\\_b\\_pixel}$$ \nwhere the second sum runs over the $N\\_p (\\bm{\\tau},{{\\mathsf{c}}})$ HEALPix pixels mapped from cluster c onto the desired (*τ**x*, *τ**y*) pixel. Radial binning, used in the remainder of this subsection, is obtained by formally identical summations,\n \n$$\\Delta F({{\\nu}}, \\tau) = \\frac{1}{N\\_c} \\sum\\_{{{\\mathsf{c}}}=1}^{N\\_c} \\sum\\_{{{\\mathsf{h}}}=1}^{N\\_p (\\tau,{{\\mathsf{c}}})} \\Delta F({{\\nu}},{{\\mathsf{c}}},{{\\mathsf{h}}}) \\coma\n\\label{eq:excess\\_b}$$ \nwhere the second sum here runs over the *N**p*(*τ*, c) HEALPix pixels that fall in the radial *τ* bin of cluster c. We henceforth focus mostly on the radially binned Eq. .\nThe significance *S* of the excess intensity over the background in some bin *τ* of the cluster c may be estimated as\n \n$$\\begin{aligned}\n \\label{eq:SingleBinSignificance}\n{{S}}({{\\nu}},\\tau,{{\\mathsf{c}}})\n& = \\frac{\\sum\\_{{{\\mathsf{h}}}=1}^{N\\_p (\\tau,{{\\mathsf{c}}})} \\Delta T({{\\nu}},{{\\mathsf{c}}},{{\\mathsf{h}}})}{\\eta({{\\nu}}) T\\_0({{\\nu}})\\sqrt{N\\_p (\\tau,{{\\mathsf{c}}}) {{\\mathsf{n}}}}} \\coma\\end{aligned}$$ \nwhere n is the number of (correlated, HEALPix) pixels in the beam,\n \n$${{\\mathsf{n}}}\\simeq \\frac{\\pi\\left( \\mbox{FWHM}/2\\right)^2}{\\delta\\Omega}\n\\simeq 60\\left(\\frac{\\mbox{FWHM}}{15'}\\right)^2 \\fin\n\\label{eq:LWA\\_PixPerBeam}$$ \nHere, we assumed that the noise is predominantly thermal, at the *T*0(*ν*) antenna noise temperature, and given by Poisson statistics among the ∼ *N**p*/n independent beams mapped onto the *τ* bin. The product *T*0 ⋅ FWHM varies with the PSF across the sky, so its effective, co-added value could deviate from its nominal, tabulated value. Therefore, in Eq. we introduced order-unity correction factors *η*. These factors vary among channels, but in principle should be fairly independent across the sky, as shown below, and so roughly independent of *τ* and c.\nThe factors *η* are calibrated by sampling the sky with control clusters, as discussed in §[subsec:controlsamples] and in §[append:controlsample]. In the cleared data, after CLEAN, *η* varies between ∼ 0.7 and ∼ 1.3 as a function of frequency. In the restored data, *η* tends to be larger and varies between ∼ 0.9 and ∼ 1.9 for different frequencies. In both cleared and restored data, *η* shows small, $\\lesssim 5\\%$ spatial variations across the stacked control maps. Hence, we define mean factors *η̄*, by averaging *η* in the domain 0 < *τ* < 10 uniformly in *τ*. The resulting averaged factors are used throughout the following analysis. Table [tab:mapssum] provides the averaged factors, as well as the local factors *η**s* estimated near the anticipated shock position. The factors *η* and *η**s* are consistent within $\\lesssim 6\\%$.\nThere are different ways to stack the signal over clusters. We consider two opposed methods for estimating the significance of an excess signal stacked over multiple clusters, for given *ν* and *τ*. We refer to the first, more standard method, as beam co-addition (equivalent to photon co-addition in ). Here, we compute the excess brightness summed over all clusters, and estimate its significance over random fluctuations of the summed background, leading to\n \n$${{S}}^{(bm)}({{\\nu}},\\tau) = \\frac\n{\\sum\\_{{{\\mathsf{c}}}=1}^{N\\_c} \\sum\\_{{{\\mathsf{h}}}=1}^{N\\_p (\\tau,{{\\mathsf{c}}})} \\Delta T({{\\nu}},{{\\mathsf{c}}},{{\\mathsf{h}}})}\n{\\eta({{\\nu}}) T\\_0({{\\nu}}) \\sqrt{{{\\mathsf{n}}}\\sum\\_{{{\\mathsf{c}}}=1}^{N\\_c} N\\_p(\\tau,{{\\mathsf{c}}})}} \\coma\n\\label{eq:photon\\_add}$$ \ndirectly generalizing Eq. for multiple clusters. As this method weighs the stacked excess against the stacked solid angle, it has the advantage of being less sensitive to individual clusters with a small *θ*500 that leaves fewer independent beams for determining *S*(*ν*, c, h).\nIn the second method, referred to as cluster co-addition (also used in ), we co-add the significance of the coincident excess *S*(*ν*, c, h) in individual clusters as unit-normal random variables of equal weights, leading to\n \n$${{S}}^{(cl)}({{\\nu}},\\tau) = \\frac\n{\\sum\\_{{{\\mathsf{c}}}=1}^{N\\_c} {{S}}({{\\nu}},\\tau,{{\\mathsf{c}}}) }\n{\n\\sqrt{N\\_c}} \\fin\n\\label{eq:cluster\\_add}$$ \nAs each cluster contributes equally, regardless of its angular extent, this method avoids the large weights that beam co-addition effectively assigns to both the excess and the estimated fluctuations of the most extended clusters.\nThe radially-binned Eqs. and are readily generalised for a *S*(*ν*, *τ**x*, *τ**y*) map, by replacing the *τ* bins by (*τ**x*, *τ**y*) pixels, as in Eq. . The beam and cluster co-addition methods, analysed separately below, are generally found to yield results in good agreement with each other, provided that bright NVSS sources are masked. The correction factors *η* are calibrated separately for beam and cluster co-addition after masking NVSS sources, showing small, $\\lesssim 5\\%$ differences between the two stacking methods for both cleared and restored data.\nTo demonstrate the agreement between beam and cluster co-addition, consider for example the high-frequency channel. Near the virial radius, we find $\\lesssim0.03$ ($\\lesssim0.07$) changes in *S*, which correspond to $\\lesssim 3\\%$ ($\\lesssim 6\\%$) fractional differences, for the cleared (restored) data (see §[subsub:sigprof]). The largest deviations in *S* between the two methods, $\\lesssim 0.4$ ($\\lesssim 0.3$), are found in the central, $\\tau\\lesssim 1$ regions of the clusters. Substantial differences between beam and cluster co-addition can be seen in the restored data when bright NVSS point sources are not masked (see §[append:centrerestored]).\nWe use control samples to verify (see §[subsec:controlsamples] and §[append:controlsample]) that the distribution of the excess emission above the background, stacked around mock clusters randomly placed on the sky, is approximately normal in each channel when using the cleared data. In the restored data, this distribution is slightly skew-normal but, within a ∼ 0.1 accuracy in *S*, can be approximated as normal. The control samples are used to calibrate the noise parameters *η*, as summarised in Table [tab:mapssum]. The distribution of the calibrated significance *S* attributed to the excess is then found to approximately follow a unit-normal (zero mean and unit variance) distribution. Namely, the distribution of control samples is found to be consistent with the calibrated Eqs. ([eq:photonadd]) and ([eq:clusteradd]), out to at least the ± 3*σ* confidence level of a normal distribution.\nOur analysis is largely based on the above, single-channel expressions, applied to the high-frequency channel, where the angular resolution is superior. The low-frequency channels are mostly used to measure the spectrum and verify the signal. However, we also co-add the different channels, adopting the significance estimate\n \n$${{S}}(\\tau) = \\frac\n{\\sum\\_{{{\\nu}}=1}^{N\\_{{\\nu}}} {{S}}({{\\nu}},\\tau)}\n{N\\_{{\\nu}}^{(1+\\psi)/2}} \\coma\n\\label{eq:band\\_stacking}$$ \nwhere *N**ν* = 7 is the number of co-added, low-frequency channels. Anticipating some degree of inherent correlation between the different channels, here we introduced a correlation parameter *ψ*, which can in principle range from 1 (full correlations) to 0 (no correlations) or even become negative (for anti-correlations), and is determined empirically below based on control samples.\nThese control samples indicate strong positive inter-correlations among the seven low-frequency channels. The simple correction factor in Eq. ([eq:bandstacking]) is motivated by the approximately unit-normal *S* distribution it produces once *ψ* is calibrated; we thus find that *ψ* ∼ 0.70 (*ψ* ∼ 0.85) for the cleared (restored) data. The strong correlations appear to emerge primarily from the similar small-scale structures seen across the radio sky in the different channels, rather than from any instrumentation effects, because point source restoration significantly strengthens the correlations and because the correlations among different channels roughly scale with their PSF. As such, Eq. ([eq:bandstacking]) provides a reasonable and simple alternative to the determination of the full covariance matrices. Notice that introducing a positive *ψ* provides a conservative estimate of *S*: it effectively lowers the significance attributed to the appearance of the virial excess emission in multiple channels.\nThe channel co-addition Eq. ([eq:bandstacking]) is applied to both beam and cluster stacking methods, with typically similar results. For example, the two methods show $\\lesssim 0.2$ ($\\lesssim 0.3$) differences in *S* corresponding to $\\lesssim 7\\%$ ($\\lesssim 19\\%$) fractional changes, for the cleared (restored) data around the virial radius (see §[subsub:lowfsigprof]). In terms of *ψ*, we find $\\lesssim 1\\%$ differences between beam and cluster co-addition.\nIt is instructive to compute the stacked intensity excess profile, Δ*I* ≡ *ν* Δ*I**ν*(*ν*, *τ*), and relate it to the significance *S* of the local excess. The procedure is similar to the above stacking of *S*, as discussed in §[append:intensity].\nBackground modelling\n--------------------\nThe virial signal is typically weaker than the Galactic foreground, the extragalactic background, and the emission from the centre of the cluster, so a careful removal of the background is essential. We consider several different models for the *T**b*(*ν*, c, h) background field.\nThe simplest model is a uniform background around each cluster, which we define as the mean brightness temperature *T**b*, 0(*ν*, c) measured within an angular separation *θ* < *θ**b* of the centre of the cluster. We find that *θ**b* ≃ 5∘ is sufficiently large to establish the virial ring, and not too large to be offset by the large-scale curvature of the Galactic foreground. Our results do not vary substantially with small changes in *θ**b* around this value (see §[append:backgroundremove]). This pertains both to the above uniform background and to the polynomial backgrounds discussed next. Therefore, we adopt *θ**b* = 5∘ as our nominal value.\nMore accurate models allow spatial variations, to better capture the local background. As our cluster selection avoids extended Galactic structures, namely the Galactic plane, Loop-I, and the Fermi bubbles (see selection criteria (iv) – (vi) in §[subsec:cluster]), we apply identical masks on the data, with a negligible effect on the results. After bright point sources and these Galactic structures are masked, the remaining background varies mainly on scales larger than the anticipated, $\\lesssim 1{^{\\circ}}$ extent of the virial signal. This remaining background can be fairly well approximated using a polynomial fit on larger scales. For each cluster, we thus consider the region *θ* < *θ**b*, and fit the enclosed data in each band by an order *N**b* polynomial in {*τ**x*, *τ**y*}.\nIn §[append:backgroundremove], we consider different choices of *N**b* and *θ**b*, quantifying their implications using the control samples presented in §[subsec:controlsamples] below. Overall, we find that the statistical distribution of *S*(*τ*) is approximately symmetric, normal and normalised, as constructed above, provided that *N**b* ≥ 2. In contrast, a low, *N**b* ≤ 1 order polynomial is not sufficient to capture the curvature in the background, and leads to *S*(*τ*) distributions that are neither symmetric, nor normal, nor normalised. However, higher orders *N**b* cause increasingly more of the virial excess itself to be fitted as part of the background, thus spuriously diminishing the estimated signal.\nThese considerations indicate that *N**b* = 2 or *N**b* = 3 best balance a fitting power sufficiently high to render *S*(*τ*) normal and normalised, but sufficiently low to avoid over-fitting much of the local excess due to the cluster as part of the background. Odd-order terms in a polynomial are integrated out by the radial binning, so for any odd *N**b*, the results are very similar, although not identical, to those of *N**b* − 1. Therefore, and for better accuracy in the {*τ**x*, *τ**y*} stacked maps, we adopt *N**b* = 3 as the nominal polynomial order. Interestingly, similar background properties were found by when fitting the stacked brightness of *Fermi* clusters. For our nominal, *N**b* = 3, the results are not sensitive to the fitting range for *θ**b* ≥ 5∘. For sensitivity tests and a discussion of alternative choices of *N**b*, see §[append:backgroundremove].\nControl sample tests\n--------------------\nIn order to estimate the statistics of the stacked quantities, calibrate the parameters *η* and *ψ*, test Eqs. ([eq:photonadd])–([eq:bandstacking]) for *S* (and their counterparts –, and for *I*), examine different background models, and investigate possible systematic biases, we generate control samples of mock clusters and stack the real LWA data around them with the same pipelines applied to the MCXC cluster sample. For each test, we thus generate $N\\_{\\rm mock} = 5000$ control samples, each chosen to mimic the real cluster sample. This value of $N\\_{\\rm mock}$ samples the analysed sky many ($\\gtrsim 40$) times, leading to sufficiently converged estimates ( < {2%, 2%, 3%} in {1, 2, 3}*σ* cleared-data containment around *τ* ∼ 2.4; < 1% in *η̄* and *ξ*).\nEach control sample consists of 44 control clusters with random positions on the sky, chosen under the same geometric constraints as the real sample, and in addition, constrained to lie at least $1{{\\overset{^\\circ}{.}}}5$ away from any MCXC catalogue source and any bright (1.4 GHz flux exceeding 1 Jy) point source from the NVSS catalogue. We verify that the latter constraint, avoiding bright NVSS sources for more efficient control samples, has no significant effect on our nominal results, as pixels are already masked within $0{{\\overset{^\\circ}{.}}}3$ from such bright NVSS point sources. The control clusters are assigned with the same relevant parameters (angular scale and, for modelling the virial signal, also mass and redshift) as the real clusters, and are identically analysed. The significance *S*(*ν*, *τ*) of the mock stacked excess is then estimated for each cluster sample.\nFigure [fig:mock] shows the resulting distribution of *S* around the control cluster sample, for nominal analysis parameters, in particular *N**b* = 3 and *θ**b* = 5 for background removal, and a mask of radius $0{{\\overset{^\\circ}{.}}}3$ around $F\\_{\\rm 1.4GHz} > 1\\Jy$ NVSS sources. The figure shows results for both the high-frequency (left panels) and the co-added low-frequency (right) channels, using either cleared (top row) or restored (bottom row) data. The results are stacked using both beam (thin lines) and cluster (thick lines) co-addition, found to be in good agreement with each other.\n [fig:mock]\nThe results are shown after calibrating the single-channel noise correction factors *η̄*, and the multi-channel correlation factor *ψ*. These parameters are tuned such that *S* is normalised to have unit variance, *σ*(*S*) = 1, when averaged over *τ*, in each channel separately and when the low-frequency channels are co-added; the resulting *η̄* and *ψ* values are not far from unity (see Table [tab:mapssum]). The figure then contrasts, for all relevant *τ* bins separately, the nominal values of *S* (y-axis labels) inferred from Eqs. ([eq:photonadd])–([eq:bandstacking]), against the mock confidence intervals (dashed curves) extracted from the control samples up to the maximal, ± 3*σ* levels available with $N\\_{\\rm mock}$, and against the normal-distribution intervals *μ* ± *n**σ* (dot-dashed curves) anticipated from the locally measured mean *μ* and standard deviation *σ* of the control samples for *n* = 1, 2, and 3. The respective curves should therefore coincide with each other and with integer values of *S* only when the underlying distribution of *S* is locally symmetric, normal and normalised; see §[append:describecontrolsample] and §[append:controlsample] for details.\nOverall, we find that the stacking equations for *S* [Eqs. –] and *I* (§[append:intensity]) can be applied to the cleared data, with minor errors with respect to the control sample distribution, even when the latter is extrapolated as a normal distribution to high-confidence levels. Our tests indicate that in the radial range 0.5 < *τ* < 10, we may then slightly underestimate the significance of an excess signal by $-0.06\\lesssim|\\Delta S/S| <0$ in the high-frequency channel, and overestimate it by $0<|\\Delta S/S| \\lesssim 0.02$ in the co-added low-frequency channels. In particular, around the virial radius, the nominal confidence levels are conservative (slightly exaggerated) and consistent with the extrapolated control sample distributions within $\\gtrsim -4\\%$ ($\\lesssim +3\\%$), for the high (co-added low) frequency data. For clarity, when displaying the *S* profiles measured from the real sample, we overplot the extrapolated significance levels of the corresponding control sample.\nStacking results\n================\nWe begin in §[subsec:73MHz] by presenting our nominal results, based on the high-frequency channel. To test these results and constrain the spectrum, in §[subsec:7lowerbands], we present the co-added results for the seven low-frequency channels. We focus on the cleared data, showing restored data results mainly to clarify the role of point sources. The correction factors, namely *η̄* for channel noise and *ψ* for cross-channel correlations, are used throughout, as inferred above from the control cluster catalogues.\nNominal analysis: high-frequency channel\n----------------------------------------\nIn our nominal analysis, we focus on the high-frequency, $73.15\\MHz$ channel. Here, the spatial resolution is considerably better than in other channels, complications due to PSF sidelobes are minor, and the control samples are well behaved even without any noise corrections. We inspect the brightness profile in §[subsubsec:HighFreqIntensity], and the significance of the implied excess in §[subsub:sigprof]; a pronounced excess near the virial is shown to be robust in §[subsec:virialexcess], and argued to be genuine and not associated with beam sidelobes in §[subsec:sidelobe].\nNominal parameters are used (henceforth, unless otherwise stated), including the standard cuts giving our nominal, 44 cluster sample, a radial binning with Δ*τ* = 0.25 resolution, polynomial background modelling of order *N**b* = 3 within *θ**b* = 5∘, masking data within $0{{\\overset{^\\circ}{.}}}3$ of NVSS point sources with a flux exceeding 1 Jy, and applying the (small) *η̄* correction.\n### Brightness profile\nFigure [fig:band7flux] shows, for both cleared and restored data sets, the radial, *τ*-profile of the mean excess brightness, averaged over the 44 clusters in our sample, as defined in §[append:intensity] [namely, using Eqs. ([eq:excessIntensitybm]) and ([eq:excessIntensitycl]), with 1*σ* uncertainty intervals (error bars) computed using Eqs. ([eq:errIntensitybm]) and ([eq:errIntensitycl])]. Nominal parameters are used.\n [fig:band7flux]\nBoth beam and cluster co-addition methods are shown in the figure, and are found to be in good agreement with each other, here and quite generally, except where point sources make a strong contribution. This agreement indicates that the diffuse features are fairly evenly stacked among the clusters, rather than dominated by a handful of outlier clusters. We verify this behaviour by manually checking the excess in individual clusters, estimating the mean and standard deviation of the excess, and removing the strongest contributing clusters, as demonstrated for the virial signal in §[subsub:sigprof]. For simplicity, we focus on beam co-addition for the remainder of §[subsubsec:HighFreqIntensity].\nAs expected, a strong signal is identified in the centres of clusters, and is partly removed by the modified CLEAN algorithm. The original signal, as inferred from the restored data, which more faithfully represents the signal including point sources, is peaked approximately at the centre of each cluster, with an excess intensity of Δ*I* ≃ 1.2 × 10− 10 erg s− 1 cm− 2 sr− 1 when averaged over clusters. This component, likely attributed to a combination of point sources, such as radio galaxies, and diffuse emission, such as from radio halos and minihalos, is further discussed in §[sec:centralemission].\nOutside the clusters, at large, $\\tau\\gtrsim 3$ radii, well beyond the virial radius, the signal is in general consistent with noise, with only expectedly small, $\\lesssim 1\\sigma$ fluctuations. The cleared and the restored data are not in good agreement here, indicating that deviations of the restored data from the background are largely associated with point sources. The low-significance cleared-data excess emission around *τ* ∼ 3.5 ( ∼ 2*σ*) and around *τ* ∼ 5.5 ( ∼ 2.5*σ*) are discussed below in §[subsub:sigprof].\nIn the peripheries of clusters, around $2\\lesssim\\tau\\lesssim3$, both data sets show a localised, significant excess peaked in the 2.25 ≤ *τ* < 2.5 bin, coincident with the virial shock signal found in by stacking *Fermi* data (shown as a vertical yellow band in the figures), as well as with other observations and previous theoretical predictions outlined in §[sec:Intro]. This excess, when averaged over clusters, is Δ*I* ∼ 5.0 × 10− 12 erg s− 1 cm− 2 sr− 1 in the cleared data, and ∼ 10% smaller in the restored data. The similar radius, magnitude, and shape of the excess in both data sets indicate that this signal is not associated with, nor strongly contaminated by, point sources. We henceforth refer to this signal as the virial excess.\n### Significance profile\nWe quantify the significance of the local excess above the background using both the nominal *S* derived in §[sec:StackedQuantities] for a normal distribution, and the control samples described in §[subsec:controlsamples]. Figure [fig:band7sig] shows the radial profile of the excess significance *S*, computed according to Eqs. ([eq:photonadd]) and ([eq:clusteradd]), as a function of *τ*, for both the cleared (top panel) and the restored (bottom) data, using both beam (abbreviated “bm”; symbols) and cluster (abbreviated “cl”; curves) co-addition. The sensitivity of the results to alternative parameter choices is explored in §[append:sensi]. Our analysis is based on the 0 ≤ *τ* < 10 region, sufficiently large to capture three times the radius of the -ray excess, or equivalently, more than four times the FWHM even in the lowest-frequency channel; the significance figures below focus on the 0 ≤ *τ* < 7.5 range only for visibility.\n [fig:band7sig]\nAs discussed in §[subsec:controlsamples], we overplot the confidence-level estimates based on the corresponding control samples, extrapolated to high orders in the normal-distribution approximation. Namely, the figure shows (as dotted lines) the mean of the beam co-added *S*, offset by integer multiples of its standard deviation, based on the respective control samples. The nominal confidence levels are consistent with these extrapolated confidence levels (compare dashed vs. dot-dashed curves in Fig. [fig:mock]), within the accuracy discussed in §[subsec:controlsamples]. Namely, outside the central *τ* < 0.5, our nominal significance estimates are consistent with the extrapolated levels within ∼ 4% ( ∼ 10%) for the cleared (restored) data.\nConsider first the restored data, shown in the bottom panel of Fig. [fig:band7sig]. Here, the central signal within $\\tau\\lesssim 1$ is highly significant, reaching the ∼ 18*σ* level. Beyond a local minimum (the ’dip’, henceforth) seen around *τ* ∼ 1.5, the nominal stacked signal (circles and thick dot-dashed curve) reaches the ∼ 3*σ* confidence level in the peak, 2.25 ≤ *τ* < 2.5 bin, with only $\\lesssim 1\\sigma$ fluctuations at large radii beyond *τ* ≃ 3. Both the signal and the fluctuations appear to be somewhat extended in *τ*. Using thicker, Δ*τ* = 0.5 bins (squares and thin double-dot dashed curve) raises the significance of the virial excess to the ∼ 4*σ* level in the 2 ≤ *τ* < 2.5 bin, while the fluctuations reach ∼ 2*σ*. The significance is also increased if these Δ*τ* = 0.5 bins are first offset by 0.25 in *τ* (triangles), but only to the ∼ 3.4*σ* level, in the 2.25 ≤ *τ* < 2.75 bin. The location of the peak is consistent with the stacked *Fermi* data analysis in either choice of angular binning, and in both stacking methods.\nNext, consider the cleared data (top panel), after point sources were removed by CLEAN and not restored. Here, the central peak is diminished to the ∼ 6*σ* level, as expected due to its point source contribution, whereas the virial excess becomes more significant, with a ∼ 4.2*σ* peak in the same, 2.25 ≤ *τ* < 2.5 radial bin. Thus, while point source removal does not appreciably change the magnitude or position of this virial signal (see Fig. [fig:band7flux]), it raises its significance by lowering the background noise level. Such behaviour is consistent with this excess arising from diffuse, rather than point-source, emission. The peak remains broad; using thicker, Δ*τ* = 0.5 bins again raises the significance, here to the ∼ 5*σ* confidence level both in the 2.0 ≤ *τ* < 2.5 bin and in the offset, 2.25 ≤ *τ* < 2.75 bin. The nominal excess is comprised of the cumulative contribution of ∼ 70% of the clusters in our sample, with a mean *S* ≃ 1.1 ± 0.7 in the 2 < *τ* < 3 range. No sub-sample strongly governs the signal, so removing the few most significant, {3.3, 2.4, 2.2…}*σ* contributions does not significantly alter the stacked results.\nUnlike the virial excess, discussed in some detail below in §[subsec:virialexcess], the central, $\\tau\\lesssim1.5$ signal is highly sensitive to point source removal, which substantially lowers its brightness (Fig. [fig:band7flux]) and corresponding significance (Fig. [fig:band7sig]). While most of the central flux is thus attributed to point sources, the residual stacked signal in the cleared data suggests a ∼ 6*σ* diffuse central component, extending in out to ∼ 1*R*500. This diffuse signal shows some $|\\Delta S/S|\\lesssim7\\%$, or equivalently $|\\Delta S|\\lesssim0.4$, difference between the two stacking methods (compare symbols and corresponding curves in Fig. [fig:band7sig]). In contrast, the virial peak shows only a ∼ 0.4% difference between the two stacking methods, suggesting that the virial excess is more evenly distributed among clusters than the central cleared signal.\nAlthough the noise level is reduced in the cleared data, point source removal appears to amplify some of the $\\lesssim1\\sigma$ fluctuations, found in restored data outside the clusters (*τ* > 3), resulting in apparently extended, ∼ 2*σ* excess signals in cleared data near *τ* ≃ 3.5 and near *τ* ≃ 5.5. Consider first the *τ* ∼ 3.5 excess. This signal is seen only in the high-frequency channel, and not in other frequencies (see §[subsub:lowfsigprof]), so we cannot substantiate its viability. Furthermore, this signal is dominated by only three clusters, each showing a > 2.5*σ* excess at *τ* ≃ 3.5. Two of these clusters harbour a (partly masked) extended, > 1 Jy NVSS point sources at the same radius. Excluding these three clusters lowers this excess to < 1*σ*, with only a minute effect on the virial signal (see stars in Fig. [fig:band7sig]). Hence, we disregard this *τ* ≃ 3.5 signal as a fluctuation. In contrast, the *τ* ∼ 5.5 signal — like the virial excess — reflects multiple small contributions from a large fraction ( ∼ 70%) of our clusters, and is persistent, seen in all channels. However, this signal is less significant than the virial excess and more sensitive to CLEAN, and it becomes far less significant or even vanishes entirely when the declination selection criterion is varied (see §[append:sensi]). We further discuss this *τ* ≃ 5.5 signal in §[subsec:summarydiscussion].\n### Virial excess\nFigure [fig:band7sig] and the preceding discussion indicate that the virial excess in the high-frequency channel is robust and significant at the 4–5*σ* confidence level. The signal peaks around *τ* ≃ 2.4, coincident with its -ray counterpart, identified earlier by stacking *Fermi* data. The excess is somewhat extended in *τ*, contributing to bins over the $2.0\\lesssim\\tau\\lesssim2.8$ range. The location, magnitude, and shape of the signal inferred from beam vs. cluster co-addition are in good agreement with each other. This agreement supports the robustness of these statistics, and indicates that the stacked excess arises from many of the clusters in our sample, rather than being dominated by a handful of outlier clusters. The agreement between the excess parameters in cleared vs. restored data indicates that the excess is diffuse, and not dominated or distorted by point sources. These conclusions are confirmed by inspecting the individual significance profiles of all clusters in our sample.\n [fig:2dsig]\nThe rescaled, stacked virial signal is sufficiently strong to be seen even without full radial binning. Figure [fig:2dsig] shows a partly-folded map of the significance *S*(*τ**x*, *τ**y*) of the stacked excess, prepared as in the radial plots but without radial binning. Namely, an excess flux map is first generated for each of the 44 clusters in our sample, using the cleared data. Each such map is rescaled by the respective *R*500, and rotated randomly about the centre of the cluster to wash out any co-added large-scale gradients. The resulting maps are then stacked using beam co-addition, according to Eq. ([eq:photonadd]). To better show the signal, the resulting image is then folded onto a quadrant or an octant (see labels). The nominal, high-frequency signal is folded onto quadrant 2 (upper left sector). To further emphasize the signal, we fold the same image again, onto octant 5 (just below quadrant 2). The bottom quadrant and octant show the low-frequency signal, obtained by co-adding the seven low-frequency channels as discussed in §[subsec:7lowerbands]. Quadrant 1 (upper right) combines the high-frequency channel with the co-added, low-frequency channels. Both the central, $\\tau\\lesssim1$ signal, and a virial ring peaked around *τ* ∼ 2.4 are evident. The $2.2 \\lesssim \\tau \\lesssim 2.5$ range, identified () as a virial ring in stacked -rays, is also shown (dotted orange circles). Figures with a larger *τ* extent and modified selection criteria are shown in §[append:2Dsigmaps].\nIn order to further test the robustness of the virial signal, we show (see §[subsec:LowFrequencyVirial]) that it presents in all channels, and carry out a series of sensitivity and convergence tests in §[append:backgroundremove] and §[append:sensi]. In particular, we examine the sensitivity of the signal to the cluster selection criteria, the background modelling order, the background fitting area, and different radial bin sizes. Overall, the results are found to be insensitive to reasonable variations in these parameters.\n### Ruling out a sidelobe origin\nGiven the strong central signal, especially before point sources are removed, sidelobes become a major concern: one must carefully test if the virial ring may arise simply from stacked interferometric ripples around the central signals. In principle, such a spurious effect should be ruled out by the CLEAN algorithm, as the identified point sources near the centre were convolved with the PSF before they were removed, along with their sidelobes, and the remaining central signal in the cleared data is not sufficiently strong to generate the virial excess as a sidelobe ripple. Nevertheless, putative artefacts due to inaccuracies in the algorithm should be ruled out before the virial excess can be securely established.\nIn subsequent parts of the analysis, we rule out a putative virial excess stemming from the residuals of sidelobe ripples of the central signal that were not removed by CLEAN. This conclusion is derived in several independent methods, including: (i) a significant virial excess signal is found also in those clusters that do *not* show a central signal (as discussed below and shown in Fig. [fig:band7weakstrong]); (ii) Monte-Carlo simulations of PSF-convolved central emission models do not show the observed virial ring (§[append:centrerestored]); (iii) multiple channels show the signal at similar ring radii, despite variations among their PSF patterns (§[subsec:7lowerbands]); (iv) removing one or a few outlier clusters with bright NVSS sources in their centre strongly diminishes the raw (restored, unmasked) stacked central signal, with no significant effect on the virial excess (§[append:psf]); (v) intentional attempts to generate an artificial virial excess by choosing control, centrally-bright clusters, fail to produce a significant signal (§[append:psf]); (vi) the spectrum of the virial excess is found to be a robust power-law (§[subsec:virialspec]), whereas the central signal shows a non-trivial spectrum varying spatially and among clusters (§[append:centrerestored]); and (vii) the virial excess is well-fit by the expected model: spatially coincident with the previously stacked *γ*-ray signal and of a normalisation comparable to that inferred in Coma, not far from equipartition (§[sec:summary]).\nAlthough most clusters show a bright central signal, at least before point sources are removed, some of them do not. The first method we invoke to test for sidelobe artefacts is to focus on those clusters that do not show a central excess. In such clusters, an excess signal near *τ* ≃ 2.4 clearly cannot be dismissed as sidelobes of emission from the centre. We define such centre-faint clusters as those showing an intensity deficit, Δ*I* < 0, in their central, *τ* < 0.5 region, after the large-scale background was removed. For the selection of these clusters, we use the raw data, the restored data without additional NVSS point-source masking, in order to include the full imprint of all point sources that may have left some sidelobes contamination in the cleared data even after CLEAN. We adopt nominal parameters, focusing on the high-frequency channel.\n [fig:band7weakstrong]\nAfter removing the background, 13 out of the 44 clusters in the sample are classified as centre-faint, while 31 clusters are classified as centre-bright. Figure [fig:band7weakstrong] shows the stacked excess intensity profiles of these centre-faint vs. centre-bright clusters, for both cleared (top panel, with masking) and restored (bottom, without masking) data. As the cleared data show, the centre-faint clusters make a substantial contribution to the virial ring signal, comparable to and even exceeding that of the centre-bright clusters, when accounting for the different sizes of each sub-sample. Namely, the virial excess in the cluster-faint clusters is significant at the ∼ 2.5*σ* confidence level, slightly higher than anticipated from Poisson statistics when taking into account the smaller size of this sub-sample.\nThe mean intensity of the centre-faint clusters agrees with that of the centre-bright clusters in the virial radius bin. As expected, the two sub-samples show different, $\\lesssim2\\sigma$ fluctuations outside the clusters, beyond *τ* ≃ 3. The centre-faint signal is somewhat broader than found for the full sample, extending out to *τ* ≃ 3, in resemblance of the low-frequency signal discussed in §[subsec:7lowerbands]. We conclude that the agreement between the virial signal as inferred from centre-faint vs. centre-bright clusters supports the virial signal as a genuine effect, rather than an artefact due to a putative failure of CLEAN to remove PSF sidelobes of the emission from the centres of clusters.\nLow-frequency channels\n----------------------\nNext, consider the seven low-frequency channels, in order to test the results of the nominal, high-frequency analysis in §[subsec:73MHz] and to extract some spectral information. In these frequencies, the resolution is substantially lower than in the high-frequency channel, PSF sidelobes are more pronounced, and the control samples indicate noise characteristics that require somewhat larger correction factors *η̄*, as provided in Table [tab:mapssum]. In order to obtain robust results, we therefore co-add the seven channels, taking into account the correlation factor *ψ*, unless otherwise stated.\n### Brightness profile\nFigure [fig:band0-6flux] presents the mean excess brightness profiles of the co-added low-frequency channels, for both the cleared and restored data. The figure shows Δ*I*(*τ*) = *ν*Δ*I**ν*, which is optimised for an *I**ν* ∝ *ν*− 1 spectrum, computed as described in §[append:intensity] [using Eq. ([eq:Intensitysum]), with uncertainties computed according to Eq. ([eq:errIntensitysum])]. Again, we use nominal parameters as in §[subsec:73MHz], and find that the two stacking methods yield consistent results. For simplicity, we focus here (throughout §[subsubsec:LowFreqIntensity]) on beam co-addition.\n [fig:band0-6flux]\nThe brightness profile in the low-frequency channels is similar to the nominal, high-frequency behaviour discussed in §[subsec:73MHz], with expected differences due to the poorer resolution available here. Strong central emission in the restored data is found to peak, when averaged over clusters and channels, at Δ*I* ≃ 1.1 × 10− 10 erg s− 1 cm− 2 sr− 1 (using beam co-addition), quite similar in shape and in magnitude to the high-frequency channel. Outside the clusters, at $\\tau\\gtrsim 3$, a very broad, ∼ 2*σ* excess is seen throughout the $4.5\\lesssim\\tau\\lesssim6$ range, coincident with the *τ* ∼ 5.5 high-frequency excess discussed in §[subsub:sigprof] and in §[subsec:summarydiscussion].\nIn the periphery of the clusters, a significant excess is found around *τ* ∼ 2.6, broadly consistent with the virial signal seen in the high-frequency channel in terms of location, width, and amplitude. The excess is again quite similar in cleared and restored data, suggesting that it is mostly diffuse in nature. However, some differences between the two data sets suggest that contamination by point sources is stronger and more difficult to correct at low frequencies, due to the poor resolution and a possible abundance of soft-spectrum sources.\nThis virial excess in the cleared data peaks, when averaged over clusters and channels, at Δ*I* ∼ 2.8 × 10− 12 erg s− 1 cm− 2 sr− 1 (for beam co-addition). This value is similar to its high-frequency counterpart, supporting the detection of the high-frequency signal and affirming that it is not a PSF artefact, as discussed above (in §[subsec:sidelobe]) and further below (in §[sec:summary] and §[append:psf]). The virial excess at low frequencies is slightly ( ∼ 8%) less significant than in the high-frequency channel, possibly because the signal is convolved with more extended PSFs at low frequencies, rendering the signal broader (see §[append:backgroundremove]); a more careful analysis of the signal in different channels, along with a derivation of the implied spectrum, is deferred to §[subsec:speclwa] (backward modelling) and §[subsec:virialspec] (forward modelling).\nThe virial signal is slightly broader in the low-frequency channels than it is in the high-frequency channel, as one might expect from the lower resolution. Here, the signal peaks in the 2.5 ≤ *τ* < 2.75 bin, adjacent to the 2.25 ≤ *τ* < 2.5 bin of the high-frequency peak. The broader signal at low frequencies shows a significant excess coincident with the $2.2\\lesssim\\tau\\lesssim 2.5$ range of the stacked *Fermi* signal and with the high-frequency LWA channel, but it extends slightly beyond *τ* ≃ 3. In addition to the lower resolution, this broadening could be in part attributed to the contribution of faint, soft point sources. This possibility is supported by the substantial differences between the profiles obtained from cleared vs. restored data, for example the pronounced minimum around *τ* ≃ 3.7 in the latter. We revisit the issue in more detail in §[subsec:LowFrequencyVirial].\n### Significance profile\nFollowing the same procedure applied to the nominal, high-frequency channel in §[subsub:sigprof], next we compute the significance of the excess in the low-frequency channels. Figure [fig:band0-6sig] shows the resulting significance profiles *S* for the cleared (top panel) and restored (bottom) data, both for individual channels (thin curves without symbols) and for the excess co-added over the seven low-frequency channels (symbols and thick curves). Confidence levels extrapolated from the corresponding (channel co-added) control samples are also shown (dotted curves), using the same procedure as for Fig. [fig:band7sig].\n [fig:band0-6sig]\nLike in the high-frequency case, the central signal is very significant at low frequencies, with a ∼ 25*σ* excess in the restored data. The dip here is seen at *τ* ∼ 1.75, just outside its high-frequency counterpart, presumably because of the lower spatial resolutions. When using the cleared data, the central signal is reduced to the ∼ 12*σ* level, consistent with a dominant point-source contribution as found in the high-frequency case.\nFor Δ*τ* = 0.25 resolution, both cleared and restored data sets show a virial signal peaked around *τ* = 2.6, seen separately in almost every individual channel, as well as in the co-added result. The exception is the $52.2\\MHz$ channel, where the peak is found farther out, around *τ* ≃ 3. The virial signal is not strongly affected by the removal of point sources, although it does strengthen from ∼ 2*σ* in the restored data to ∼ 3.8*σ* in the cleared data, as expected from a diffuse signal contaminated by point sources, and as seen in the nominal analysis. As in the high-frequency case, using thicker, Δ*τ* = 0.5 bins raises the significance, as expected for an extended signal. In the cleared data, we again find a ∼ 5*σ* excess in the 2.25 ≤ *τ* < 2.75 bin, just like in the high-frequency channel. Without an offset, the ∼ 5*σ* peak shifts from the 2 ≤ *τ* < 2.5 in the high-frequency channel, to the adjacent, 2.5 ≤ *τ* < 3 bin at low frequencies. The nominal excess is comprised of the cumulative contribution of ∼ 60% of the clusters in our sample.\nBeyond *τ* ≃ 3.5, the significance profile is consistent with noise, with $\\lesssim 1\\sigma$ fluctuations in the restored data, and the aforementioned ∼ 2*σ* excess at $4.0 \\lesssim \\tau \\lesssim 6.5$ in the cleared data. This excess, peaked around *τ* ∼ 5.5, is robust, like its high-frequency counterpart, and is comprised of small contribution from most of the clusters in our sample.\n### Virial excess\nFocusing in more detail on the virial excess, we examine its properties based on the low-frequency channels, alone and in relation to the high-frequency results. All low-frequency channels show some virial excess, in significance levels ranging between ∼ 2.0*σ* and ∼ 3.7*σ* in the cleared data. Co-adding the cleared, low-frequency channels yields a significant virial signal, reaching the ∼ (3.8--5)*σ* level. This is comparable to the (4–5)*σ* level obtained using the cleared high-frequency channel alone, thanks to its superior resolution. The low-frequency virial signal is, like in the high-frequency channel, robust and not sensitive to selection criteria or analysis details (see §[append:sensi]).\nAs in the high-frequency case, the co-added low-frequency signal is sufficiently strong to be visible even without full radial binning, as illustrated in the folded maps in Fig. [fig:2dsig]. The virial, *τ* ∼ 2.5 arc is visible in the low-frequency quadrant 4 (bottom right), although it is somewhat more diffuse and less pronounced than in the high-frequency quadrant 2 (top left), due to the lower resolution. As expected, further folding the low-frequency result onto octant 6 (bottom triangle) or co-adding it to the high-frequency channel (top right quadrant 1) gives a more pronounced signal.\nThe low-frequency virial signal is somewhat less robust than it is in the high-frequency channel, presenting larger differences between beam and cluster co-addition methods. In particular, we find ∼ 7% differences between the two stacking methods in the cleared data. However, these discrepancies are not significant, and they appear to be at least in part associated with residual point sources. As expected, the cleared data signals show better agreement between the two stacking methods than found in the restored data.\nIn low frequencies, the virial signal peaks consistently at radii somewhat larger than the *τ* ∼ 2.4 peak found in the high-frequency channel, an effect attributed in part to the better resolution of the latter. Six of the seven low-frequency channels peak in the 2.5 ≤ *τ* < 2.75 bin when using the cleared data, just outside the adjacent, 2.25 ≤ *τ* < 2.5 bin of the high-frequency peak. With a finer, Δ*τ* ≃ 0.167 (6 bins per *τ*; see §[append:sensi]) resolution, we again find the low-frequency peak adjacent to the high-frequency peak with the same high resolution, indicating that the effect is small. We conclude that the peak shifts from *τ* ≃ 2.6 in the co-added low-frequencies, to *τ* ≃ 2.4 in the better resolved, high-frequency.\nInterestingly, when using the restored data, some of the peaks shift to slightly larger radii. This suggests that the discrepancy with respect to the high-frequency channel may in part be associated with low-frequency contamination from soft point sources, not fully removed by CLEAN. Another possibility is that the virial emission from individual clusters has a broad distribution, extending out to *τ* ≃ 3 or even beyond that, with some preference around *τ* ≃ 2.4 that is picked up when sufficient high-resolution data are stacked. Such an interpretation could explain why the virial signals both for the low-frequency results and for the centre-faint, high-frequency results (see §[subsec:sidelobe]) have a similarly broad *τ* distribution, with a peak similarly shifted to *τ* ≃ 2.6.\nThe comparable peak radii found in the different low-frequency channels provide additional evidence that the virial excess is unlikely to be an artefact due to uncleaned sidelobe contamination from the central signal. Indeed, such putative ripples would show in different radii, corresponding to the frequency-dependent PSF, as we verify directly in §[append:centrerestored].\n### Spectra\nNext, consider a crude estimate of the spectra of the central and virial signals, before deprojecting and modelling them in §[sec:centralemission] and §[sec:virialemission]. The specific brightness of the two signals in all eight channels is shown in Fig. [fig:Spectra], as a function of frequency. For this purpose, we define the central signal as the *τ* < *τ**m**a**x* excess, exploring different choices of *τ**m**a**x*; the figure focuses on the most central, *τ**m**a**x* = 0.17 region accessible with present resolution. As the virial signal is extended radially at low frequencies, we define it for spectral purposes as the mean excess in the broad, 2.25 ≤ *τ* < 3.0 range. When considering the central signal, we examine both the raw excess, in the restored data without NVSS masking, and the cleaned excess, in the cleared data with masking. The former includes the contribution of point sources, and is useful as it more accurately reflects the spectrum of any putative sidelobe effects due to the central signal. The latter is a closer representation of the underlying, diffuse central signal, although some point source contamination is likely to persist.\n [fig:Spectra]\nIn principle, the spectrum should be estimated by forward modelling, where different model signals are injected into control samples, which are then analysed with the real pipeline in order to determine the best-fitting model. We use this method to analyse the spectrum of the central signal in §[sec:centralemission], and of the virial signal in §[sec:virialemission]. Here, we use a crude backward modelling to directly fit the binned spectrum in Fig. [fig:Spectra]. The results of such modelling may be useful in regimes where projection effects are favourable and PSF corrections can be approximately accounted for. As in §[subsec:controlsamples], let us define the spectral index *α* through *I**ν* ∝ *ν*− *α*.\nThe raw central excess is the strongest among the three signals in the figure, and its statistical uncertainties (red error bars on the top curve) are negligible. At frequencies above $50\\MHz$, where the PSF no longer varies strongly with frequency (see Table [tab:mapssum]) and the resolution is sufficient for *τ**m**a**x* = 0.17, the data are found to be well-fit by a pure power-law, *α* = 0.8 ± 0.1, with *χ**ν*2 ≃ 0.4 chi-squared per degree of freedom. At lower frequencies, the spectrum appears to be harder, but this is likely due to the strong frequency dependence of the PSF. For larger *τ**m**a**x*, the spectrum remains well-fit by a pure power law above $50\\MHz$, but the integrated spectrum gradually softens. Such softening suggests a considerable contribution from underlying diffuse emission, possibly in the form of radio halos and minihalos which often exhibit such a behaviour. At larger scales, the steepening appears to saturate at some $1.5\\lesssim\\alpha\\lesssim2$; for *τ**m**a**x* = 1, we find *α* = 1.5 ± 0.1 (*χ**ν*2 ≃ 1.5). At small scales, extrapolation to *τ**m**a**x* = 0 gives *α* = 0.7 ± 0.1, consistent with a population of point sources.\nIt is interesting to compare the spectra of possible sidelobe ripples from the central signal, to the *α* ∼ 1 spectrum of the virial shock, as expected based on theory and on previous high-energy studies, and favoured in §[sec:virialemission] based on forward modelling. The raw, steep, central signal can be approximated as multiple point sources stacked close to the centre of the cluster, so a strong, sharp ripple is expected to show *α* ≃ 0.7, substantially harder than the virial signal. In contrast, the cumulative emission from an extended central region can give rise to a more diffuse ripple with a much softer, *α* ∼ 1.5 spectrum. Therefore, mistaking a ripple for an *α* ≃ 1 virial signal is unlikely. Such contamination cannot, however, be ruled out entirely based on spectral arguments alone, as the inferred spectrum is sensitive to the precise handling of point sources. For instance, removing one cluster with a bright NVSS source in its centre changes the raw spectrum in Fig. [fig:Spectra] from 0.8 ± 0.1 to 1.0 ± 0.2, which is similar to the expected virial signal. Naively, the ripples should remain softer than *α* = 1, but making such a determination requires a more careful analysis.\nThe cleared central signal is much weaker than the raw signal, its statistical errors are larger, and some spectral curvature emerges in backward modelling. Consequently, while the signal can still be fit by a power-law spectrum, the uncertainty in the resulting spectral index, ranging from *α* = 0.9 ± 1.0 at *τ**m**a**x* = 0.17 to *α* = 1.4 ± 0.7 at *τ**m**a**x* = 1, is substantial. Note that this mostly diffuse signal is quite similar to the expectation from stacked radio halos, in terms of its spatial and spectral distributions. The virial signal is even weaker, strongly dominated by statistical uncertainties, and the location of its peak radius is frequency-dependent, so a backward determination of the spectrum is not useful. The spectra of the two signals in the cleared data are addressed more carefully below, using forward modelling (see §[sec:centralemission], §[subsec:virialspec] and §[append:CentralResults]).\nCentral excess\n==============\nBefore studying the virial excess, it is important to separate out the central, $\\tau\\lesssim 1.5$ excess, to minimize its risk of biasing the measurement of the virial signal. We examine the central signal in both restored and cleared data. As in §[subsec:speclwa], the raw excess is studied using the restored data without masking NVSS sources, while the cleaned excess after point source removal is studied using the cleared data after masking NVSS sources.\nThe virial signal could in principle contribute, in projection and especially where the PSF is extended, to the central excess. As we show in §[append:mutualeffects], this contribution is only at the $\\lesssim 5\\%$ level for the raw data, and so can be neglected. For the cleared signal, this contribution can reach ∼ 20%, so the central and virial signals are simultaneously modelled in §[subsec:virialmodel]. For simplicity, and to ∼ 20% accuracy, in this section, we neglect the virial contribution even for the cleared data.\nWe utilise the different appearances of the central signal in restored and cleared data to better model the emission and its effect on the virial signal. Importantly, we examine if sidelobes of the central signal may be able to mimic the virial excess. In §[sec:CentralModels], we outline models for the central component in each data set. The fitting procedure, through *χ*2 minimisation, is described in §[subsec:Fittingprocedure]. We outline the results in §[subsec:centralexcessresults] and provide the technical details in §[append:CentralResults].\nCentral excess models\n---------------------\nWe model the central emission as a combination of a point source (subscript ’p’) and an extended (subscript ’e’) component, with respective power-law spectral indices *α**p* and *α**e*. The range of redshifts spanned by the clusters in our sample is small, so the specific flux projected along the line of sight in normalised coordinates $\\bm{\\tau}=\\{\\tau\\_x,\\tau\\_y\\}$ can be approximately described by a simple one-dimensional model,\n \n$$\\label{eq:central\\_emision\\_point}\nF\\_{cen} \\equiv F\\_{{{p}}} \\tilde{\\nu}^{-\\alpha\\_p} \\delta(\\tau) + A\\_{{{e}}} \\tilde{\\nu}^{-\\alpha\\_e} \\frac{\\Theta(\\tau\\_{\\rm cut}-\\tau)}{(\\tau^2 + \\tau\\_c^2)^{\\zeta}} \\coma$$ \nwhere *ν̃* ≡ *ν*/*ν**h* is the normalised (to $\\nu\\_h=73.152\\MHz$) frequency, *τ**c* is the core radius, and Θ is the Heaviside step function. The normalizations *F**p* and *A**e*, and the normalised core radius *τ**c* and cutoff radius $\\tau\\_{\\rm cut}$, are for simplicity assumed to be identical for all clusters. As the typical core radius, 0.1*R*500, is smaller than the FWHM of the LWA PSF even in the high-frequency channel, model results are not sensitive to the exact choice of *τ**c* around this value; hence, we henceforth use *τ**c* = 0.1. For illustrative purposes, we define the total flux in the high-frequency channel, *F*73 ≡ *F**p* + *F**e*, where *F**p* and $F\\_{{{e}}}\\equiv 2\\pi \\tau\\_{\\rm cut}^{2-\\zeta}A\\_{{{e}}}/(2-\\zeta)$ are the specific fluxes of the point-like and extended components, respectively, integrated over *τ* at *ν* = *ν**h*.\nRecall that bright sources in the LWA maps are deconvolved with the PSF, and the associated emission is either removed or restored with a Gaussian restoring beam (for restored data; see typical FWHM in Table [tab:mapssum]). When modelling the cleared data, Gaussian restoration is irrelevant, and we simply convolve the model with the PSF.\nIn contrast, in the restored data, which shows a much brighter central excess than in the cleared data, most ( ∼ 75%; see §[append:CentralResults]) of this excess has been identified by CLEAN and restored. Indeed, the central signal is stronger and more sharply peaked in the restored data than it is in the cleared data, as seen from the stacked intensity (by a factor of ∼ 4; Figs. [fig:band7flux] and [fig:band0-6flux]) and significance (Figs. [fig:band7sig] and [fig:band0-6sig]) profiles. Therefore, our model for the restored central excess ([eq:centralemisionpoint]) is convolved with the Gaussian restoring beam and then fit to the data. This model can then be convolved with the PSF, to constrain the maximal putative sidelobe artefacts.\nFitting procedure\n-----------------\nFor both restored and cleared data sets, we run Monte-Carlo simulations and minimize *χ*2 to determine the best-fitting parameters. The procedure involves sampling a grid of possible model parameter values, and testing each parameter set against $N\\_{\\rm mock}$ mock samples. Each mock sample contains 44 mock clusters, like the real sample, distributed randomly across the allowed regions of the sky as described in §[subsec:controlsamples]. For each mock cluster, we inject the model emission Eq. , convolve the mock sky image with either the PSF or a restoring Gaussian beam, as explained above, and pass the resulting mock image through our nominal stacking and analysis pipelines. We use $N\\_{\\rm mock} = 30$ such mock samples for each choice of parameter set, sufficient for convergence within $\\lesssim0.1\\%$ in *χ*2.\nEach control sample *j* yields a significance distribution M*j*(*ν*, *τ*, c) = *S*(*ν*, *τ*, c), as described in §[sec:StackedQuantities]. We average among the $N\\_{\\rm mock}$ samples of a given parameter set in order to estimate its model outcome, $\\mathcal{M}(\\nu,\\tau,{{\\mathsf{c}}}) \\equiv N\\_{\\rm mock}^{-1} \\Sigma\\_{j=1}^{N\\_{\\rm mock}} \\mathcal{M}\\_{j}(\\nu,\\tau,{{\\mathsf{c}}})$. The *χ*2 value can then be computed, as usual, by summing over all relevant channels, radial bins, and clusters,\n \n$$\\chi^2 =\n \\sum\\_{{{\\nu}}}\n \\sum\\_{\\tau} \\sum\\_{{{\\mathsf{c}}}=1}^{N\\_c} \\left[{{S}}(\\nu,\\tau,{{\\mathsf{c}}}) - \\mathcal{M}(\\nu,\\tau,{{\\mathsf{c}}})\\right]^2\n \\coma\n \\label{eq:chi2\\_0}$$ \nas *S*(*ν*, *τ*, c) is approximately a unit normal random variable. As in §[sec:result], we separately analyse the high-frequency channel and the combined low-frequency channels. The seven low-frequency channels are jointly fitted with the same parameters of the model. For the central excess modelling, we compute *χ*2 in the radial range 0 ≤ *τ* ≤ 1.5 (and verify that the results are not sensitive to exact choice of outer radius).\nA simple alternative to Eq. is obtained by choosing the normalised stacked signal *S*(*ν*, *τ*) as the unit normal random variable, instead of its individual cluster counterpart *S*(*ν*, *τ*, c). This choice leads to\n \n$$\\chi^2 =\n \\sum\\_{{{\\nu}}}\n \\sum\\_{\\tau} \\left[{{S}}(\\nu,\\tau) - \\mathcal{M}(\\nu,\\tau)\\right]^2\n \\coma\n \\label{eq:chi2}$$ \nreducing the number N of degrees of freedom (DOF) by a factor ∼ *N**c*. Here, we again averaged over mock samples, $\\mathcal{M}(\\nu,\\tau) \\equiv N\\_{\\rm mock}^{-1} \\Sigma\\_{j=1}^{N\\_{\\rm mock}} \\mathcal{M}\\_{j}(\\nu,\\tau)$, where M*j*(*ν*, *τ*) = *S*(*ν*, *τ*) is obtained by stacking over clusters, as defined in §[sec:StackedQuantities], in either beam-weighted or cluster-weighted methods. This prescription aims to better fit the stacked signals (shown in Figs. [fig:band7sig] and [fig:band0-6sig], typically giving a lower *χ*2 per DOF), at the expense of the sensitivity to the mass-dependence of individual cluster models. The method is more conservative than Eq. , typically providing similar parameter estimates at lower confidence levels, so we primarily use Eq. . Quantitatively, Eqs. and typically yield $\\lesssim 20\\%$ differences between their best fit parameter estimates. The values of *ζ* and $\\tau\\_{\\rm cut}$ change slightly more, by ∼ 30%, but these variations typically lie within the 1*σ* uncertainty of these parameters. A larger and more significant difference is obtained for the normalisation $\\dot{m}\\xi\\_e\\xi\\_B^{(2+p)/4}$ of the virial signal. Here, using Eq. lowers the best fit by ∼ (20–55)%; see discussion in §[subsec:TS].\nThe frequency summations in Eqs. ([eq:chi20]) and ([eq:chi2]) do not take into account correlations between channels, and so tend to overestimate the *χ*2 expected for co-adding independent low-frequency channels. Indeed, these equations reasonably identify the best-fitting parameters, but the correlations can lead to unreasonably small parameter uncertainties. We may crudely correct for correlations by choosing the corrected *S*(*τ*), defined in Eq. ([eq:bandstacking]), as the random variable. Along with the corresponding, corrected model M(*τ*) = *N**ν*− (1 + *ψ*)/2Σ*ν* = 1*N**ν*M(*ν*, *τ*), we then obtain\n \n$$\\chi^2 =\n \\sum\\_{\\tau} \\left[{{S}}(\\tau) - \\mathcal{M}(\\tau)\\right]^2\n \\coma\n \\label{eq:chi2\\_2}$$ \nfurther reducing N by a factor of ∼ *N**ν*. Equation ([eq:chi22]) cannot be used to model the spectrum, and so is mainly invoked below only to gauge the correlations.\nFor model parameters other than the spectrum, we can compare the best fits obtained from Eqs. ([eq:chi2]) and ([eq:chi22]). While the best-fit values agree, the former equation leads to consistently and predictably underestimated uncertainties, due to channel correlations. The results become fully consistent with each other if one multiplies the *χ*2 of Eq. ([eq:chi2]) by a constant correction factor C. When co-adding cleared data in all low-frequency channels, this correction yields consistent results for all model parameters, provided that C ≃ 0.25 (see §[append:controlsample]).\nAn alternative way to correct for correlations is by generalising Eq. ([eq:bandstacking]) to all model parameters. Namely, assuming that the uncertainty in any model parameter *a* should satisfy *σ**a*, *N* = *N*(*ψ* − 1)/2*σ**a*. Here, *σ**a* (*σ**a*, *N*) denotes the uncertainty obtained from one channel (*N* channels), and *ψ* was calibrated in §[sec:StackedQuantities]. We find that Eq. ([eq:chi2]) yields a similar relation, *σ**a*, *N*\\* = *N*(Ψ − 1)/2*σ**a*\\*, but with a different correlation parameter Ψ affected by the numerical procedure, found to be Ψ ≃ 0.45 for the restored data and Ψ ≃ 0.01 for the cleared data. Hence, when deriving the uncertainty *σ**a*, *N*\\* in *a* from Eq. ([eq:chi2]) with summation over *N* channels, we correct the result as\n \n$$\\label{eq:uncertainty\\_corr}\n\\sigma\\_{a,N} = N^{(\\psi - \\Psi)/2} \\sigma\\_{a,N}^{\\*} \\coma$$ \nand again recover a good agreement with the outcome of Eq. ([eq:chi22]) (see demonstrations in §[append:controlsample]). Anticipating parabolic *χ*2 minima near the best-fit values, we find that C ≃ *N*(Ψ − *ψ*) ≃ 0.26 for the cleared data, consistent with the independent finding above, and C ≃ 0.46 for the restored data.\nUnless otherwise stated, we henceforth compute *χ*2 using Eq. and correct the co-added results for the low-frequency channels using Eq. . Other methods are occasionally demonstrated to yield similar results. We verify that the results are not sensitive to the choice of $N\\_{\\rm mock}$. For brevity, in the following text, we sometimes quote *χ*2 (or, when relevant, C*χ*2) values only for beam co-addition, but provide both beam (denoted ’bm’) and cluster (’cl’) values in the tables and most figures.\nCentral modelling results\n-------------------------\nFor the raw data, the Gaussian-convolved central model, including both point-like and extended components, provides a good fit in both high-frequency (*χ*2 per DOF value of *χ**n*2 ≃ 0.8) and co-added low-frequency (C*χ**n*2 ≃ 0.6) channels, but the large parameter uncertainties suggest that the model is oversimplified. Here, the spectral indices of the two components typically fall in the range $0.7\\lesssim \\alpha\\lesssim 1.3$, but the point-like component softens considerably in the higher-frequency channels, and we are unable to determine the spectrum of each component more precisely, as the results are sensitive to the co-addition method and channel selection. The detailed modelling is outlined in §[append:centrerestored].\nIn the cleared data, the extended emission component alone, convolved with the PSF, reproduces the central signal nicely in both high-frequency (*χ**n*2 ≃ 0.2) and co-added low frequency (C*χ**n*2 ≃ 0.1) channels, with a spectral index *α**e* = 1.08− 0.24+ 0.24 (beam co-addition), consistent with the backward estimate in §[subsec:speclwa]. This extended excess likely arises from a combination of faints sources, diffuse emission from the cluster cores, and bright point-source residuals that were not fully removed by CLEAN. The detailed modelling is provided in §[append:centrecleared].\nWe find that in both raw and cleared data, residuals due to the central signal cannot account for the virial excess, thus supporting the validity of the latter. The worst-case putative post-CLEAN sidelobe artefacts can be estimated from the PSF-convolved model inferred (with restoring beam convolution) from the raw data. The spurious ripples thus introduced in the stacked quantities are frequency-dependent and inconsistent with the data, failing to mimic the virial excess; see §[append:centrerestored] for details. In general, the first ripple is found at radii too small to account for the virial excess, whereas the second ripple is too extended and too weak for confusion with the virial excess. Similarly, the spectrum of the raw central excess appears to vary with *τ* and with *ν*, so its putative sidelobe residuals are very unlikely to reproduce the robust flat spectrum (see §[subsec:virialspec]) of the virial excess.\nVirial excess signal\n====================\nWith the central signal quantified in §[sec:centralemission], we now turn to the virial excess, focusing on the cleared data, where much of the point-source contamination has been removed. We test if the observed signal agrees with simple models for the synchrotron emission from the virial shock, which are briefly provided in §[subsec:virialmodel] based on a derivation in §[append:model]. To this end, we combine linear regression with TS tests, as described in §[subsec:TS]. The results are then presented in §[subsec:virialp2] for the special case of a flat injection spectrum, *p* = 2, giving rise to a cooled photon spectrum *α**v**i**r* ≃ 1 from the virial shock. The results are then generalised for an arbitrary spectrum in §[subsec:virialspec]. We examine the dependence of the virial excess signal upon the cluster mass in §[subsec:massdepen]. Finally, the magnetic field downstream of the shock is estimated in §[subsec:magnetic].\nVirial shock synchrotron model\n------------------------------\nOur virial shock modelling assumes that the shock deposits a fraction *ξ**e* of the downstream thermal energy in CREs, and a fraction *ξ**B* in magnetic fields. While CRE cooling is dominated by Compton losses off the CMB for the relevant, $B<1\\muG$ fields, a fraction ∼ *B*2/(8*π**u**c**m**b*) of the energy is radiated as synchrotron emission, where *u**c**m**b* is the CMB energy density. The CREs cool rapidly, so their radiative signature is expected to be well-localised near the virial shock, reflecting the temporal and spatial changes in energy injection rate across the shock. Therefore, while we model the emission from a virial shock, for simplicity, by invoking a stationary flow in spherical symmetry, the actual shock morphology can be highly irregular, and the energy injection rate through the shock can show strong variations with position and time.\nThe co-addition procedures we apply to the LWA data effectively average out these spatial and temporal variations, preferentially picking up the component of the virial shock signal that is most conducive to our stacking and circular binning. The stacking and analysis procedures can also pick up foreground and background contaminations, such as the changes in background due to the quenching of star-forming galaxies that crossed inside the virial shock. We consider two different spatial models for the virial-shock component of the stacked radio signal: emission from a thin spherical shell at a normalised radius *τ**s* (henceforth the shell model), and emission from a thin ring of the same radius *τ**s* on the plane of the sky (henceforth the planar model).\nFor concreteness, and following, we adopt an isothermal *β*-model to describe the gas distribution in a cluster, and assume hydrostatic equilibrium. The anticipated signal, derived for both shell and planar models in §[append:model], then follows the scaling\n \n$$\\nu I\\_\\nu \\propto \\dot{m} \\xi\\_e \\xi\\_B^\\frac{2+p}{4} \\tau\\_{{{s}}}^{-\\frac{p+4}{2}} \\times \\frac{M\\_{14}^\\frac{8 + p}{6}}{(1+z)^4} \\times\n \\nu^{-\\frac{p-2}{2}} \\tilde{f}(\\tau/\\tau\\_{{{s}}})\n \\coma\n\\label{eq:shock\\_prop}$$ \nwhere we defined *M*14 ≡ *M*500/1014*M*⊙ and the normalised angular dependence *f̃* of the relevant (shell or planar) model. Given the cluster mass and redshift, the free parameters of the model are the normalised shock radius *τ**s*, the CRE injection spectral index *p*, and the product $\\dot{m}\\xi\\_e\\xi\\_B^{(2+p)/4}$, which weighs the gas accretion rate, the CRE acceleration efficiency, and the magnetisation efficiency. The proportionality constant in Eq. depends on *p*, the cosmology, and the range {*γ*1, *γ*2} of CRE Lorentz factors [see Eq. ]. In the anticipated limit of a strong shock, *p* = 2, so *ν**I**ν* becomes frequency independent, the free parameters reduce to *τ**s* and $\\dot{m}\\xi\\_e\\xi\\_B$, and the proportionality constant depends on the CRE energy range only through the combination ln(*γ*2/*γ*1) [see Eq. ].\nFitting procedure\n-----------------\nTo determine the best-fitting parameters and their uncertainties, we apply the same procedures here as in §[subsec:Fittingprocedure]. Namely, for each model and set of parameters, we average over multiple Monte-Carlo simulations of control samples incorporating the injected model, compare the outcome to the data, and minimize the resulting *χ*2. Again, the analysis is performed separately for the high-frequency channel and for the co-added seven low-frequency channels.\nThe *χ*2 minimisation is equivalent to maximising the likelihood L, given by\n \n$$\\ln \\mathcal{L} = - \\frac{1}{2} \\chi^2.\n \\label{eq:likelyL}$$ \nFor the co-added low-frequency channels, we use the corrected value C*χ*2 instead of *χ*2, as defined and tested in §[subsec:Fittingprocedure], to crudely correct for inter-channel correlations. Again, this has no effect on the estimated parameters, and only corrects their estimated uncertainty intervals.\nTo quantify the significance of the virial excess, we use the test statistic \n \n$${\\rm TS} \\equiv -2 \\ln \\frac{\\mathcal{L}\\_{-}}{\\mathcal{L}\\_{+}} = \\chi^2\\_{-} - \\chi^2\\_{+} \\coma$$ \nwhere subscript ’ − ’ (’ + ’) refers to the model without (with) the virial shock. After maximising the likelihood of each model over its free parameters, the resulting TS is assumed to approximately have a *χ*2 distribution with N = N+ − N− degrees of freedom, corresponding to the extra parameters added by the virial shock. The implied significance of the excess is denoted $\\sigma\\_{\\rm TS}$.\nBeam and cluster co-additions give similar results for the virial excess; for brevity, the following text focuses mainly on beam co-addition. In particular, quoted values of TS, *χ**n*2, and, when relevant, also C*χ**n*2, refer to beam co-addition unless otherwise stated. Both beam-weighted and cluster-weighted results are provided in most figures and in the supplementary material.\nFigures [fig:band7sig] and [fig:band0-6sig] show that the virial excess is localised in the $2 \\lesssim \\tau \\lesssim 3$ bins in the high-frequency data, and in the $2 \\lesssim \\tau \\lesssim 3.5$ bins in the low-frequency data. Therefore, our nominal *χ*2 analysis focuses on the radial range $0 \\leq \\tau \\leq \\tau\\_{\\rm fit}$ with $\\tau\\_{\\rm fit} = 5$, including both central and virial signals. We verify that the results are not sensitive to reasonable variations in $\\tau\\_{\\rm fit}$ (ranging from 4 to at least 7.5).\nThe parameters inferred for the central and virial models are not entirely independent of each other. In §[sec:centralemission], we modelled the central source parameters while focusing on small radii and neglecting the virial contribution. In §[subsec:virialp2] below, we simultaneously model the virial excess and the main parameters of the central signal. Comparing the results indicates that the virial excess has a small but noticeable, $\\lesssim15\\%$ effect on the inferred parameters of the central model. The opposite effect — a putative bias in the virial model parameter estimates due to contamination by the central component — leads to a somewhat larger effect, with $\\lesssim25\\%$ changes in best-fitting parameters. Such effects are discussed in §[append:mutualeffects].\nIn addition to the statistical errors associated with the real and control samples, propagated into the model-parameter estimates, there are also systematic effects associated with the model assumptions, the properties of the cluster catalogue, etc. Based on the study of different model and analysis variants, we find that the systematic uncertainties are typically no larger than the statistical errors. An exception is the radio signal normalisation, $\\dot{m}\\xi\\_e\\xi\\_B^{(2+p)/4}$, where the systematic error may be as large as a factor of ∼ 4.\nWhile replacing our nominal *χ*2 estimate in Eq. by the more standard, non-stacked Eq. , does not significantly change the best-fitting values of most parameters, it does tend to lower the best-fitting normalisation $\\dot{m}\\xi\\_e\\xi\\_B^{(2+p)/4}$ of the virial signal by ∼ 20–55%. Hence, we present results based on both equations in the following sections. The difference in the best-fitting $\\dot{m}\\xi\\_e\\xi\\_B^{(2+p)/4}$ can be attributed at least in part to the dispersion of excess significance among individual clusters, and the partial correlation between this excess and cluster mass.\n [fig:model]\nResults for a strong virial shock (*p* = 2)\n-------------------------------------------\nccccccccccc Model & Channel & *F*73 & *α**e* & *τ**s* & $\\dot{m}\\xi\\_e\\xi\\_B^{\\frac{2+p}{4}}$ (10− 4) & *p* & *χ*2 & N & TS (*σ*) \n(1) & (2) & (3) & (4) & (5) & (6) & (7) & (8) & (9) & (10) \nplanar& high & 1.63− 0.20+ 0.23 & — & 2.45− 0.09+ 0.11 & 2.85− 0.65+ 0.66 & **2** & 17.2 & 15 & 18.5 (3.9*σ*) \n& low & 2.01− 0.20+ 0.25 & 1.20− 0.26+ 0.27 & 2.63− 0.10+ 0.09 & 1.95− 0.35+ 0.36 & **2** & 137.1 & 134 & 30.5 (5.2*σ*) \n& low & **2.01** & **1.20** & 2.62− 0.10+ 0.10 & 2.27− 0.50+ 0.71 & 2.02− 0.16+ 0.17 & 137.1 & 133 & 29.7 (4.8*σ*) \nshell & high† & 1.65− 0.23+ 0.23 & — & 2.78− 0.21+ 0.31 & 2.29− 0.95+ 1.17 & **2** & 13.2 & 7 & 7.6 (2.3*σ*) \n& low & 1.95− 0.23+ 0.25 & 1.14− 0.26+ 0.28 & 2.99− 0.13+ 0.13 & 2.31− 0.47+ 0.49 & **2** & 162.0 & 134 & 24.0 (4.5*σ*) \n& low & **1.95** & **1.14** & 3.00− 0.13+ 0.13 & 2.73− 0.59+ 0.63 & 2.04− 0.11+ 0.12 & 160.0 & 133 & 23.8 (4.2*σ*) \n[tab:fittingvirial]\n**Columns:** (1) Shock model; (2) The high-frequency (high) or the seven co-added low-frequency (low) channels; (3) Flux of the central emission normalised to the 73MHz frequency, in units of 10− 23 erg s− 1 cm− 2 Hz− 1; (4) Spectral index of the extended central emission; (5) Shock radius normalised to *R*500; (6) Virial shock normalisation, in 10− 4 units; (7) Injected spectral index of CREs; (8) *χ*2 values of the fit (without the C correction; see §[subsec:Fittingprocedure]); (9) Number of degrees of freedom; (10) TS value (equivalent significance in parenthesis). See text (§[subsec:virialp2]) for technical details. Values in **boldface** are fixed rather than fitted. \n† — Using $\\tau\\_{\\rm fit}=3$ to avoid a *τ* ∼ 3.5 fluctuation; see §[subsec:modelhighnu].\nWe begin with the simple case where the shock is strong. Here, CREs are injected with a flat, *p* = 2 spectrum, so the resulting *ν**I**ν* of synchrotron emission is frequency independent. We fit the cleared data to the joint model, comprised of both a central extended source and virial shock emission. For the virial shock component, we fit the parameters *τ**s* and $\\dot{m}\\xi\\_e\\xi\\_B$. For the central component of the model, we fit the parameters *F*73 and *α**e*, but fix the parameters *ζ* = 0.9 (1.2) and $\\tau\\_{\\rm cut}=0.9$ (1.6) for high (low) frequency data to their best-fit values, for simplicity and because these values are poorly constrained. Our results — including the virial shock parameters — are not sensitive to the exact choices of *ζ* and $\\tau\\_{\\rm cut}$ (see §[append:centrecleared] and Table [tab:fittingvirialdetail]). We find that biases introduced by the central signal to the virial modelling and vice versa are small (see §[append:mutualeffects]).\nFigure [fig:model] shows the best-fitting significance profiles (left column) and parameter estimates (middle and right columns) for the high-frequency (top and middle rows) and co-added low-frequency (bottom row) channels. Confidence contours in such figures are (unless otherwise stated) based on fitting *χ*2 as a fourth-order polynomial in the two free parameters, for better visibility. Models are shown both in the absence (dashed significance profiles) and in the presence (other dashings) of the virial shock, in both planar (dot-dashed) and shell (double dot-dashed) model variants. Table [tab:fittingvirial] demonstrates the best-fitting parameters and the implied TS significance levels of the virial shock, for different model variants (more details and models are provided in Table [tab:fittingvirialdetail]). In the low-frequency co-addition results shown in the table, parameter uncertainties are corrected using Eq. ([eq:uncertaintycorr]) and the TS values are based on C*χ*2 (where C ≃ 0.25; see §[subsec:Fittingprocedure]).\nWe discuss the results for the high-frequency channel in §[subsec:modelhighnu] and for the combined low-frequency channels in §[subsec:modellownu], obtained in both cases by simultaneously fitting the central and virial signals with *p* = 2.\n### High-frequency channel\nThe high-frequency models are shown in the top and middle rows of Fig. [fig:model]. The top row uses the nominal data, featuring in particular a ∼ 2*σ* excess around *τ* ≃ 3.5, shown above (see Fig. [fig:band7sig]) to arise from two clusters with extended NVSS point sources. To reduce the sensitivity of the model to these and other structures outside the virial radius, the middle row of Fig. [fig:model] shows high-frequency channel results when limiting the fitting radial range to $\\tau\\_{\\rm fit} = 3$.\nThe nominal planar model (orange dot-dashed curves) fits the virial shock signal well, with *χ**n*2(*τ* ≤ 5) ≃ 1.0, and is not sensitive to the inclusion (top row) or exclusion (middle row) of *τ* > 3 data. The best-fitting parameters are (for $\\tau\\_{\\rm fit}=5$, unless otherwise stated)\n \n*τ**s* = 2.45− 0.09+ 0.11 (2.44− 0.09+ 0.11) \nand\n \n$$\\dot{m}\\xi\\_e\\xi\\_B=2.85^{+0.66}\\_{-0.65} ~ (2.79^{+0.70}\\_{-0.64}) \\times 10^{-4}\n \\label{eq:best\\_xi}$$ \nfor beam (cluster; henceforth omitted) co-addition. The best-fitting shock position coincides with that inferred by based on the *Fermi* -ray signal. We find that TS ≃ 18.5, corresponding to a 3.9*σ* detection of the virial shock for N = 2 degrees of freedom. If we were to use the *τ**s* = 2.4 of as a prior, the significance would increase to TS ≃ 18.2 (4.3*σ* for N = 1).\nThe above results are based on the nominal *χ*2 of Eq. ([eq:chi2]). If we use, instead, the more standard, non-stacked least-squares of Eq. ([eq:chi20]), the best-fitting shock position *τ**s* = 2.40− 0.08+ 0.08 is practically unchanged, but $\\dot{m}\\xi\\_e\\xi\\_B = 1.48^{+0.27}\\_{-0.27} \\times 10^{-4}$ is nearly halved (blue dashed contours), and the significance of the virial shock increases to TS ≃ 33.2 (5.4*σ* for N = 2, or 5.8*σ* for N = 1 with the prior). Figure [fig:band7modelTS] presents the radial significance profiles based on Eq. ([eq:chi20]). Also shown is the TS-based significance profile (thick red solid curve) corresponding to the nominal planar model, as a function of *τ**s*.\n [fig:band7modelTS]\nLike the planar model, the shell model is not sensitive to the choice of $\\tau\\_{\\rm fit}$, but only when using the standard least-squares Eq. ([eq:chi20]). In this case, we find both *τ**s* = 2.70− 0.14+ 0.17 and $\\dot{m}\\xi\\_e\\xi\\_B = 1.85^{+0.31}\\_{-0.31} \\times 10^{-4}$ somewhat larger than their planar case counterparts in this method, with a high-significance, *χ**n*2(*τ* ≤ 5) ≃ 1.2 and TS ≃ 41.1 (6.1*σ* for N = 2; no priors used henceforth) detection of the virial shock. The nominal Eq. ([eq:chi2]) is more sensitive to confusion with the *τ* ∼ 3.5 artefact, so using $\\tau\\_{\\rm fit}\\gtrsim5$ leads to an exaggerated *τ**s* ≃ 4. For $\\tau\\_{\\rm fit}=3$, the best fit is *τ**s* = 2.83− 0.22+ 0.26 and $\\dot{m}\\xi\\_e\\xi\\_B = 2.55^{+0.88}\\_{-0.81}$, consistent with the above but with a weaker, *χ**n*2(*τ* ≤ 3) ≃ 1.9 and TS ≃ 7.6 (2.3*σ* for N = 2) detection due to the smaller radial range used.\nWe conclude that the virial shock signal is identified at a high confidence level in the high-frequency channel. Both planar and shell models give good fits to the data, and with similar normalisations $\\dot{m}\\xi\\_e\\xi\\_B$. While the two models reproduce the same peak locations in flux, in the shell model this peak radius is smaller than the shock radius, due to the combined effect of projection and PSF convolution. Note that in the high-frequency channel, neither model reproduces the deep local minimum at *τ* ≃ 1.5.\n### Low-frequency channels\nThe low-frequency model is demonstrated in the bottom row of Fig. [fig:model] and in Table [tab:fittingvirial] (and in more detail in Table [tab:fittingvirialdetail]). The fitted models show similar behaviour to that found in the high-frequency channel, with comparable parameters and a similarly high-confidence detection of the virial shock.\nThe nominal planar model gives\n \n*τ**s* = 2.63− 0.10+ 0.09 (2.62− 0.10+ 0.10) \nand\n \n$$\\dot{m}\\xi\\_e\\xi\\_B=1.95^{+0.36}\\_{-0.35}~(1.83^{+0.36}\\_{-0.36}) \\times 10^{-4}\n \\label{eq:best\\_xi\\_lowf}$$ \nfor beam (cluster, henceforth omitted) co-addition. The C*χ**n*2(*τ* ≤ 5) ≃ 0.3 fit yields TS ≃ 30.5 (5.1*σ* for N = 2). Again, replacing Eq. ([eq:chi2]) by the standard Eq. ([eq:chi20]) leaves *τ**s* unchanged, halves $\\dot{m}\\xi\\_e\\xi\\_B = 0.84^{+0.15}\\_{-0.13} \\times 10^{-4}$, and increases the significance to TS ≃ 37.5 (5.8*σ* for N = 2).\nThe shell model yield *τ**s* = 2.99− 0.13+ 0.13 and $\\dot{m}\\xi\\_e\\xi\\_B = 2.31^{+0.49}\\_{-0.47} \\times 10^{-4}$ with C*χ**n*2 ≃ 0.3 and TS ≃ 24.0 (4.5*σ* for N = 2) when using the nominal Eq. ([eq:chi2]). Like the high-frequency case, the shell model gives a somewhat larger *τ**s* compared to its planar counterpart. Using the standard Eq. ([eq:chi20]) instead of Eq. ([eq:chi2]) gives a slightly larger *τ**s* = 3.14− 0.09+ 0.06, a ∼ 30% smaller $\\dot{m}\\xi\\_e\\xi\\_B = 1.58^{+0.19}\\_{-0.20} \\times 10^{-4}$, and substantially raises the significance to TS ≃ 70.0 (8.1*σ* for N = 2).\nWe conclude that the low-frequency channels support the high-frequency behaviour, again indicating the presence of the virial shock at a very high confidence level, and with parameter estimates and confidence levels consistent with their high-frequency counterparts. Co-adding the two virial signals, namely in the high-frequency channel and in the co-added low-frequency channels, with equal weights, the significance estimated by the standard Eq. ([eq:chi20]) indicates a detection of the virial shock at confidence levels reaching TS ≃ 69.2 (8.0*σ* for N = 2) for the planar model and TS ≃ 106.3 (10.0*σ* for N = 2) for the shell model.\nResults for virial shock with arbitrary *p*\n-------------------------------------------\n [fig:specp]\nWe now relax the strong shock assumption, modelling the virial signal with *p* as a free parameter rather than fixed to its strong-shock, *p* = 2 limiting value. For this purpose, we use the cleared data in all low-frequency channels, as in §[subsec:modellownu]; the results are again summarised in Table [tab:fittingvirial] and in Table [tab:fittingvirialdetail]. Due to the significant uncertainties surrounding the virial shock emission, the spectrum estimated through forward modelling is more accurate than the direct estimate in §[subsec:speclwa].\nTo simplify the analysis, the central emission parameters are fixed to their best-fitting values inferred for the respective *p* = 2 case, leaving only the virial shock signal parameters *p*, *τ**s*, and the generalised normalisation $\\dot{m}\\xi\\_e\\xi\\_B^{(2+p)/4}$. The best-fitting shock radius again yields *τ**s* = 2.62− 0.10+ 0.10 (2.62− 0.11+ 0.11) for the planar (shell) model with nominal *χ*2, consistent with the results in §[subsec:modellownu], so we focus on the relation between the remaining two parameters, *p* and $\\dot{m}\\xi\\_e\\xi\\_B^{(2+p)/4}$.\nFigure [fig:specp] shows the confidence level contours of these parameters, for the planar (top panel) and shell (bottom) model. Here, contours are produced by interpolating and smoothing the *χ*2 data, instead of fitting it to a polynomial, as the latter would distort the contours in this particular case. We find the spectral index\n \n$$p = 2.02^{+0.17}\\_{-0.16}~(2.04^{+0.12}\\_{-0.11}) \\coma\n \\label{eq:spec\\_p}$$ \nfor the planar (shell) model, with TS ≃ 29.7 (4.8*σ* for N = 3) for the planar model and TS ≃ 23.8 (4.2*σ* for N = 3) for the shell model. The best-fit spectral index is consistent with the strong shock limit *p* = 2, justifying the assumption of a flat, *d**N*/*d**E* ∝ *E*− 2 injected CRE spectrum. Consequently, the associated normalisation $\\dot{m}\\xi\\_e\\xi\\_B^{(2+p)/4} =\n2.27^{+0.71}\\_{-0.50} \\times 10^{-4}~(2.73^{+0.63}\\_{-0.59} \\times 10^{-4})$ for the planar (shell) model agrees with the findings in §[subsec:modellownu]. By adopting the standard *χ*2 of Eq. instead of the nominal Eq. , the best-fitting result for *p* is again consistent with 2.0, and the other parameters and confidence levels are unchanged with respect to the *p* = 2 results.\nThe flat spectrum of the virial excess is robust, showing no sensitivity to the co-addition method or to cluster selection, nor any evidence for spectral curvature. For example, for the planar (shell) model, the four lowest frequency, 37–52 MHz channels yield *p* = 2.04− 0.15+ 0.15 (2.04− 0.10+ 0.10), while the three higher frequency, 57–68 MHz channels give a similar *p* = 2.04− 0.29+ 0.24 (2.02− 0.22+ 0.21). This robustness contrasts with the central-signal spectrum in our modelling, which strongly depends on assumptions, co-addition method, choice of channels, etc. While this sensitivity may reflect our inaccurate modelling of the cluster centre, it may well arise from a spatially-dependent spectral curvature of the combined point-like and extended components of the central excess. In such a case, the robustness of the virial excess spectrum provides further evidence that it does not arise from putative sidelobe artefacts.\nMass dependence\n---------------\nThe synchrotron brightness of the virial shock is expected to be a strong, *I**ν* ∝ *M*5/3 function of the cluster mass; see Eq. . We test whether massive clusters indeed show a stronger virial excess signal by splitting our sample into two sub-samples according to cluster mass. Choosing $M\\_{500} = 5 \\times 10^{13} M\\_{\\sun}$ as the approximate median gives a sub-sample of 23 more massive clusters with a median *M*500 ≃ 1.0 × 1014*M*⊙, and a sub-sample of 21 less massive clusters with a median *M*500 ≃ 3.3 × 1013*M*⊙.\nWe stack the two sub-samples in the same method as in §[sec:StackedQuantities], using nominal parameters. Figure [fig:masstest] shows the significance of the excess brightness of the two sub-samples, using both high (top panel) and low (bottom) frequency cleared data with beam co-addition.\n [fig:masstest]\nIn both high and low frequencies, we find a significant, ∼ 4*σ* excess in the stacked massive clusters, but only a ∼ 2*σ* excess in the stacked low mass clusters, in qualitative agreement with the expected mass dependence. The high-mass excess is of comparable significance to that of the full sample, despite having only half the clusters. Due to the smaller size of the sub-samples, some spurious signals (such as a ∼ 2.5*σ* excess around *τ* ≃ 3.5 in the high-frequency channel) become stronger than in the full sample.\nThe estimated virial excess brightness per cluster in the high-frequency channel (stacked low-frequency channels) is $\\Delta I \\simeq 6.7 \\times 10^{-12} \\erg\\se^{-1}\\cm^{-2}\\sr^{-1}$ ($3.6 \\times 10^{-12} \\erg\\se^{-1}\\cm^{-2}\\sr^{-1}$) in the massive-cluster sub-sample, which is higher by a factor of 1.3 with respect to the full sample, and by a factor of ∼ 2.0 (1.5) with respect to the low-mass sub-sample. These results are again consistent with a strong mass-dependence of the signal. For the low-mass sub-sample, the significance of the signal is too low for a forward modelling that could reliably test the *M*5/3 dependence.\nFitting the central and virial models to the massive sub-sample alone yields results consistent with those derived for the full sample. The inferred spectral indices and shock position do not change appreciably, while the central flux *F*73 increases by $\\lesssim75\\%$ ($\\lesssim45\\%$) for the restored (clear) data of the massive sub-sample. The best-fitting virial shock normalisation $\\dot{m}\\xi\\_e\\xi\\_B^{(2+p)/4}$ in the massive sub-sample is the same as in the full sample when using the standard *χ*2 of Eq. , but smaller by ∼ 30% ( ∼ 20%) for the planar (shell) model when adopting the nominal *χ*2 of Eq. , as expected from the dispersion among clusters and the partial correlation with mass. In conclusion, the results are consistent with massive clusters dominating the signal, but a larger sample is needed to quantify the mass dependence of the model parameters.\nwidth=1\n| | | | | | | | | | | | | |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |\n| Name | *β* | *n*0 | *r**c* | $k\\_{\\rm B}T$ | *z* | *R*500 | $\\dot{m}\\xi\\_e$ | *ξ**B* | *B* | $\\dot{m}$ | *ξ**B*ʹ | *B*ʹ |\n| | | (10− 3 cm− 3) | (kpc) | (keV) | | (Mpc) | (%) | (%) | (${\\rm \\mu G}$) | | (%) | (${\\rm \\mu G}$) |\n| (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) | (12) | (13) |\n| A 376 | 0.57 | 5.85 | 119.1 | 3.69 | 0.05 | 0.81 | 0.6 | 4.8 ± 1.1 | 0.24 ± 0.03 | 0.9 ± 0.1 | 1.0 ± 0.1 | 0.10 ± 0.00 |\n| A 576 | 0.57 | 7.61 | 105.6 | 3.67 | 0.04 | 0.83 | | | 0.24 ± 0.03 | 0.9 ± 0.1 | 1.0 ± 0.1 | 0.10 ± 0.00 |\n| A 779 | 0.34 | 10.35 | 11.5 | 1.76 | 0.02 | 0.45 | | | 0.24 ± 0.03 | 2.5 ± 0.2 | 0.6 ± 0.1 | 0.08 ± 0.00 |\n| A 1795 | 0.63 | 29.55 | 86.4 | 5.76 | 0.06 | 1.22 | | | 0.26 ± 0.03 | 0.7 ± "},"data_id":{"kind":"string","value":"arxiv_0184766"}}},{"rowIdx":46195,"cells":{"content":{"kind":"string","value":" As a simple example of cross-model fairness consider the classification scenario depicted in Figure [fig:lineramodelmultiplicity] shown in Section [sec:background] where three distinct predictors from the family of linear classifiers achieve 100% *accuracy* with respect to the displayed *validation set*. While models perfect in this respect do not make any *observable* mistakes, they may still suffer from *disputable regions* within which they disagree, giving rise to possible claims of cross-model unfair treatment voiced by previously unseen individuals residing in these spaces. Whenever we cannot guarantee perfect classification on the dedicated validation data, these considerations become more immediate as certain individuals from within this set may be treated differently across apparently equivalent models, as seen in Figures [fig:lineramodelmultiplicity1] and [fig:lineramodelmultiplicity2]. Notably, as we move towards more *expressive* families of predictive algorithms, their complexity and parameterisation space grow, possibly making the observed phenomenon more prominent, especially for workflows whose optimisation is *greedy* or *stochastic*.\nFrom the viewpoint of predictive performance there may be no objective reason to favour one model over another. This observation extends to confusion matrices (also known as contingency tables) since in certain scenarios they offer identical classification summaries despite distinct errors being made (as shown later in Table [tab:contingency]). In this setting a predictor could possibly be selected at random, however such an arbitrary choice poses ethical dilemmas as it may be dismissed by (adversely affected) individuals who are classified differently across this collection of models. Specifically, they can argue to be treated with the predictor granting them the most favourable outcome (especially if the selection procedure was ad hoc in the first place). Nonetheless, as the models under consideration are intrinsically different, many choices will inevitably result in some people benefiting at the expense of others, which in any case is difficult to justify. Even if a collection of models boasts perfect performance with respect to a designated validation set, we still have to account for any disputable regions that are not covered by these data, hence unobserved, yet allow individuals who are placed therein to claim unfair treatment in view of model multiplicity. This line of reasoning becomes especially important in the context of criminal justice, where a person might be entitled to be processed by the most advantageous model as long as it achieves certain performance for the (validation) population as a whole. This concept of fairness can be linked to the Blackstone’s ratio or the “presumption of innocence” , where lack of convincing evidence (or a unanimous vote) warrants the most favourable treatment – “It is better that ten guilty persons escape than that one innocent suffers.”\nSuch observations ought to encourage developers of predictive models to implement various selection heuristics and criteria that account for properties beyond predictive performance; e.g., overall complexity or coverage of a model may be considered to increase the likelihood of its uniqueness, and classification with a reject or abstain option can be adopted to mitigate inconsistent predictions across a collection of models . Another concern are cases where only relatively few classifiers that exhibit a given level of performance present an individual with a favourable outcome. For example, consider such a curated ensemble of predictive models taking the role of jurors where an overwhelming majority offers the less desirable decision. While the unfavourable ratio can be ignored with just a single model creating a precedent for a positive prediction, accounting for the proportion of classifiers and individual reasons for their respective outputs may provide important insights. However, with an abundance of predictive pipelines composed of diverse data processing techniques and model families, deriving bounds on the least and most favourable treatment of each individual within a dedicated *fairness* validation data set may be impractical or even infeasible. Understanding the stability of each prediction under certain modelling assumptions – even if limited to an empirical analysis – could nonetheless enhance, or even guarantee, fairness, trustworthiness, accountability and robustness of important data-driven decisions, or delegate them to a human supervisor if deemed necessary.\nTo the best of our knowledge such considerations found limited recognition in fairness literature, albeit they have been reported in machine learning . Our contributions aim to establish solid foundations as well as provide initial empirical investigation and theoretical analysis of *prediction-based individual fairness under model multiplicity*, which we call *cross-model fairness*, thus fill the aforementioned research gap . Specifically, this paper discusses relevant literature and positions our novel conceptualisation of *cross-model fairness* within its scope (Section [sec:relatedwork]). Next, it formally defines our notion of fairness based on *disputable regions* that arise under two distinct types of *utility-based model multiplicity* stemming from either a *strict* or *relaxed* predictive performance criterion (Section [sec:background]). These findings outline concepts fundamental to cross-model fairness. We introduce them by investigating diverse crisp binary classification scenarios where one outcome is universally preferred over the other, e.g., grant or decline parole, for a two-dimensional synthetic data set, nonetheless, as we show later, our contributions generalise beyond this restrictive setting. In the process we uncover caveats and assumptions relevant to both the modelling task and the approach used to evaluate predictive performance; these findings help us to identify the core principles of cross-model fairness under utility-based predictive multiplicity and devise strategies to address the (unintended) consequences of this phenomenon. After establishing these foundations, the paper proceeds to a comprehensive, large-scale, empirical study of cross-model fairness on three real-life data sets popular in the literature: Credit Approval, German Credit and Adult (Section [sec:experiments]). Our experiments are based on top-performing classification workflows published in the OpenML repository – a setup that captures a realistic diversity of models that arise naturally in the machine learning pipeline instead of being artificially created with the sole purpose of studying ramifications of model multiplicity – which ensures credibility of our results. We publish an open source implementation of our experiments on GitHub[1](#fn1) to facilitate their reproducibility and allow others to easily apply our evaluation protocol to the wide selection of data sets available in OpenML or elsewhere. We report cross-model fairness with two model multiplicity metrics – *discrepancy* and *ambiguity* – showing the prevalence and severity of this phenomenon. To aid in this analysis and help to uncover the degree of unfairness across the multiplicity spectrum – both as a high-level overview and an in-detail, instance-specific perspective – we design and present a novel visualisation toolkit. We also study the utility of the fair-by-design ensemble model achieved by allowing an individual to choose the (most favourable) classifier. Our results show the detrimental effects of such an approach on the overall predictive performance of the underlying classification system, particularly so when the employed AI or ML model is relatively expressive, therefore its decision boundary is flexible. Next, we offer an analytical treatment of cross-model fairness (Section [sec:multiplicity]), investigating the influence of the expressiveness (flexibility) of predictive models on their fairness, and consequences of granting each individual the best possible outcome using a fair-by-design ensemble model. Specifically, we observe that achieving this notion of fairness may require limiting the number of admissible predictors by imposing upon them (modelling) restrictions consistent with the natural process governing the generation of the underlying data. Such an approach is in line with strong notions of ante-hoc interpretability that are necessary for predictive models deployed in high stakes domains . Finally, we conclude our work and outline future research directions (Section [sec:conclusionfuture]). To the best of our knowledge, we are the first to empirically study the extent and severity of cross-model unfairness for realistic classification workflows in view of predictive multiplicity. Other pieces of research in this space offer theoretical or algorithmic contributions that operate in quite restrictive settings, which limits their real-world impact. In contrast, our work:\n1. provides a flexible definition of fairness in view of model multiplicity, called cross-model fairness;\n2. reports results of a large-scale, comprehensive, empirical study of this phenomenon for real-life data sets and predictive workflows;\n3. demonstrates how naïve enforcement of cross-model fairness in the form of a fair-by-design ensemble model may adversely impact predictive performance;\n4. analyses and discusses the ramifications of such a notion of fairness; and\n5. provides an open source implementation of our evaluation pipeline and a collection of graphical investigative tools to allow others to easily replicate our results and apply the same analysis to different data sets.\nRelated Work[sec:relatedwork]\n=============================\nFairness of artificial intelligence and machine learning algorithms has attracted considerable attention in recent years following proliferation of data-driven automated decision-making systems across real-life applications . Two main themes dominate this research field: individual and group-based fairness, both focusing on strategies to identify and mitigate disparate impact that manifests itself through discriminatory behaviour and diverse biases . Nonetheless, complementary viewpoints also emerge, investigating this topic under substantially different assumptions such as social and population changes over time, which bring to light considerations of the delayed impact of enforcing fair decisions . Regardless, a nearly universal assumption found in the literature is to presuppose a *fixed predictive model*, thus overlooking its provenance, evolution as well as any implicit choices related to it. This is at odds with the model *multiplicity* phenomenon – i.e., possible existence of a *collection* of equally capable classifiers – which has been largely neglected by the fairness community .\n*Individual fairness* deals with disparate treatment of a single person based on selected comparison criteria such as relevant *protected characteristics*. This can be captured by a dedicated *similarity metric* – assuming an intuitive notion that similar people should be treated comparably – however defining such a measurement strategy is non-trivial . An alternative approach to individual fairness is formulated via *counterfactuals*, whereby changing a (protected) attribute should not yield a different prediction. This framing is also very intuitive and relatively easy to test, however the concept of a “counterfactual individual” may be ill-defined. For example, altering ethnicity while preserving the remaining features intact may not be representative given that all these other personal traits and attributes are (implicitly) influenced by the single personal characteristic being manipulated. Notably, causality may be employed to partially alleviate such shortcomings . *Group-based fairness*, on the other hand, deals with disparate treatment of sub-populations determined, for example, by partitions drawn across protected attributes, in which case parity may be achieved by tweaking the underlying model separately for each group, e.g., via their respective classification thresholds . Importantly, many such notions of fairness are inherently incompatible , and enforcing some of them may require trading off a degree of predictive performance . Multiplicity of data-driven predictive models is a well-observed phenomenon – sometimes called the Rashōmon effect of statistics – where a group of classifiers exhibits comparable utility despite intrinsic differences . This collection of models may rely on different subsets of attributes or patterns found in the training data, with some arguing that exogenous information may be necessary to narrow down its scope . The ramifications of this phenomenon have only recently been studied for fairness, accountability and transparency of machine learning models. explored the implications of model multiplicity on veracity of counterfactual explanations, which property may be lost when switching between different models whose performance is comparable. A recent empirical study of multiplicity in ML explainability has shown the prevalent inconsistency of attribution methods across a range of different modelling scenarios . In relation to fairness, model multiplicity has been used to understand the dependence of (inaccessible) classifiers on selected (protected) attributes, which can help to robustly identify discriminatory practices . More recently, model multiplicity was suggested as an additional metric for evaluating accountability of classifiers. Specifically, showed how this phenomenon may be problematic (from ethical and fairness perspectives) when individuals receive conflicting predictions, and offered to address this issue by granting them the most favourable decision. While this line of reasoning is close to ours, the aforementioned study is limited to linear models based on Mixed-Integer Programming – restricting the modelling setting allows the authors to derive theoretical results and propose algorithmic solutions, nonetheless it curtails real-life applicability of their findings. Our work, in contrast, is rooted directly in fairness, thus provides a more fundamental and general treatment of model multiplicity in this context. We complement these contributions with a comprehensive analysis of real-life predictive workflows built for data sets commonly used in fairness studies, which constitutes the core of this paper. Our approach is unique when compared to related work in both the fairness and multiplicity domains in so far as it studies these aspects for predictive models that arise naturally (in the OpenML environment) in contrast to *artificial multiplicity* constructed specifically for research and experimentation purposes. Fundamentally, cross-model unfairness may be abated (to an extent) by employing a predictive paradigm that is more complex than crisp (binary) classification. The previous section briefly discussed strategies that can be used to increase the chances of a model being unique, e.g., by enforcing various heuristics and criteria with respect to its properties other than predictive performance. Adopting *classification with a reject or abstain option* is another approach to mitigate predictions that are unfair from a perspective of an individual in view of model multiplicity . Dedicated techniques such as *prediction intervals* or *conformal predictions* can also be deployed. The former marks any instance falling within a specified area constructed around a decision boundary as unreliable, allowing it to be assigned the most favourable decision; the latter follows a similar strategy, instead outputting a set of admissible classes, which can be further processed as desired. A possible alternative is to use *ensemble learning*, e.g., a random forest , where instead of a majority vote the prediction is derived through maximising the output of the contributing models. Machine learning algorithms with prediction margins, such as support vector machines , as well as probabilistic classifiers that offer confidence scores or trustworthiness measures are another option. While in most of such cases the acute signs of cross-model unfairness can be eliminated, the underlying problem may not necessarily be fully resolved, causing troubles down the line. This has been shown for probabilistic classification, where model multiplicity remains problematic .\nNavigating Model Multiplicity in View of Cross-model Fairness[sec:background]\n=============================================================================\nThe notion of fairness studied by this paper is built upon the *model multiplicity* phenomenon understood here as existence of a collection of data-driven models that are indistinguishable in terms of their predictive performance under a fixed evaluation strategy . Furthermore, in this work we are interested in *crisp binary classification* in which one outcome is universally preferred to the other by individuals whose case is being decided. Therefore, a model *f* : X ↦ Y classifies an instance *x* ∈ X as *f*(*x*) = *ŷ*, where *ŷ* ∈ Y ≡ {0, 1} and 1 represents a favourable outcome. Predictive performance of such a model is measured on a predetermined validation data subset $\\widetilde{X} = \\{ x\\_i \\}\\_{i=1}^n $ of size *n* ∈ N, where *x**i* ∈ X, accompanied by annotations $\\widetilde{Y} = \\{y\\_i\\}\\_{i=1}^n$, where *y**i* ∈ Y, using a selected metric *m* : Y × Y ↦ R calculated as $ m \\left( f(\\widetilde{X}), \\widetilde{Y} \\right) $. In our initial analysis we rely on linear, polynomial, *k*-nearest neighbours and decision tree classifiers applied to a synthetic two-dimensional data set, which helps us to demonstrate the principles of cross-model fairness that arises under utility-based predictive multiplicity through visual inspection and hand-crafted examples; all of our results, nonetheless, can be easily generalised beyond this restrictive setting. While in principle model multiplicity may span a diverse range of classifiers, we restrict our considerations to a *confined family of predictive functions* (inspired by such a formalisation of curves). This is desirable as distinct data-driven algorithms exhibit unique characteristics that translate into different shapes of their respective decision boundaries, allowing for diverse mistakes to be made, as seen when comparing Figures [fig:lineramodelmultiplicity2] and [fig:polynomialmodelmultiplicity2]. Therefore, a **family of data-driven models** F consists of classifiers *f* ∈ F based upon a single predictive algorithm and trained on a fixed data set, i.e., F ⊆ {*f* ∣ *f* : X ↦ Y}. It can further be constrained by imposing restrictions on the parameterisation space or optimisation approach, among other properties, used with the selected method, e.g., a collection of decision trees no deeper than 7. Notably, algorithms whose training procedure is stochastic or greedy may yield distinct models from a single specification when run multiple times. A family of models can also, by extension, include more complex predictive workflows built with pre- or post-processing steps such as input normalisation or output calibration. It is therefore undesirable to further constrain or formalise our notion of data-driven model family given the diversity of predictive pipelines that can be derived from fundamental algorithmic building blocks.\n[t].32 \n[t].32 \n[t].32 \nPolynomial model multiplicity with two mistakes.[fig:polynomialmodelmultiplicity2]\nTo address cross-model unfairness under predictive multiplicity we further identify a subset of classifiers from a single family that all share a fixed level of predictive performance (i.e., *utility*). To this end, we need to select an evaluation metric; for simplicity, we employ accuracy throughout this paper, which is defined as $m\\_{\\text{acc}}(\\hat{Y}, Y) = \\sum\\_{\\hat{y}, y \\in \\hat{Y}, Y} \\mathds{1}\\_{\\hat{y} = y} \\; / \\; |Y|$, where *Ŷ* = *f*(*X*) are predictions and *Y* are annotations for a data set *X*. Moreover, we require a designated collection of labelled instances $(\\widetilde{X}, \\widetilde{Y})$ to serve as a dedicated *predictive performance validation set*. This adds to the existing training and validation data used to fit the model and tune its hyper-parameters, both of which constitute an integral part of any AI or ML workflow. Notably, the performance validation set $\\widetilde{X}$ can also be employed to evaluate our notion of fairness, however in certain cases – reviewed in Section [sec:multiplicity] – separating the two may be beneficial. In this setting, *utility-based model multiplicity* F*ε* is determined by a fixed level of predictive performance *ε* ∈ R shared by classifiers from a single family of data-driven models F. In particular, there are two *meaningful* viewpoints on multiplicity:\npopulation-based\nwhere the performance of each model is computed for the entire data space X, which may be infeasible given a possibly infinite number of qualifying models (e.g., slight alterations of a linear classifier); and\nvalidation-based\nwhere the performance is measured on a dedicated validation set $\\widetilde{X}$, making the problem tractable.\nIn view of the former, the three models shown in Figure [fig:lineramodelmultiplicity] are distinct, whereas based on the latter they are indistinguishable, i.e., they belong to a single multiplicity class F*ε*. Notably, either of these two notions is distinct from a purely *theoretical non-uniqueness* of models within a single family, where the same decision boundary – thus unobservable changes – can be achieved with different realisations of a predictive pipeline; for example, see Figure [fig:decisiontreemultiplicity] depicting two structurally different decision trees classifying the entire data space in the same fashion. Throughout this paper we are predominantly concerned with the validation-based multiplicity, which is outlined in Definition [def:multiplicity]. Such a setting simplifies our considerations since, in practice, it limits the number of models that we need to consider.\nTheoretical multiplicity of decision trees. While the models are distinct their predictions are identical for the entire data space.[fig:decisiontreemultiplicity]\n[def:multiplicity] **Utility-based model multiplicity** F*ε* is captured by a set of classifiers from a single family of models F, all of which offer fixed predictive performance *ε* for a given metric *m* computed on a dedicated validation data set and its labels $(\\widetilde{X}, \\widetilde{Y})$: \n$$\\mathcal{F}\\_\\epsilon \\coloneqq \\left\\{ f \\in \\mathcal{F} : m\\left( f(\\widetilde{X}), \\widetilde{Y} \\right) = \\epsilon \\right\\} \\text{.}$$ \n[b]\n0.30\nrrr & red & blue \nred & 48 & 0 \nblue & 2 & 50 \n0.30\nrrr & red & blue \nred & 49 & 1 \nblue & 1 & 49 \n0.30\nrrr & red & blue \nred & 50 & 2 \nblue & 0 & 48 \nIt is important to observe that the choice of the performance metric determines which models are considered equivalent. For example, all of the classifiers shown in Figure [fig:lineramodelmultiplicity2] make two mistakes, hence are indistinguishable with respect to *accuracy*; however, a different metric, e.g., *precision*, *recall* or *specificity*, would make the violet, orange and green models distinct. Since all of the predictive performance metrics for classification tasks are derived from confusion tables, instead of relying on calculated numerical values we can refer directly to the underlying contingency matrices. Table [tab:contingency] lists specific errors made for the red and blue classes by the models shown in Figure [fig:lineramodelmultiplicity2], where the green predictors incorrectly classify two blue points, the violet model errs on two red examples and the orange classifier mistakes one blue and one red instance. However, even such a fine-grain approach is insufficient to capture the *same type of a mistake made on different data points*, i.e., individual predictions, motivating our notion of cross-model fairness under utility-based multiplicity. We are therefore interested in classifiers that are indistinguishable in terms of performance measured on a dedicated validation data set, but provide distinct class assignment for certain people. Our approach does not account for ground truth since the individuals affected by cross-model unfairness may not necessarily be included in the initial collection of instances used to assess this property, posing challenges – the need to collect new labels – when expanding this data stet.\nIn its simplest form, cross-model fairness can be studied by defining a designated (unlabelled) **fairness validation data set** *X̊* = {*x**i*}*i* = 1*n* of size *n* ∈ N, where *x**i* ∈ X, through which we can identify inconsistent predictions across a selected collection of models. A generalisation of such individual mistakes – inspired by the *population-based* model multiplicity viewpoint – are *disputable spaces* $\\mathring{\\mathcal{X}}\\_{\\mathcal{F}\\_\\epsilon}$, an example of which is shown in Figure [fig:lineramodelmultiplicity]. This extension – outlined by Definition [def:disputable] – is useful when the selected fairness validation data set is sparse and cannot capture inconsistent predictions at the desired level of detail, i.e., $\\mathring{X} \\cap \\mathring{\\mathcal{X}}\\_{\\mathcal{F}\\_\\epsilon} = \\emptyset $, hence no instance from *X̊* is placed within the disputable spaces $\\mathring{\\mathcal{X}}\\_{\\mathcal{F}\\_\\epsilon}$. Notably, the shape of these regions is influenced by the model family F; for example, consider the polynomial classifiers shown in Figure [fig:polynomialmodelmultiplicity2] for which such spaces are much more complex than in the case of linear classification (refer to Figure [fig:lineramodelmultiplicityall]). Logical models, on the other hand, constrain disputable regions to (hyper-)rectangles since they impose axis-parallel splits on the data space (see Figure [fig:decisiontreemultiplicity]).\n[def:disputable] A **disputable space** (or region) $\\mathring{\\mathcal{X}}\\_{\\mathcal{F}\\_\\epsilon} \\subseteq \\mathcal{X}$ for utility-based model multiplicity F*ε* is given by: \n$$\\mathring{\\mathcal{X}}\\_{\\mathcal{F}\\_\\epsilon} \\coloneqq \\left\\{ x \\in \\mathcal{X} : \\exists \\; f\\_i, f\\_j \\in \\mathcal{F}\\_\\epsilon \\;\\; \\text{s.t.} \\;\\; f\\_i(x) \\neq f\\_j(x) \\right\\} \\text{,}$$ \n for a chosen predictive metric *m* and labelled (performance) validation set $(\\widetilde{X}, \\widetilde{Y})$, where $\\forall \\; f \\in \\mathcal{F}\\_\\epsilon \\;\\; m\\left( f(\\widetilde{X}), \\widetilde{Y} \\right) = \\epsilon$. It is a generalisation of instance-based cross-model unfairness determined by a designated fairness validation data set *X̊*, which is necessarily limited in scope.\nIf the main property considered while choosing a classifier *f* ∈ F*ε* is predictive performance, then from the perspective of utility-based model multiplicity F*ε* all such predictors may be viewed as equivalent, without any particular preference for a given classifier. Lacking some further, well-defined selection criteria, an arbitrary choice can nonetheless lead to unfair cross-model treatment of individuals given the existence of an equally suitable predictor that may provide these people with a more favourable outcome. For example, consider the green classifiers shown in Figure [fig:lineramodelmultiplicity2]; both of them have identical confusion matrices (Table [tab:contingency:green]) yet only one of the two borderline blue individuals may be assigned the preferred outcome – the red class – depending on the model choice. Framing such a scenario as an automated decision between granting (red) or denying (blue) parole in an (algorithmically-supported) judicial hearing, performance-based indistinguishability of models becomes an important (ethical) factor. While ideally the task should be to minimise the scope of any disputable region, i.e., to deal with $\\mathring{\\mathcal{X}}\\_{\\mathcal{F}\\_\\epsilon}$, in this paper we focus on a designated fairness validation set *X̊*, a setting formalised by Definition [def:fairnessdefinition], which specifies cross-model fairness through predictive consistency of a model family for a fairness validation data set. Notably, such a notion of fairness can be expanded from individuals to *groups* by comparing the impact of multiplicity, e.g., ratios of affected instances, across them.\n[def:fairnessdefinition] A classifier *f* ∈ F*ε* is **cross-model fair** towards individuals *x* ∈ *X̊* in view of utility-based model multiplicity F*ε* iff ∀ *f*′ ∈ F*ε* ∀ *x* ∈ *X̊* *f*(*x*) = *f*′(*x*).\n![ Maximising the feature space predicted with the more favourable outcome (red) by joining (at intersections) the two green and one purple (top left and out of view) models f \\in \\mathcal{F}_\\epsilon shown in Figure [fig:lineramodelmultiplicity1] gives a cross-model individually fair classifier f^\\star under utility-based predictive multiplicity \\mathcal{F}_\\epsilon.[fig:lineramodelmultiplicity1merged]](figure/linear_model_multiplicity_1_joined) Maximising the feature space predicted with the more favourable outcome (red) by joining (at intersections) the two green and one purple (top left and out of view) models *f* ∈ F*ε* shown in Figure [fig:lineramodelmultiplicity1] gives a cross-model individually fair classifier *f*⋆ under utility-based predictive multiplicity F*ε*.[fig:lineramodelmultiplicity1merged]\nIn such a setting, eliminating unfairness caused by disparate decisions found across admissible models entails treating each individual with the most beneficial predictor *f* ∈ F*ε*. This observation leads us to define a new, fair-by-design model *f*⋆ built by aggregating all of the classifiers that are equivalent under utility-based model multiplicity F*ε*. We do so by incorporating disputable regions into the decision space predicted with the more favourable outcome, therefore maximally growing its coverage. This approach may be considered as a simple model ensemble where we are interested in the best, rather than the average, result. Figure [fig:lineramodelmultiplicity1merged], for example, shows such a composition of classifiers for the models depicted in Figure [fig:lineramodelmultiplicity1], assuming that the red class (1) is preferred to the blue class (0). The new decision boundary is constructed by joining linear models at their intersection – the two green segments in the visible frame – as to maximise the area predicted with the favourable outcome. Notably, it is possible that the fair-by-design model itself does not belong to the family, i.e., $f^\\star \\notin \\mathcal{F}$, which is captured by the aforementioned example – the decision boundary of *f*⋆ is not linear. The cross-model individually fair classifier *f*⋆ is formalised in Definition [def:fairmodel]. In theory, achieving fairness in such a way requires access to all of the classifiers that are equivalent under utility-based model multiplicity F*ε* since we are concerned with disputable spaces. While, effectively, this may be impractical or impossible, identifying cross-model unfairness for an individual is contingent on having at least one model whose prediction disagrees with another available classifier, allowing us to rectify this undesirable behaviour in practice. Alternatively, a representative sample of predictors – either created intentionally for this purpose or available as a by-product of model development, optimisation and refinement – could be used to this end.\n[def:fairmodel] A **cross-model individually fair classifier** *f*⋆ under utility-based model multiplicity F*ε* is defined as: \n$$f^\\star(x) \\coloneqq \\max\\_{f \\in \\mathcal{F}\\_\\epsilon} f(x)\n \\text{,}$$ \n for any instance *x* ∈ X. (Recall that 1 is the preferred prediction in our binary classification setting.)\nThus far we operated under strict utility regime *ε*, nonetheless this notion can be relaxed by allowing a certain deviation from the desired level of predictive performance. We can argue in favour of such an approach given that the utility is measured on a data subset, an expansion or contraction of which is likely to yield some variation in predictive performance. We can specify the allowed *tolerance* through an additional parameter *δ* ∈ R+ – where the performance band is denoted with *ε* ± *δ* – which extends the utility-based model multiplicity F*ε* provided in Definition [def:multiplicity] to F*ε* ± *δ* as outlined in Definition [def:multiplicitydelta]. An alternative operationalisation of this concept is to round the predictive performance *ε* to a specified decimal point *δ*, written as *ε* ≃ *δ*; for example, *ε* ≃ 10− 2 indicates *ε* rounded to the second decimal place. We use the latter approach throughout this paper to streamline our exploration of real-life data sets.\n[def:multiplicitydelta] **Relaxed utility-based model multiplicity** F*ε* ± *δ* is a collection of classifiers from across a single family of models F that exhibit a level of predictive performance within a fixed range [*ε* − *δ*, *ε* + *δ*] for a chosen metric *m*: \n$$\\mathcal{F}\\_{\\epsilon\\pm\\delta} \\coloneqq \\left\\{ f \\in \\mathcal{F} : \\epsilon-\\delta \\leq m\\left( f(\\widetilde{X}), \\widetilde{Y} \\right) \\leq \\epsilon+\\delta \\right\\} \\text{,}$$ \n where ($\\widetilde{X}, \\widetilde{Y})$ is a dedicated validation data set with labels, and *δ* is the tolerance of the predictive performance level *ε*.\nReal-life Impact of Predictive Multiplicity on Cross-model Fairness[sec:experiments]\n====================================================================================\nHaving explored the fundamentals of cross-model fairness under predictive multiplicity, we shift our attention to the real-life occurrence and impact of this phenomenon, which we investigate by studying pre-existing collections of ML pipelines. In particular, we focus on a common scenario when such considerations may arise, namely iterating over ML models during their development. To reconstruct this setting we turn to OpenML: a reproducibility repository for machine learning experiments . Specifically, we concentrate on three real-life data sets popular in fairness research – *Credit Approval*, *German Credit* and *Adult* – investigating the numerous executions of the top-performing binary classification workflow available therein for these data – histogram-based gradient boosting classification tree, random classification forest and decision tree-based AdaBoost respectively. In each case, the ML task is (supervised) crisp binary classification run on 10-fold cross validation, thus every workflow yields 10 distinct training and test data sets, and the same number of models. When reporting *stability* and *fairness profiles* – Figures [fig:experiments:stabilityprofile] and [fig:experiments:fairnessprofile] – we select the model trained on folds 2–10 whose performance is evaluated on fold 1, which also serves as the fairness validation set. We provide these visualisations for only one fold since our investigation of *ambiguity* and *discrepancy* – which are measures of multiplicity shown in Figure [fig:ambdics] – across *all* the folds shows a consistent behaviour. A summary of this setup given as identification numbers (IDs) of OpenML data sets, tasks and flows is provided in Table [tab:openml].\n[t]\nrrrrr Data Set & Data ID & Task ID & Flow ID & Execution Count \n German Credit & 31 & 31 & 6794 & 102,306 \nAdult & 1590 & 7592 & 6970 & 411 \nUsing OpenML in lieu of building, training and testing our own classifiers allows us to perform a first-of-a-kind analysis of multiplicity based on realistic predictive models that arise naturally instead of being artificially constructed with the sole purpose of studying this phenomenon. Such an approach makes our results more credible, especially that we choose the top-performing models available for each data set. To ensure a large sample of classifiers we opt for well-established data sets; while this means that we are unable to investigate more modern data sets that current fairness research relies on, such an analysis can be easily done in the future using the open source implementation of our evaluation framework. The code needed to reproduce our experiments and the results reported in this section is published on GitHub[2](#fn2). As we have observed earlier in Section [sec:background], standard performance metrics derived from confusion matrices are insufficient to study the extent of individuals being classified inconsistently for a chosen family of models F*ε* – see Table [tab:contingency:green] for an example. To fill this gap we design a toolkit capable of:\n* measuring the severity of this phenomenon,\n* identifying the affected instances, and\n* effectively communicating these findings at the desired level of detail.\nTo this end, we employ the numerical measures of multiplicity – *ambiguity* and *discrepancy* – introduced by, and build bespoke graphical analytic instruments around them. Additionally, we develop a comprehensive visualisation tool that summarises the prediction structure for each individual performance band *ε* of a model family F – a classifier perspective called *stability profile*. We complement this approach with a fine-grain inspection plot that captures precise classification results for all data points across chosen performance bands *ε* – a prediction view named *fairness profile*. We first introduce and explain these tools; next, we apply them to the chosen data sets and model families F from the OpenML repository; finally, we interpret our findings and discuss their implications. We complement our analysis by investigating the same selection of properties when the performance bands are relaxed (Definition [def:multiplicitydelta]), and inspecting the change in utility (accuracy in our case) for fair-by-design ensemble models *f*⋆ (Definition [def:fairmodel]) built for the aforementioned data sets.\n[t]0.31![Credit Approval – top 3 performance bands spanning 119 models.[fig:experiments:stabilityprofile:creditaproval]](figure/model_profile_29_12736_0_None \"fig:\")\n[t]0.31![German Credit – top 2 performance bands spanning 32 models.[fig:experiments:stabilityprofile:germancredit]](figure/model_profile_31_6794_0_None \"fig:\")\n[t]0.31![Adult – top 16 performance bands spanning 25 models.[fig:experiments:stabilityprofile:adult]](figure/model_profile_7592_6970_0_None \"fig:\")\n[t]0.48![Credit Approval – top 3 performance bands with 69 instances.[fig:experiments:fairnessprofile:creditaproval]](figure/prediction_profile_29_12736_0_True_None \"fig:\")\n[t]0.48![German Credit – top 2 performance bands with 100 instances.[fig:experiments:fairnessprofile:germancredit]](figure/prediction_profile_31_6794_0_True_None \"fig:\")\n \n[t]0.96![Adult. For clarity, the profile shows a sample of 250 instances out of 586 individuals treated unfairly by any of the 25 models spanning the top 16 performance bands depicted in Figure [fig:experiments:stabilityprofile:adult]. There are a total of 4,885 data points in this validation set.[fig:experiments:fairnessprofile:adult]](figure/prediction_profile_7592_6970_0_True_None \"fig:\")\n*Stability profiles* are our first visual diagnostic tool; they provide an overview of predictive volatility with respect to the entire fairness validation set from the perspective of a group of classifiers drawn from the chosen model family F – see Figure [fig:experiments:stabilityprofile]. They capture the consistency of individual predictions across models belonging to distinct performance bands *ε* by grouping them in colour-coded pyramids. Each such stack informs us of the number – quantity of horizontal segments – and frequency – width thereof – of unique prediction vectors (i.e., different class assignments) across the entire fairness validation data set. For example, if the evaluation yields the following prediction vectors for a given performance band *ε*: [0, 0, 0, 0], [1, 1, 1, 1], [0, 0, 0, 0] and [0, 1, 1, 0], the corresponding pyramid is constructed of three segments whose widths are 2, 1 and 1. The most desirable, i.e., the most cross-model fair, shape of such a pyramid is one amassing all the models in a single segment at the bottom as this configuration indicates that every classifier offers the same set of predictions. The least desirable shape is a stack of segments whose width is 1, i.e., each model returns a different prediction vector. *Fairness profiles* complement stability profiles by focusing on the behaviour of each model with respect to individual instances as shown in Figure [fig:experiments:fairnessprofile]. They visualise how distinct classifiers – one per row – grouped together by predictive performance bands – captured by colour-coding – assign a (binary) decision – differentiated by the saturation of each cell – to every instance – one per column – in the designated validation data set. For example, the collection of 36 rows coloured in orange in Figure [fig:experiments:fairnessprofile:creditaproval] visually depicts the binary predictions – light or dark shading – output by 36 models whose accuracy is 93% for 69 instances represented by individual columns. These plots shed light on the volatility of predictions across individuals, hence cross-model fairness of their classification. The stability and fairness profiles are linked through shared colouring of the predictive performance bands, for example, compare Figures [fig:experiments:stabilityprofile:creditaproval] and [fig:experiments:fairnessprofile:creditaproval]. A fairness profile can be plotted *faithfully*, thus accurately depicting the classification of every instance for all the models; alternatively, a *summary* variant that aggregates distinct predictions across models within a performance band (i.e., a coloured segment) by sorting them for each instance (i.e., a column) separately may instead be preferred due to its improved readability – we use this approach in Figure [fig:experiments:fairnessprofile]. Ideally, each column ought to be in a single – light or dark – shade, which indicates a consistent, thus fair, prediction output by models in a single level or throughout different levels of predictive performance.\n[t]0.96 \n \n[t]0.96 \n[t]0.96 \nThe remaining instrument in our toolkit – shown in Figure [fig:ambdics] – is a visual depiction of two multiplicity metrics: ambiguity and discrepancy . As explained earlier, each execution of an OpenML workflow is run on 10-fold cross validation, which yields 10 distinct pairs of training and test data sets together with the same number of the corresponding models. By calculating these metrics separately for each fold-specific test data (using its respective model), treating them as the fairness validation set, we get a more detailed view of cross-model fairness stability. *Ambiguity* quantifies the (percentage) proportion of instances treated unfairly among models with a given level of predictive performance from a single family. *Discrepancy* measures the (percentage) proportion of instances for which predictions change between any two models with a given level of predictive performance from a single family. We present the former as scatter plot-based trajectories, with flat shapes hovering near 0% being optimal for cross-model fairness under predictive multiplicity – they indicate consistent classification of individuals. We depict the latter as violin plots, showing the overall distribution as well as the minimum and maximum number of instances treated unfairly when switching between two arbitrary models with a given predictive performance level – short violins capture more stable, hence cross-model fair, treatment of individuals. Both figures are accompanied by a bar plot illustrating (on a logarithmic scale) the number of distinct models, i.e., workflow executions, across all utility bands to help better assess the trustworthiness and reliability of these results. Prior to analysing cross-model fairness under predictive multiplicity, we inspect the influence of using non-zero tolerance *δ* (Definition [def:multiplicitydelta]), i.e., relaxing the performance bands, on the quantity of unique utility levels (computed with accuracy). We investigate this phenomenon by rounding predictive performance *ε* to the 3rd (*ε* ≃ 10− 3) and 2nd (*ε* ≃ 10− 2) decimal place, with the results reported in Table [tab:datadetails]. While applying these tolerance values has no effect on Credit Approval and German Credit – given that their performance validation sets have only up to 100 instances – it affects Adult with its 4,885 data points. This observation suggests that for large validation sets we may need to relax the performance bands to get digestible and meaningful results – as we will see later when analysing our findings – however, such an approach makes it easier to identify alternative models and challenge cross-model fairness. In general, this procedure is likely to degrade our notion of fairness as it combines predictions from across strict, i.e., non-relaxed, performance bands, creating a more diverse sample that possibly disagrees on a larger subset of data points.\n[t]\nrrrrr & & \n German Credit & 100 & 26 & 26 & 26 \nAdult & 4,885 & 185 & 57 & 9 \nEquipped with the right insights and tools, we are in a position to assess cross-model fairness under utility-based predictive multiplicity for the selected data sets. Figures [fig:experiments:stabilityprofile] and [fig:experiments:fairnessprofile] show their *stability* and *fairness profiles*; Figure [fig:ambdics] depicts the *ambiguity* and *discrepancy* metrics of model multiplicity. As explained earlier, the former two display results only for the 1st fold of each workflow execution, whereas the latter spans all of the folds. Given that our analysis produces a large number of (accuracy-based) performance bands, the stability and fairness profiles show only a collection of top-performing models. First, we focus on the stability profile of each selected data set shown in Figure [fig:experiments:stabilityprofile]. While for the Credit Approval and German Credit data sets (Figures [fig:experiments:stabilityprofile:creditaproval] and [fig:experiments:stabilityprofile:germancredit]) the high-utility classifiers appear cross-model fair – the stack corresponding to the most accurate models (left-most, depicted in blue) has only one segment – this ceases to hold for classifiers with lower accuracy. In contrast, the predictive workflow used for the Adult data set (Figure [fig:experiments:stabilityprofile:adult]) generates a large collection of models whose accuracy differs beyond the second decimal point – multiple stacks composed of at most 3 segments of width 1 – highlighting potential cross-model fairness issues. One contributing factor may be significantly larger predictive performance and fairness validation sets: 4,885 instances as compared to 69 and 100 for the other two data sets; however, as we proceed with our investigation we see that this size discrepancy is not entirely to blame. Next, we move on to the summary variant of the fairness profiles generated for the selected data sets – shown in Figure [fig:experiments:fairnessprofile] – to paint a more accurate picture of their cross-model fairness. Recall that consistent shading within each column, i.e., for individual data points, both within a single and across the colour-coded performance bands, conveys a stable, thus cross-model fair, prediction. A vertical bar spanning a single performance band that changes its saturation in the middle signifies the maximum disagreement between the models of this utility. While the profiles for the Credit Approval and German Credit data sets (Figures [fig:experiments:fairnessprofile:creditaproval] and [fig:experiments:fairnessprofile:germancredit]) appear relatively consistent, the one for Adult (Figures [fig:experiments:fairnessprofile:adult]) is more jittery (note that for legibility reasons the plot only shows a subset of 250 instances out of 586 individuals who are treated unfairly, with the entire data set composed of 4,885 points). To further investigate cross-model fairness under predictive multiplicity we analyse ambiguity and discrepancy of our predictive workflows; the results are shown in Figure [fig:ambdics] and corroborate the insights gathered thus far. Note that for the Credit Approval and German Credit data sets (Figures [fig:ambdics:creditapproval] and [fig:ambdics:germancredit]) we work with strict model multiplicity F*ε* (Definition [def:multiplicity]), whereas Adult (Figure [fig:ambdics:adult2]) is investigated with relaxed performance bands F*ε* ≃ *δ* (Definition [def:multiplicitydelta]), which in this case is achieved by rounding utility to the second decimal point (*ε* ≃ 10− 2). We apply this procedure since the stability (Figure [fig:experiments:stabilityprofile:adult]) and fairness (Figure [fig:experiments:fairnessprofile:adult]) profiles for this particular data set suggest that ambiguity and discrepancy may otherwise be uninformative. Based on the former we can see an overwhelming number of unique prediction vectors – visible as stacked segments of width 1 – in each colour-coded performance band, signifying cross-model unfair treatment of many instances; the latter plot paints a similar picture. We validate these observations by calculating the number of performance bands as well as computing ambiguity without and with rounding to the third (*ε* ≃ 10− 3) and second (*ε* ≃ 10− 2) decimal points – refer to Table [tab:datadetails] and compare Figures [fig:ambdics:adult2] and [fig:adultamb].\n[t].42 \n[t]0.42 \n[t]0.32 \n[t]0.32 \n \n[t]0.32 \n[t]0.32 \n[t]0.32 \nFor all the data sets, ambiguity raises quite sharply as we move away from an optimal model, i.e., the top performance band, reaching between 40 and 90% at its peak, which draws attention to a surprisingly high degree of cross-model unfairness for these classifiers. Discrepancy shows that while in many cases it is possible to switch between two (performance-wise) equivalent models without affecting individual predictions, a non-negligible proportion of instances – captured by tall violin plots – is likely to be treated differently. More generally, the predictive workflows used with the Credit Approval and German Credit data sets seem to behave consistently without the need of relaxing performance bands, whereas the one used with Adult only provides discernible patterns when utility is rounded to the third (*ε* ≃ 10− 3) or second (*ε* ≃ 10− 2) decimal point. We suspect that in the latter case the culprits are volatility and high expressiveness of the underlying predictive model – decision tree-based AdaBoost – however this intuition cannot be confirmed without further analysis. Notably, relaxing performance bands merges distinct levels of utility, thereby introducing more disagreements on the level of individual predictions. This phenomenon can be observed in the ambiguity range being stretched from 0–25% without rounding the predictive performance to 0–35% and 0–50% when it is rounded to the third and second decimal places respectively (see Figures [fig:ambdics:adult2] and [fig:adultamb] for reference).\nFinally, we analyse the utility – measured with accuracy – of the fair-by-design ensemble models *f*⋆ (Definition [def:fairmodel]) constructed for every performance band of each selected data set. In addition to strict model multiplicity F*ε* (Definition [def:multiplicity]), we investigate relaxed performance bands F*ε* ≃ *δ* (Definition [def:multiplicitydelta]) exclusively for Adult, rounding accuracy to the third (*ε* ≃ 10− 3) and second (*ε* ≃ 10− 2) decimal points. The results – presented in Figure [fig:perfectmodel] – show a nearly universal drop in predictive performance when dealing with *f*⋆. An interesting exception is German Credit – Figure [fig:perfectmodel:germancredit] – for which the proportion of the favourable class is 70% (based on the ground truth labels). This class imbalance improves the accuracy of individually fair models for performance bands (x-axis) below 70%, whereas after this mark the utility drops as expected. The insights provided by our comprehensive inspection toolkit lead to a few important observations. When a classifier is sub-optimal in terms of its predictive performance – which we may not necessarily know at the time – its cross-model unfairness is especially concerning. In our experiments this can be seen in the stability profiles (Figure [fig:experiments:stabilityprofile]), where the height and width of the pyramids rapidly increase as we move towards lower utility bands; the same behaviour is reflected in the sharp raise in ambiguity as we move away from an optimal model (Figure [fig:ambdics]). Plotting the latter metric across *all* the utility bands allows us to pick up another interesting aspect of fairness in view of model multiplicity: incremental improvements of predictive performance may lack measurable influence on cross-model fairness (especially for suboptimal models). This phenomenon is captured by the initial rate of change in ambiguity and the wide span of the worst (highest) value of this metric across numerous performance bands. While specific instances affected by cross-model unfairness may change as we improve the predictive performance of a model, metrics measuring this property may remain relatively unchanged until we cross a certain utility threshold – a phenomenon that may be misleading for model developers. Another observation is the prevalence of cross-model unfairness for large fairness validation sets, which can be seen across our inspection toolkit for the Adult data set. While this is not necessarily a universal characteristic, such a behaviour highlights the importance of considering the difficulty of classifying a particular data set given the capabilities and expressiveness of the selected family of models. More specifically, we should inspect whether certain individuals are more prone than others to be subject to cross-model unfairness either within a subset of performance bands or across all of them – discrepancy plots (Figure [fig:ambdics]) communicate this phenomenon on a high level and fairness profiles (Figure [fig:experiments:fairnessprofile]) allow to zoom in and see a more fine-grain perspective. Identifying the affected data points and generalising them into representative cohorts can help to devise decision heuristic to combat cross-model unfairness; their development can additionally account for domain-specific properties such as the anticipated class imbalance and the selected evaluation metric to deliver better predictive performance than a naïve operationalisation of our notion of cross-model fairness, such as through the fair-by-design ensemble model. The following section offers a more theoretical discussion of cross-model fairness and outlines its broader implications.\nTowards Cross-model Fairness of Individual Predictions[sec:multiplicity]\n========================================================================\nOur experiments show the prevalence and severity of cross-model unfairness in view of utility-based predictive multiplicity for real-life machine learning workflows, which prompts us to explore this problem on a more fundamental level. Such an investigation is especially important given that the naïve operationalisation of the proposed notion of fairness brings about various challenges, in particular a possible drop in predictive performance that curtails the utility of the underlying data-driven modelling effort. This phenomenon is acutely visible for the fair-by-design ensemble model given by Definition [def:fairmodel] whose example is constructed in Figure [fig:lineramodelmultiplicity1merged], where *f*⋆ misclassifies two instances whereas the base models *f* ∈ F*ε* make just one mistake each. Among others, we therefore should pay special attention to the composition of the selected performance and fairness validation sets since they are respectively a key to model equivalence and identification of unfair behaviour. In particular, their representativeness, density and relative distribution should be carefully considered and potentially expanded or augmented over time. In addition to the modelled data space, we ought to scrutinise inherent characteristics of the classifiers themselves. Overall stability of the model family, which may depend on the stochasticity or greediness of the corresponding training procedure, as well as proximity of instances to class boundaries – for example, conveyed by prediction uncertainty and model (over- or under-) confidence – and behaviour of a classifier in sparse data regions all play a role in the volatility of automated decision-making. The inherent expressiveness and flexibility of the chosen model, and more broadly any data modelling workflow that can be built upon it by incorporating new steps such as data pre-processing or feature engineering, can also be problematic. In particular, it may lead to *pathological* instantiations of the fair-by-design ensemble model where *every* individual is offered the most favourable outcome, which we discuss in the next paragraph. All of these observations create opportunities (predominantly on technical grounds) for individuals adversely affected by an automated decision to easily challenge cross-model fairness of this process in view of utility-based predictive multiplicity. One conceivable attack on our notion of cross-model fairness can come from *under-specification* of the selected family of classifiers F or any predictive workflow built around it. Expressive classifiers may be flexible enough to single out individual instances and assign an arbitrary prediction to them, for example, due to their inherent complexity or parameterisation scope. Such freedom can adversely influence cross-model fairness by permitting any two instances to swap predictions – regardless of their placement in the data space – while maintaining the desired level of performance *ε*, i.e., operating within the designated utility-based model multiplicity class F*ε*. This can be easily achieved for classifiers with considerable parameterisation scope such as deep neural networks, but it is also relevant to simpler models, e.g., *k*-nearest neighbours, when they are misconfigured as shown in Figure [fig:neighboursmodelmultiplicity1].\n[t]0.485 \nOne red instance is misclassified.[fig:neighboursmodelmultiplicity1:2]\n[t]0.485 \nOne blue instance is misclassified.[fig:neighboursmodelmultiplicity1:1]\nFlexibility of a model *f* from a given family F can be defined through a proxy such as complexity Ω(*f*) or Vapnik–Chervonenkis dimension , and imposed as an additional constraint. The precise specification of such a metric may be unique to each model family; for example, the number of non-zero parameters for linear models, the highest coefficient degree for polynomial classifiers and the depth, width or number of instances per leaf for decision trees. More broadly, expressiveness, hence flexibility, of models and workflows built upon them may be controlled by ensuring diversity of training data, fixing a lower bound on confidence of each decision, abstaining from predictions, restricting model parameterisation, enforcing regularisation such as pruning for trees and LASSO for linear models, or limiting overfitting via alternative mechanisms. Otherwise, with a budget of errors given by the required level of predictive performance, an excessive number of people could abuse this notion and claim unfair treatment. The aforementioned strategies tasked with constraining the model space can be complemented by (use case-specific) operational requirements. For example, instead of a single performance metric employed to determine model equivalence, their hierarchy can be implemented, measuring accuracy first, and followed by precision and recall to address any ties. More broadly, such an approach could be applied directly to the confusion matrix of a classifier by sequentially imposing restrictions on its individual entries. Arguably, one could optimise exclusively for *recall* and *specificity* to maximise the number of favourable decisions; however, any such course of action is just a stopgap as it is likely to lack a solid foundation rooted in the domain-specific aspects of the modelled problem. A more meaningful heuristic should therefore rely on well-defined properties of the utilised model – for example, monotonicity of a particular feature with respect to the prediction – which strategy is consistent with some definitions of AI and ML explainability . A particularly promising solution that marries technical properties and relevant aspects of the underlying data domain is imposing constraints on the model form . Such an approach ensures a strong notion of (ante-hoc) interpretability and address the problems posed by cross-model fairness when faced with predictive multiplicity, which are especially pronounced for flexible model classes. This solution is particularly promising given that both of these desiderata are of paramount importance for high stakes domains, and constrained model forms can help us to fulfil both of them.\nLacking a (close to) *unique* classifier – obtained by imposing various desiderata to narrow down the scope of the chosen family of predictive models, thus making any alternative difficult or impossible to find – we may need to instead rely on the fair-by-design classifier *f*⋆ outlined by Definition [def:fairmodel]. Treating each individual with the best available model in this fashion may however degrade the overall predictive performance for the task at hand, making the fair application of data-driven decisions on a par with less effective classifiers to begin with. For example, this phenomenon can be observed for the fair-by-design model *f*⋆ shown in Figure [fig:lineramodelmultiplicity1merged]. Notably, any model family F*ε* subject to *population-* or *validation-based* multiplicity, i.e., with meaningful alternatives, is bound to suffer from disputable spaces. Depending on the density of data in these regions and their size, the fair-by-design model *f*⋆ will offer *no worse recall* but *no better specificity* than any individual model *f* ∈ F*ε* as stated by Proposition [col:fairmodel]. This change in predictive performance is especially prominent (reaching its limit) when the chosen family of models F is expressive enough to single out individual data points, in which case the cross-model fair-by-design classifier *f*⋆ will assign the most favourable outcome to every instance – 0% specificity and 100% recall – as given by Proposition [col:fairmodelexpressive].\n[col:fairmodel] **Specificity** *m**s* of a cross-model individually fair-by-design classifier *f*⋆ under utility-based predictive multiplicity F*ε* is no better than that of any other classifier *f* ∈ F*ε*: \n∀ *f* ∈ F*ε* *m**s*(*f*⋆(X), Y) ≤ *m**s*(*f*(X), Y), \n where (X, Y) is the labelled data space. Additionally, **recall** *m**r* of *f*⋆ is no worse than that of any *f* ∈ F*ε*: \n∀ *f* ∈ F*ε* *m**r*(*f*⋆(X), Y) ≥ *m**r*(*f*(X), Y). \n[col:fairmodelexpressive] When a family of classifiers F*ε* under utility-based predictive multiplicity is expressive enough to assign a selected class *c* ∈ Y to any data point *x* ∈ X, i.e., \n∀ *x* ∈ X ∃ *f* ∈ F*ε* s.t. *f*(*x*) = *c*, \n the corresponding cross-model individually fair-by-design classifier *f*⋆ achieves 0% **specificity** (*m**s*) and 100% **recall** (*m**r*): \n*m**s*(*f*⋆(X), Y) = 0% *m**r*(*f*⋆(X), Y) = 100%, \n in which case every point is assigned the most favourable outcome.\nThe results above, specifically Proposition [col:fairmodel], can be interpreted geometrically as expanding the space predicted with the more favourable outcome (red in our case) to the entirety of disputable regions – see Figure [fig:lineramodelmultiplicity1merged] for an example. It follows directly from observing individual predictions under *f*⋆.\n* Each data point from the preferred class (red) predicted as such (true positive) will not be affected, but misclassified positive instances (false negatives) may be corrected by *f*⋆ – a possible improvement in **recall**.\n* Each data point from the undesirable class (blue) predicted as such (true negative) may be (mis)classified as positive, but misclassified negative instances (false positives) will not be affected by *f*⋆ – a possible drop in **specificity**.\nFurthermore, the fair-by-design classifier *f*⋆ for models *f* ∈ F*ε* from a family that is flexible enough to predict any individual with an arbitrary class will assign the preferred output to all data, from which Proposition [col:fairmodelexpressive] follows. These observations alone should encourage the developers of AI and ML systems to meaningfully constrain the model form and strive for high utility computed on a comprehensive and representative validation set to minimise the scope of any disputable regions. Given the benefits of finding a classifier *f* ∈ F*ε* that is favourable for a specific individual, the concept of cross-model fairness under utility-based predictive multiplicity can be framed as an adversarial challenge. In such a setting people disparately affected by an automated decision can confront the owner of the underlying predictive model by building an equivalent classifier (performance-wise) that offers them the desired outcome instead. These “adversaries” attempt to identify a predictor *f*′ from within the employed family of models F*ε* – ensuring that it complies with all of the restrictions imposed by F*ε*, including the predetermined level of predictive performance *ε* – that assigns a selected individual the most favourable class. The party responsible for building and deploying the challenged model, on the other hand, ought to minimise the possible number of such claims by reducing the size of any disputable regions. This tug-of-war is bound to have a positive effect on the predictive model in question by iteratively improving its accountability, robustness and overall quality. Two possible realisations of this process are through *collective action* or by establishing dedicated *oversight bodies*. The former can accumulate and track cross-model unfairness though strategies such as crowdsourcing, the success stories of which are the Ad Observatory[3](#fn3) and Gender Shades projects; the latter can be given unrestricted access to all the required components and attempt to build alternative models in pursuit of uncovering adversely affected individuals. Reporting requirements are another possibility that found success in different areas of AI and ML, e.g., Model Cards . Facilitating this back-and-forth process may require releasing (a subset of) relevant training data as well as performance and fairness validation sets (in addition to the specification of the utilised model family F) – either exclusively to oversight bodies or to the broader public – which may be problematic due to inherent trade secrets. While distributing training data cannot be easily sidestepped, predictive performance of a model may be assessed without access to evaluation data as the classifier itself can instead be submitted for testing, akin to how ML competitions are run. Notably, having a comprehensive performance validation set that faithfully represents all of the individual cases is advantageous for the model creators as it restricts the number of admissible classifiers. Lastly, publishing a fairness validation set may also be beneficial to the owner of the model under investigation, encouraging a broader community to identify unfair predictions – and, more generally, disputable spaces – as well as engendering trust in the deployed model itself. This decoupling of the two – predictive performance validation data and fairness validation set – may therefore be desirable, especially that the latter *does not need to be labelled* since as it stands we are only interested in disparate treatment of these instances across equivalent classifiers *f* ∈ F*ε* to assess cross-model individual fairness. Just as predictive robustness is an important desideratum of a data-driven model for AI and ML engineers, cross-model fairness ought to be of concern to people being affected by automated decision-making. Guaranteeing this property engenders trust in model-makers who in such a scenario have to make the best effort to offer each individual an undisputable outcome (from a technical perspective). Additional reassurance can be had by giving people the benefit of the doubt whenever they fall into a disputable region; this can either be the most favourable prediction when operationalising our fair-by-design ensemble model or a bespoke decision heuristic chosen based on the context and application domain. Notably, such a scenario is a specific instance of favouring the user in the face of uncertainty, quantified here through predictive multiplicity. This approach is in line with augmenting classification with a reject or abstain option to account for scenarios where a model lacks confidence to predict an instance. The definition of cross-model fairness relies heavily on the selected family of data-driven classifiers and any modelling restrictions imposed upon it. This has wide-reaching implications: the fairness of the selected model is not only contingent on its own predictive behaviour but also on the behaviour of other classifiers within the same utility-based multiplicity regime, regardless of whether the model-maker is aware of their existence. While this is a relatively strong requirement that imposes most of the burden on the creators of data-driven models, it is intended to instil a high level of confidence in people subject to algorithmic decision-making, who are relatively powerless in comparison to the organisations collecting the data and building the models. It incentivises the model-makers to impose constraints on the model form as well as continuously monitor and revise their modelling pipelines – which are especially important for high stakes domains – and forgo relying purely on data-driven methods (which tend to exhibit high stochasticity) and ad hoc selection of the final model since such practices are likely to broaden the scope of the underlying model family. As each individual prediction may necessitate an explanation or justification depending on its operational context, or even require facilitating an opportunity for contestation and recourse, cross-model fairness frees people from having to pursue the aspect of this process linked to classification non-uniqueness. Additionally, the difficulty in satisfying this notion of fairness is likely to promote designing human-in-the-loop predictive systems or operating them in the decision support mode instead, thus encouraging responsible adoption of data-driven automation. The challenges of implementing cross-model fairness – such as having to precisely define the admissible family of models and searching through it to identify any disputable spaces – may nonetheless only warrant all this effort when the stakes are high enough.\nConclusion and Future Work[sec:conclusionfuture]\n================================================\nIn this paper we formalised the notion of cross-model fairness that arises due to utility-based model multiplicity – a scenario in which a number of classifiers assign different labels to selected data points despite being considered equivalent based on their predictive power. Such a situation is pertinent to distinct types of models, diverse parameterisation of the same model, or classifiers that result from multiple training runs of a given model when this procedure is, for example, greedy or stochastic. Additionally, high-dimensional and sparse data may contribute to this phenomenon given the curse of dimensionality (i.e., everything being far away from each other). In particular, we built these concepts around a user-specified *family of predictive models*, taking into consideration its *expressiveness*, which in the extreme may lead to awarding the most favourable prediction to every individual. We then defined *two meaningful notions of model multiplicity*: one determined by the entire data space and another measured with respect to a designated *fairness validation data set*. We generalised the former into *disputable spaces* – regions where at least two models disagree on a prediction – which may not necessarily be identified even with a comprehensive fairness validation set. Next, we showed how to combine (performance-wise) equivalent models to present each individual with the best possible decision. After establishing these foundations, we reported results of a comprehensive, large-scale, empirical study of cross-model fairness on three real-life data sets popular in the literature using top-performing classification workflows published in the OpenML repository. Such an experimental setup captures a realistic diversity of models that arises naturally in the machine learning workflow instead of being artificially concocted specifically to study model multiplicity, which ensures credibility of our findings. To help identify cross-model unfairness stemming from predictive multiplicity we adapted two relevant metrics – *discrepancy* and *ambiguity* – and introduced a bespoke visualisation toolkit. Our findings highlight the prevalence and severity of this phenomenon, illustrating the importance of considering utility-based model multiplicity as a new dimension of algorithmic fairness. We also demonstrated that employing the fair-by-design ensemble model, which grants each individual the most favourable outcome, may adversely affect the overall predictive power of the classification task at hand, especially for highly expressive models. As an alternative, we discussed imposing restrictions on the model space to limit the number of viable alternatives, arguing for enforcing (operational) constraints that are meaningful to each individual modelling problem, e.g., prediction monotonicity with respect to a chosen attribute, which is also recognised as a strategy for introducing (ante-hoc) interpretability whereby predictions are aligned with human values and can be easily justified. Otherwise, being able to determine disputable spaces – “grey areas” or “edge cases” from a classification standpoint – can allow to abstain from making a decision or engage a human expert in the process, thus partially alleviate the prevalence of cross-model unfair predictions. In future work, we will investigate numerical metrics and analytical tools to assess cross-model unfairness in a broader context, extending the notion beyond crisp binary classification. For example, we will study some of the techniques discussed briefly in Section [sec:relatedwork]. We will also look into strategies for integrating ground truth labels into calculation of cross-model fairness to capture the quality of each prediction. Moreover, we will investigate developing methods to derive bounds on the most and least favourable treatment of each individual from within a given data set for a selection of predictive models. This should facilitate systematic identification of disputable and stable regions, possibly leading to a new optimisation objective – minimise the former or maximise the latter – for training cross-model fair, robust and effective predictors. Finally, we will explore the lower limit of utility for a fair-by-design classifier that always uses the most beneficial model from their collection determined by a given level of predictive performance.\nThis research was supported by the Estonian Research Council (projects number PUT1458 and PRG1604); the Dora Plus programme, sponsored by the European Regional Development Fund and Estonian government; and the ARC Centre of Excellence for Automated Decision-Making and Society, funded by the Australian Government through the Australian Research Council (project number CE200100005).\n---\n1. <https://github.com/So-Cool/CrossModelFairness>[↩](#fnref1)\n2. <https://github.com/So-Cool/CrossModelFairness>[↩](#fnref2)\n3. <https://adobservatory.org>[↩](#fnref3)\n\nCross-model Fairness:Empirical Study of Fairness and Ethics Under Model Multiplicity\n====================================================================================\nWhile data-driven predictive models are a strictly technological construct, they may operate within a social context in which benign engineering choices entail implicit, indirect and unexpected real-life consequences. Fairness of such systems – pertaining both to individuals and groups – is one relevant consideration in this space; it arises when data capture *protected* characteristics upon which people may be discriminated. To date, this notion has predominantly been studied for a *fixed* model, often under different classification thresholds, striving to identify and eradicate undesirable, discriminative and possibly unlawful aspects of its operation. Here, we backtrack on this fixed model assumption to propose and explore a novel definition of *cross-model fairness* where individuals can be harmed when one predictor is chosen ad hoc from a group of equally-well performing models, i.e., in view of utility-based *model multiplicity*. Since a person may be classified differently across models that are otherwise considered equivalent, this individual could argue for a predictor granting them the most favourable outcome, employing which may have adverse effects on others. We introduce this scenario with a two-dimensional example and linear classification; then, we present a comprehensive empirical study based on real-life predictive models and data sets that are popular with the algorithmic fairness community; finally, we investigate analytical properties of cross-model fairness and its ramifications in a broader context. Our findings suggest that such unfairness can be readily found in the real life and it may be difficult to mitigate by technical means alone as doing so is likely to degrade predictive performance.\n**Highlights**\n* Cross-model fairness guarantees that a person receives the same prediction for a collection of classifiers with identical or comparable predictive performance, i.e., under utility-based model multiplicity.\n* When at least one prediction output by multiple equivalent models for a single individual is perceived as favourable, the person might argue for the precedence of this outcome.\n* Cross-model fairness is consistent with the Blackstone’s ratio and “presumption of innocence” – lack of convincing evidence ought to warrant the most favourable treatment.\n* Granting each person the most favourable decision afforded by a collection of equivalent models may degrade the overall predictive performance on the underlying task, especially when the employed family of models is highly expressive.\n* A possible solution is to limit the number of admissible predictors by imposing appropriate modelling restrictions that are consistent with the social context and the natural process governing the generation of the underlying data.\n**Source Code**<https://github.com/So-Cool/CrossModelFairness>\nCross-model Fairness: A New Notion of Algorithmic Fairness in Machine Learning[sec:intro]\n=========================================================================================\nData-driven predictive models are making impressive strides across numerous domains, leading to their proliferation throughout businesses and society, where they either support decision-making or outright automate relevant tasks. This speedy adoption of Artificial Intelligence (AI) and Machine Learning (ML) algorithms, however, outpaces research investigating the potential harm of these techniques across different aspects of everyday life. While excluding human operators from the decisive process bears the promise of faster, more precise as well as consistent and replicable outputs that lack implicit human biases, pre-existing (historical) patterns concealed in training data may easily overshadow these benefits. The ubiquity of oftentimes erratically and narrowly tested and validated models can therefore contribute to and amplify problems with fairness, accountability and robustness of the predictive tasks being addressed. This is of particular concern when AI and ML models affect people – with possibly long-term or legally binding decisions – which has been documented in various contexts including banking, school admission, job screening and judiciary ruling . It is thus critical, among other desiderata, to ensure fairness of the resulting predictions with respect to protected characteristics, e.g., ethnicity or gender, when dealing with both individuals and population subgroups.\nWhile mitigating disparate treatment of groups and individuals in view of protected attributes is of paramount importance, other notions of algorithmic fairness should not be neglected. Here, we explore its novel conceptualisation – which we call *cross-model fairness* – where instead of focusing on bias exhibited by a single model we deal with their collection characterised by equal (or comparable) predictive performance according to a given evaluation strategy and metric . Our notion of fairness is intended to engender trust in automated decision-making by guaranteeing that each individual is offered the most favourable outcome they can get given a fixed family of models pre-selected to solve the underlying predictive task. A specific scenario where cross-model fairness plays an important role is credit scoring; it is not uncommon for such models to be refined and tuned, yielding different realisations of the chosen modelling pipeline, which may affect the consistency of individual decisions. From a technical perspective, such a conceptualisation of fairness can be viewed as a notion of predictive *robustness* when the model in question evolves while being constrained by its level of predictive performance. As a simple example of cross-model fairness consider the classification scenario depicted in Figure [fig:lineramodelmultiplicity] shown in Section [sec:background] where three distinct predictors from the family of linear classifiers achieve 100% *accuracy* with respect to the displayed *validation set*. While models perfect in this respect do not make any *observable* mistakes, they may still suffer from *disputable regions* within which they disagree, giving rise to possible claims of cross-model unfair treatment voiced by previously unseen individuals residing in these spaces. Whenever we cannot guarantee perfect classification on the dedicated validation data, these considerations become more immediate as certain individuals from within this set may be treated differently across apparently equivalent models, as seen in Figures [fig:lineramodelmultiplicity1] and [fig:lineramodelmultiplicity2]. Notably, as we move towards more *expressive* families of predictive algorithms, their complexity and parameterisation space grow, possibly making the observed phenomenon more prominent, especially for workflows whose optimisation is *greedy* or *stochastic*.\nFrom the viewpoint of predictive performance there may be no objective reason to favour one model over another. This observation extends to confusion matrices (also known as contingency tables) since in certain scenarios they offer identical classification summaries despite distinct errors being made (as shown later in Table [tab:contingency]). In this setting a predictor could possibly be selected at random, however such an arbitrary choice poses ethical dilemmas as it may be dismissed by (adversely affected) individuals who are classified differently across this collection of models. Specifically, they can argue to be treated with the predictor granting them the most favourable outcome (especially if the selection procedure was ad hoc in the first place). Nonetheless, as the models under consideration are intrinsically different, many choices will inevitably result in some people benefiting at the expense of others, which in any case is difficult to justify. Even if a collection of models boasts perfect performance with respect to a designated validation set, we still have to account for any disputable regions that are not covered by these data, hence unobserved, yet allow individuals who are placed therein to claim unfair treatment in view of model multiplicity. This line of reasoning becomes especially important in the context of criminal justice, where a person might be entitled to be processed by the most advantageous model as long as it achieves certain performance for the (validation) population as a whole. This concept of fairness can be linked to the Blackstone’s ratio or the “presumption of innocence” , where lack of convincing evidence (or a unanimous vote) warrants the most favourable treatment – “It is better that ten guilty persons escape than that one innocent suffers.”\nSuch observations ought to encourage developers of predictive models to implement various selection heuristics and criteria that account for properties beyond predictive performance; e.g., overall complexity or coverage of a model may be considered to increase the likelihood of its uniqueness, and classification with a reject or abstain option can be adopted to mitigate inconsistent predictions across a collection of models . Another concern are cases where only relatively few classifiers that exhibit a given level of performance present an individual with a favourable outcome. For example, consider such a curated ensemble of predictive models taking the role of jurors where an overwhelming majority offers the less desirable decision. While the unfavourable ratio can be ignored with just a single model creating a precedent for a positive prediction, accounting for the proportion of classifiers and individual reasons for their respective outputs may provide important insights. However, with an abundance of predictive pipelines composed of diverse data processing techniques and model families, deriving bounds on the least and most favourable treatment of each individual within a dedicated *fairness* validation data set may be impractical or even infeasible. Understanding the stability of each prediction under certain modelling assumptions – even if limited to an empirical analysis – could nonetheless enhance, or even guarantee, fairness, trustworthiness, accountability and robustness of important data-driven decisions, or delegate them to a human supervisor if deemed necessary.\nTo the best of our knowledge such considerations found limited recognition in fairness literature, albeit they have been reported in machine learning . Our contributions aim to establish solid foundations as well as provide initial empirical investigation and theoretical analysis of *prediction-based individual fairness under model multiplicity*, which we call *cross-model fairness*, thus fill the aforementioned research gap . Specifically, this paper discusses relevant literature and positions our novel conceptualisation of *cross-model fairness* within its scope (Section [sec:relatedwork]). Next, it formally defines our notion of fairness based on *disputable regions* that arise under two distinct types of *utility-based model multiplicity* stemming from either a *strict* or *relaxed* predictive performance criterion (Section [sec:background]). These findings outline concepts fundamental to cross-model fairness. We introduce them by investigating diverse crisp binary classification scenarios where one outcome is universally preferred over the other, e.g., grant or decline parole, for a two-dimensional synthetic data set, nonetheless, as we show later, our contributions generalise beyond this restrictive setting. In the process we uncover caveats and assumptions relevant to both the modelling task and the approach used to evaluate predictive performance; these findings help us to identify the core principles of cross-model fairness under utility-based predictive multiplicity and devise strategies to address the (unintended) consequences of this phenomenon. After establishing these foundations, the paper proceeds to a comprehensive, large-scale, empirical study of cross-model fairness on three real-life data sets popular in the literature: Credit Approval, German Credit and Adult (Section [sec:experiments]). Our experiments are based on top-performing classification workflows published in the OpenML repository – a setup that captures a realistic diversity of models that arise naturally in the machine learning pipeline instead of being artificially created with the sole purpose of studying ramifications of model multiplicity – which ensures credibility of our results. We publish an open source implementation of our experiments on GitHub[1](#fn1) to facilitate their reproducibility and allow others to easily apply our evaluation protocol to the wide selection of data sets available in OpenML or elsewhere. We report cross-model fairness with two model multiplicity metrics – *discrepancy* and *ambiguity* – showing the prevalence and severity of this phenomenon. To aid in this analysis and help to uncover the degree of unfairness across the multiplicity spectrum – both as a high-level overview and an in-detail, instance-specific perspective – we design and present a novel visualisation toolkit. We also study the utility of the fair-by-design ensemble model achieved by allowing an individual to choose the (most favourable) classifier. Our results show the detrimental effects of such an approach on the overall predictive performance of the underlying classification system, particularly so when the employed AI or ML model is relatively expressive, therefore its decision boundary is flexible. Next, we offer an analytical treatment of cross-model fairness (Section [sec:multiplicity]), investigating the influence of the expressiveness (flexibility) of predictive models on their fairness, and consequences of granting each individual the best possible outcome using a fair-by-design ensemble model. Specifically, we observe that achieving this notion of fairness may require limiting the number of admissible predictors by imposing upon them (modelling) restrictions consistent with the natural process governing the generation of the underlying data. Such an approach is in line with strong notions of ante-hoc interpretability that are necessary for predictive models deployed in high stakes domains . Finally, we conclude our work and outline future research directions (Section [sec:conclusionfuture]). To the best of our knowledge, we are the first to empirically study the extent and severity of cross-model unfairness for realistic classification workflows in view of predictive multiplicity. Other pieces of research in this space offer theoretical or algorithmic contributions that operate in quite restrictive settings, which limits their real-world impact. In contrast, our work:\n1. provides a flexible definition of fairness in view of model multiplicity, called cross-model fairness;\n2. reports results of a large-scale, comprehensive, empirical study of this phenomenon for real-life data sets and predictive workflows;\n3. demonstrates how naïve enforcement of cross-model fairness in the form of a fair-by-design ensemble model may adversely impact predictive performance;\n4. analyses and discusses the ramifications of such a notion of fairness; and\n5. provides an open source implementation of our evaluation pipeline and a collection of graphical investigative tools to allow others to easily replicate our results and apply the same analysis to different data sets.\nRelated Work[sec:relatedwork]\n=============================\nFairness of artificial intelligence and machine learning algorithms has attracted considerable attention in recent years following proliferation of data-driven automated decision-making systems across real-life applications . Two main themes dominate this research field: individual and group-based fairness, both focusing on strategies to identify and mitigate disparate impact that manifests itself through discriminatory behaviour and diverse biases . Nonetheless, complementary viewpoints also emerge, investigating this topic under substantially different assumptions such as social and population changes over time, which bring to light considerations of the delayed impact of enforcing fair decisions . Regardless, a nearly universal assumption found in the literature is to presuppose a *fixed predictive model*, thus overlooking its provenance, evolution as well as any implicit choices related to it. This is at odds with the model *multiplicity* phenomenon – i.e., possible existence of a *collection* of equally capable classifiers – which has been largely neglected by the fairness community .\n*Individual fairness* deals with disparate treatment of a single person based on selected comparison criteria such as relevant *protected characteristics*. This can be captured by a dedicated *similarity metric* – assuming an intuitive notion that similar people should be treated comparably – however defining such a measurement strategy is non-trivial . An alternative approach to individual fairness is formulated via *counterfactuals*, whereby changing a (protected) attribute should not yield a different prediction. This framing is also very intuitive and relatively easy to test, however the concept of a “counterfactual individual” may be ill-defined. For example, altering ethnicity while preserving the remaining features intact may not be representative given that all these other personal traits and attributes are (implicitly) influenced by the single personal characteristic being manipulated. Notably, causality may be employed to partially alleviate such shortcomings . *Group-based fairness*, on the other hand, deals with disparate treatment of sub-populations determined, for example, by partitions drawn across protected attributes, in which case parity may be achieved by tweaking the underlying model separately for each group, e.g., via their respective classification thresholds . Importantly, many such notions of fairness are inherently incompatible , and enforcing some of them may require trading off a degree of predictive performance . Multiplicity of data-driven predictive models is a well-observed phenomenon – sometimes called the Rashōmon effect of statistics – where a group of classifiers exhibits comparable utility despite intrinsic differences . This collection of models may rely on different subsets of attributes or patterns found in the training data, with some arguing that exogenous information may be necessary to narrow down its scope . The ramifications of this phenomenon have only recently been studied for fairness, accountability and transparency of machine learning models. explored the implications of model multiplicity on veracity of counterfactual explanations, which property may be lost when switching between different models whose performance is comparable. A recent empirical study of multiplicity in ML explainability has shown the prevalent inconsistency of attribution methods across a range of different modelling scenarios . In relation to fairness, model multiplicity has been used to understand the dependence of (inaccessible) classifiers on selected (protected) attributes, which can help to robustly identify discriminatory practices . More recently, model multiplicity was suggested as an additional metric for evaluating accountability of classifiers. Specifically, showed how this phenomenon may be problematic (from ethical and fairness perspectives) when individuals receive conflicting predictions, and offered to address this issue by granting them the most favourable decision. While this line of reasoning is close to ours, the aforementioned study is limited to linear models based on Mixed-Integer Programming – restricting the modelling setting allows the authors to derive theoretical results and propose algorithmic solutions, nonetheless it curtails real-life applicability of their findings. Our work, in contrast, is rooted directly in fairness, thus provides a more fundamental and general treatment of model multiplicity in this context. We complement these contributions with a comprehensive analysis of real-life predictive workflows built for data sets commonly used in fairness studies, which constitutes the core of this paper. Our approach is unique when compared to related work in both the fairness and multiplicity domains in so far as it studies these aspects for predictive models that arise naturally (in the OpenML environment) in contrast to *artificial multiplicity* constructed specifically for research and experimentation purposes. Fundamentally, cross-model unfairness may be abated (to an extent) by employing a predictive paradigm that is more complex than crisp (binary) classification. The previous section briefly discussed strategies that can be used to increase the chances of a model being unique, e.g., by enforcing various heuristics and criteria with respect to its properties other than predictive performance. Adopting *classification with a reject or abstain option* is another approach to mitigate predictions that are unfair from a perspective of an individual in view of model multiplicity . Dedicated techniques such as *prediction intervals* or *conformal predictions* can also be deployed. The former marks any instance falling within a specified area constructed around a decision boundary as unreliable, allowing it to be assigned the most favourable decision; the latter follows a similar strategy, instead outputting a set of admissible classes, which can be further processed as desired. A possible alternative is to use *ensemble learning*, e.g., a random forest , where instead of a majority vote the prediction is derived through maximising the output of the contributing models. Machine learning algorithms with prediction margins, such as support vector machines , as well as probabilistic classifiers that offer confidence scores or trustworthiness measures are another option. While in most of such cases the acute signs of cross-model unfairness can be eliminated, the underlying problem may not necessarily be fully resolved, causing troubles down the line. This has been shown for probabilistic classification, where model multiplicity remains problematic .\nNavigating Model Multiplicity in View of Cross-model Fairness[sec:background]\n=============================================================================\nThe notion of fairness studied by this paper is built upon the *model multiplicity* phenomenon understood here as existence of a collection of data-driven models that are indistinguishable in terms of their predictive performance under a fixed evaluation strategy . Furthermore, in this work we are interested in *crisp binary classification* in which one outcome is universally preferred to the other by individuals whose case is being decided. Therefore, a model *f* : X ↦ Y classifies an instance *x* ∈ X as *f*(*x*) = *ŷ*, where *ŷ* ∈ Y ≡ {0, 1} and 1 represents a favourable outcome. Predictive performance of such a model is measured on a predetermined validation data subset $\\widetilde{X} = \\{ x\\_i \\}\\_{i=1}^n $ of size *n* ∈ N, where *x**i* ∈ X, accompanied by annotations $\\widetilde{Y} = \\{y\\_i\\}\\_{i=1}^n$, where *y**i* ∈ Y, using a selected metric *m* : Y × Y ↦ R calculated as $ m \\left( f(\\widetilde{X}), \\widetilde{Y} \\right) $. In our initial analysis we rely on linear, polynomial, *k*-nearest neighbours and decision tree classifiers applied to a synthetic two-dimensional data set, which helps us to demonstrate the principles of cross-model fairness that arises under utility-based predictive multiplicity through visual inspection and hand-crafted examples; all of our results, nonetheless, can be easily generalised beyond this restrictive setting. While in principle model multiplicity may span a diverse range of classifiers, we restrict our considerations to a *confined family of predictive functions* (inspired by such a formalisation of curves). This is desirable as distinct data-driven algorithms exhibit unique characteristics that translate into different shapes of their respective decision boundaries, allowing for diverse mistakes to be made, as seen when comparing Figures [fig:lineramodelmultiplicity2] and [fig:polynomialmodelmultiplicity2]. Therefore, a **family of data-driven models** F consists of classifiers *f* ∈ F based upon a single predictive algorithm and trained on a fixed data set, i.e., F ⊆ {*f* ∣ *f* : X ↦ Y}. It can further be constrained by imposing restrictions on the parameterisation space or optimisation approach, among other properties, used with the selected method, e.g., a collection of decision trees no deeper than 7. Notably, algorithms whose training procedure is stochastic or greedy may yield distinct models from a single specification when run multiple times. A family of models can also, by extension, include more complex predictive workflows built with pre- or post-processing steps such as input normalisation or output calibration. It is therefore undesirable to further constrain or formalise our notion of data-driven model family given the diversity of predictive pipelines that can be derived from fundamental algorithmic building blocks.\n[t].32 \n[t].32 \n[t].32 \nPolynomial model multiplicity with two mistakes.[fig:polynomialmodelmultiplicity2]\nTo address cross-model unfairness under predictive multiplicity we further identify a subset of classifiers from a single family that all share a fixed level of predictive performance (i.e., *utility*). To this end, we need to select an evaluation metric; for simplicity, we employ accuracy throughout this paper, which is defined as $m\\_{\\text{acc}}(\\hat{Y}, Y) = \\sum\\_{\\hat{y}, y \\in \\hat{Y}, Y} \\mathds{1}\\_{\\hat{y} = y} \\; / \\; |Y|$, where *Ŷ* = *f*(*X*) are predictions and *Y* are annotations for a data set *X*. Moreover, we require a designated collection of labelled instances $(\\widetilde{X}, \\widetilde{Y})$ to serve as a dedicated *predictive performance validation set*. This adds to the existing training and validation data used to fit the model and tune its hyper-parameters, both of which constitute an integral part of any AI or ML workflow. Notably, the performance validation set $\\widetilde{X}$ can also be employed to evaluate our notion of fairness, however in certain cases – reviewed in Section [sec:multiplicity] – separating the two may be beneficial. In this setting, *utility-based model multiplicity* F*ε* is determined by a fixed level of predictive performance *ε* ∈ R shared by classifiers from a single family of data-driven models F. In particular, there are two *meaningful* viewpoints on multiplicity:\npopulation-based\nwhere the performance of each model is computed for the entire data space X, which may be infeasible given a possibly infinite number of qualifying models (e.g., slight alterations of a linear classifier); and\nvalidation-based\nwhere the performance is measured on a dedicated validation set $\\widetilde{X}$, making the problem tractable.\nIn view of the former, the three models shown in Figure [fig:lineramodelmultiplicity] are distinct, whereas based on the latter they are indistinguishable, i.e., they belong to a single multiplicity class F*ε*. Notably, either of these two notions is distinct from a purely *theoretical non-uniqueness* of models within a single family, where the same decision boundary – thus unobservable changes – can be achieved with different realisations of a predictive pipeline; for example, see Figure [fig:decisiontreemultiplicity] depicting two structurally different decision trees classifying the entire data space in the same fashion. Throughout this paper we are predominantly concerned with the validation-based multiplicity, which is outlined in Definition [def:multiplicity]. Such a setting simplifies our considerations since, in practice, it limits the number of models that we need to consider.\nTheoretical multiplicity of decision trees. While the models are distinct their predictions are identical for the entire data space.[fig:decisiontreemultiplicity]\n[def:multiplicity] **Utility-based model multiplicity** F*ε* is captured by a set of classifiers from a single family of models F, all of which offer fixed predictive performance *ε* for a given metric *m* computed on a dedicated validation data set and its labels $(\\widetilde{X}, \\widetilde{Y})$: \n$$\\mathcal{F}\\_\\epsilon \\coloneqq \\left\\{ f \\in \\mathcal{F} : m\\left( f(\\widetilde{X}), \\widetilde{Y} \\right) = \\epsilon \\right\\} \\text{.}$$ \n[b]\n0.30\nrrr & red & blue \nred & 48 & 0 \nblue & 2 & 50 \n0.30\nrrr & red & blue \nred & 49 & 1 \nblue & 1 & 49 \n0.30\nrrr & red & blue \nred & 50 & 2 \nblue & 0 & 48 \nIt is important to observe that the choice of the performance metric determines which models are considered equivalent. For example, all of the classifiers shown in Figure [fig:lineramodelmultiplicity2] make two mistakes, hence are indistinguishable with respect to *accuracy*; however, a different metric, e.g., *precision*, *recall* or *specificity*, would make the violet, orange and green models distinct. Since all of the predictive performance metrics for classification tasks are derived from confusion tables, instead of relying on calculated numerical values we can refer directly to the underlying contingency matrices. Table [tab:contingency] lists specific errors made for the red and blue classes by the models shown in Figure [fig:lineramodelmultiplicity2], where the green predictors incorrectly classify two blue points, the violet model errs on two red examples and the orange classifier mistakes one blue and one red instance. However, even such a fine-grain approach is insufficient to capture the *same type of a mistake made on different data points*, i.e., individual predictions, motivating our notion of cross-model fairness under utility-based multiplicity. We are therefore interested in classifiers that are indistinguishable in terms of performance measured on a dedicated validation data set, but provide distinct class assignment for certain people. Our approach does not account for ground truth since the individuals affected by cross-model unfairness may not necessarily be included in the initial collection of instances used to assess this property, posing challenges – the need to collect new labels – when expanding this data stet.\nIn its simplest form, cross-model fairness can be studied by defining a designated (unlabelled) **fairness validation data set** *X̊* = {*x**i*}*i* = 1*n* of size *n* ∈ N, where *x**i* ∈ X, through which we can identify inconsistent predictions across a selected collection of models. A generalisation of such individual mistakes – inspired by the *population-based* model multiplicity viewpoint – are *disputable spaces* $\\mathring{\\mathcal{X}}\\_{\\mathcal{F}\\_\\epsilon}$, an example of which is shown in Figure [fig:lineramodelmultiplicity]. This extension – outlined by Definition [def:disputable] – is useful when the selected fairness validation data set is sparse and cannot capture inconsistent predictions at the desired level of detail, i.e., $\\mathring{X} \\cap \\mathring{\\mathcal{X}}\\_{\\mathcal{F}\\_\\epsilon} = \\emptyset $, hence no instance from *X̊* is placed within the disputable spaces $\\mathring{\\mathcal{X}}\\_{\\mathcal{F}\\_\\epsilon}$. Notably, the shape of these regions is influenced by the model family F; for example, consider the polynomial classifiers shown in Figure [fig:polynomialmodelmultiplicity2] for which such spaces are much more complex than in the case of linear classification (refer to Figure [fig:lineramodelmultiplicityall]). Logical models, on the other hand, constrain disputable regions to (hyper-)rectangles since they impose axis-parallel splits on the data space (see Figure [fig:decisiontreemultiplicity]).\n[def:disputable] A **disputable space** (or region) $\\mathring{\\mathcal{X}}\\_{\\mathcal{F}\\_\\epsilon} \\subseteq \\mathcal{X}$ for utility-based model multiplicity F*ε* is given by: \n$$\\mathring{\\mathcal{X}}\\_{\\mathcal{F}\\_\\epsilon} \\coloneqq \\left\\{ x \\in \\mathcal{X} : \\exists \\; f\\_i, f\\_j \\in \\mathcal{F}\\_\\epsilon \\;\\; \\text{s.t.} \\;\\; f\\_i(x) \\neq f\\_j(x) \\right\\} \\text{,}$$ \n for a chosen predictive metric *m* and labelled (performance) validation set $(\\widetilde{X}, \\widetilde{Y})$, where $\\forall \\; f \\in \\mathcal{F}\\_\\epsilon \\;\\; m\\left( f(\\widetilde{X}), \\widetilde{Y} \\right) = \\epsilon$. It is a generalisation of instance-based cross-model unfairness determined by a designated fairness validation data set *X̊*, which is necessarily limited in scope.\nIf the main property considered while choosing a classifier *f* ∈ F*ε* is predictive performance, then from the perspective of utility-based model multiplicity F*ε* all such predictors may be viewed as equivalent, without any particular preference for a given classifier. Lacking some further, well-defined selection criteria, an arbitrary choice can nonetheless lead to unfair cross-model treatment of individuals given the existence of an equally suitable predictor that may provide these people with a more favourable outcome. For example, consider the green classifiers shown in Figure [fig:lineramodelmultiplicity2]; both of them have identical confusion matrices (Table [tab:contingency:green]) yet only one of the two borderline blue individuals may be assigned the preferred outcome – the red class – depending on the model choice. Framing such a scenario as an automated decision between granting (red) or denying (blue) parole in an (algorithmically-supported) judicial hearing, performance-based indistinguishability of models becomes an important (ethical) factor. While ideally the task should be to minimise the scope of any disputable region, i.e., to deal with $\\mathring{\\mathcal{X}}\\_{\\mathcal{F}\\_\\epsilon}$, in this paper we focus on a designated fairness validation set *X̊*, a setting formalised by Definition [def:fairnessdefinition], which specifies cross-model fairness through predictive consistency of a model family for a fairness validation data set. Notably, such a notion of fairness can be expanded from individuals to *groups* by comparing the impact of multiplicity, e.g., ratios of affected instances, across them.\n[def:fairnessdefinition] A classifier *f* ∈ F*ε* is **cross-model fair** towards individuals *x* ∈ *X̊* in view of utility-based model multiplicity F*ε* iff ∀ *f*′ ∈ F*ε* ∀ *x* ∈ *X̊* *f*(*x*) = *f*′(*x*).\n![ Maximising the feature space predicted with the more favourable outcome (red) by joining (at intersections) the two green and one purple (top left and out of view) models f \\in \\mathcal{F}_\\epsilon shown in Figure [fig:lineramodelmultiplicity1] gives a cross-model individually fair classifier f^\\star under utility-based predictive multiplicity \\mathcal{F}_\\epsilon.[fig:lineramodelmultiplicity1merged]](figure/linear_model_multiplicity_1_joined) Maximising the feature space predicted with the more favourable outcome (red) by joining (at intersections) the two green and one purple (top left and out of view) models *f* ∈ F*ε* shown in Figure [fig:lineramodelmultiplicity1] gives a cross-model individually fair classifier *f*⋆ under utility-based predictive multiplicity F*ε*.[fig:lineramodelmultiplicity1merged]\nIn such a setting, eliminating unfairness caused by disparate decisions found across admissible models entails treating each individual with the most beneficial predictor *f* ∈ F*ε*. This observation leads us to define a new, fair-by-design model *f*⋆ built by aggregating all of the classifiers that are equivalent under utility-based model multiplicity F*ε*. We do so by incorporating disputable regions into the decision space predicted with the more favourable outcome, therefore maximally growing its coverage. This approach may be considered as a simple model ensemble where we are interested in the best, rather than the average, result. Figure [fig:lineramodelmultiplicity1merged], for example, shows such a composition of classifiers for the models depicted in Figure [fig:lineramodelmultiplicity1], assuming that the red class (1) is preferred to the blue class (0). The new decision boundary is constructed by joining linear models at their intersection – the two green segments in the visible frame – as to maximise the area predicted with the favourable outcome. Notably, it is possible that the fair-by-design model itself does not belong to the family, i.e., $f^\\star \\notin \\mathcal{F}$, which is captured by the aforementioned example – the decision boundary of *f*⋆ is not linear. The cross-model individually fair classifier *f*⋆ is formalised in Definition [def:fairmodel]. In theory, achieving fairness in such a way requires access to all of the classifiers that are equivalent under utility-based model multiplicity F*ε* since we are concerned with disputable spaces. While, effectively, this may be impractical or impossible, identifying cross-model unfairness for an individual is contingent on having at least one model whose prediction disagrees with another available classifier, allowing us to rectify this undesirable behaviour in practice. Alternatively, a representative sample of predictors – either created intentionally for this purpose or available as a by-product of model development, optimisation and refinement – could be used to this end.\n[def:fairmodel] A **cross-model individually fair classifier** *f*⋆ under utility-based model multiplicity F*ε* is defined as: \n$$f^\\star(x) \\coloneqq \\max\\_{f \\in \\mathcal{F}\\_\\epsilon} f(x)\n \\text{,}$$ \n for any instance *x* ∈ X. (Recall that 1 is the preferred prediction in our binary classification setting.)\nThus far we operated under strict utility regime *ε*, nonetheless this notion can be relaxed by allowing a certain deviation from the desired level of predictive performance. We can argue in favour of such an approach given that the utility is measured on a data subset, an expansion or contraction of which is likely to yield some variation in predictive performance. We can specify the allowed *tolerance* through an additional parameter *δ* ∈ R+ – where the performance band is denoted with *ε* ± *δ* – which extends the utility-based model multiplicity F*ε* provided in Definition [def:multiplicity] to F*ε* ± *δ* as outlined in Definition [def:multiplicitydelta]. An alternative operationalisation of this concept is to round the predictive performance *ε* to a specified decimal point *δ*, written as *ε* ≃ *δ*; for example, *ε* ≃ 10− 2 indicates *ε* rounded to the second decimal place. We use the latter approach throughout this paper to streamline our exploration of real-life data sets.\n[def:multiplicitydelta] **Relaxed utility-based model multiplicity** F*ε* ± *δ* is a collection of classifiers from across a single family of models F that exhibit a level of predictive performance within a fixed range [*ε* − *δ*, *ε* + *δ*] for a chosen metric *m*: \n$$\\mathcal{F}\\_{\\epsilon\\pm\\delta} \\coloneqq \\left\\{ f \\in \\mathcal{F} : \\epsilon-\\delta \\leq m\\left( f(\\widetilde{X}), \\widetilde{Y} \\right) \\leq \\epsilon+\\delta \\right\\} \\text{,}$$ \n where ($\\widetilde{X}, \\widetilde{Y})$ is a dedicated validation data set with labels, and *δ* is the tolerance of the predictive performance level *ε*.\nReal-life Impact of Predictive Multiplicity on Cross-model Fairness[sec:experiments]\n====================================================================================\nHaving explored the fundamentals of cross-model fairness under predictive multiplicity, we shift our attention to the real-life occurrence and impact of this phenomenon, which we investigate by studying pre-existing collections of ML pipelines. In particular, we focus on a common scenario when such considerations may arise, namely iterating over ML models during their development. To reconstruct this setting we turn to OpenML: a reproducibility repository for machine learning experiments . Specifically, we concentrate on three real-life data sets popular in fairness research – *Credit Approval*, *German Credit* and *Adult* – investigating the numerous executions of the top-performing binary classification workflow available therein for these data – histogram-based gradient boosting classification tree, random classification forest and decision tree-based AdaBoost respectively. In each case, the ML task is (supervised) crisp binary classification run on 10-fold cross validation, thus every workflow yields 10 distinct training and test data sets, and the same number of models. When reporting *stability* and *fairness profiles* – Figures [fig:experiments:stabilityprofile] and [fig:experiments:fairnessprofile] – we select the model trained on folds 2–10 whose performance is evaluated on fold 1, which also serves as the fairness validation set. We provide these visualisations for only one fold since our investigation of *ambiguity* and *discrepancy* – which are measures of multiplicity shown in Figure [fig:ambdics] – across *all* the folds shows a consistent behaviour. A summary of this setup given as identification numbers (IDs) of OpenML data sets, tasks and flows is provided in Table [tab:openml].\n[t]\nrrrrr Data Set & Data ID & Task ID & Flow ID & Execution Count \n German Credit & 31 & 31 & 6794 & 102,306 \nAdult & 1590 & 7592 & 6970 & 411 \nUsing OpenML in lieu of building, training and testing our own classifiers allows us to perform a first-of-a-kind analysis of multiplicity based on realistic predictive models that arise naturally instead of being artificially constructed with the sole purpose of studying this phenomenon. Such an approach makes our results more credible, especially that we choose the top-performing models available for each data set. To ensure a large sample of classifiers we opt for well-established data sets; while this means that we are unable to investigate more modern data sets that current fairness research relies on, such an analysis can be easily done in the future using the open source implementation of our evaluation framework. The code needed to reproduce our experiments and the results reported in this section is published on GitHub[2](#fn2). As we have observed earlier in Section [sec:background], standard performance metrics derived from confusion matrices are insufficient to study the extent of individuals being classified inconsistently for a chosen family of models F*ε* – see Table [tab:contingency:green] for an example. To fill this gap we design a toolkit capable of:\n* measuring the severity of this phenomenon,\n* identifying the affected instances, and\n* effectively communicating these findings at the desired level of detail.\nTo this end, we employ the numerical measures of multiplicity – *ambiguity* and *discrepancy* – introduced by, and build bespoke graphical analytic instruments around them. Additionally, we develop a comprehensive visualisation tool that summarises the prediction structure for each individual performance band *ε* of a model family F – a classifier perspective called *stability profile*. We complement this approach with a fine-grain inspection plot that captures precise classification results for all data points across chosen performance bands *ε* – a prediction view named *fairness profile*. We first introduce and explain these tools; next, we apply them to the chosen data sets and model families F from the OpenML repository; finally, we interpret our findings and discuss their implications. We complement our analysis by investigating the same selection of properties when the performance bands are relaxed (Definition [def:multiplicitydelta]), and inspecting the change in utility (accuracy in our case) for fair-by-design ensemble models *f*⋆ (Definition [def:fairmodel]) built for the aforementioned data sets.\n[t]0.31![Credit Approval – top 3 performance bands spanning 119 models.[fig:experiments:stabilityprofile:creditaproval]](figure/model_profile_29_12736_0_None \"fig:\")\n[t]0.31![German Credit – top 2 performance bands spanning 32 models.[fig:experiments:stabilityprofile:germancredit]](figure/model_profile_31_6794_0_None \"fig:\")\n[t]0.31![Adult – top 16 performance bands spanning 25 models.[fig:experiments:stabilityprofile:adult]](figure/model_profile_7592_6970_0_None \"fig:\")\n[t]0.48![Credit Approval – top 3 performance bands with 69 instances.[fig:experiments:fairnessprofile:creditaproval]](figure/prediction_profile_29_12736_0_True_None \"fig:\")\n[t]0.48![German Credit – top 2 performance bands with 100 instances.[fig:experiments:fairnessprofile:germancredit]](figure/prediction_profile_31_6794_0_True_None \"fig:\")\n \n[t]0.96![Adult. For clarity, the profile shows a sample of 250 instances out of 586 individuals treated unfairly by any of the 25 models spanning the top 16 performance bands depicted in Figure [fig:experiments:stabilityprofile:adult]. There are a total of 4,885 data points in this validation set.[fig:experiments:fairnessprofile:adult]](figure/prediction_profile_7592_6970_0_True_None \"fig:\")\n*Stability profiles* are our first visual diagnostic tool; they provide an overview of predictive volatility with respect to the entire fairness validation set from the perspective of a group of classifiers drawn from the chosen model family F – see Figure [fig:experiments:stabilityprofile]. They capture the consistency of individual predictions across models belonging to distinct performance bands *ε* by grouping them in colour-coded pyramids. Each such stack informs us of the number – quantity of horizontal segments – and frequency – width thereof – of unique prediction vectors (i.e., different class assignments) across the entire fairness validation data set. For example, if the evaluation yields the following prediction vectors for a given performance band *ε*: [0, 0, 0, 0], [1, 1, 1, 1], [0, 0, 0, 0] and [0, 1, 1, 0], the corresponding pyramid is constructed of three segments whose widths are 2, 1 and 1. The most desirable, i.e., the most cross-model fair, shape of such a pyramid is one amassing all the models in a single segment at the bottom as this configuration indicates that every classifier offers the same set of predictions. The least desirable shape is a stack of segments whose width is 1, i.e., each model returns a different prediction vector. *Fairness profiles* complement stability profiles by focusing on the behaviour of each model with respect to individual instances as shown in Figure [fig:experiments:fairnessprofile]. They visualise how distinct classifiers – one per row – grouped together by predictive performance bands – captured by colour-coding – assign a (binary) decision – differentiated by the saturation of each cell – to every instance – one per column – in the designated validation data set. For example, the collection of 36 rows coloured in orange in Figure [fig:experiments:fairnessprofile:creditaproval] visually depicts the binary predictions – light or dark shading – output by 36 models whose accuracy is 93% for 69 instances represented by individual columns. These plots shed light on the volatility of predictions across individuals, hence cross-model fairness of their classification. The stability and fairness profiles are linked through shared colouring of the predictive performance bands, for example, compare Figures [fig:experiments:stabilityprofile:creditaproval] and [fig:experiments:fairnessprofile:creditaproval]. A fairness profile can be plotted *faithfully*, thus accurately depicting the classification of every instance for all the models; alternatively, a *summary* variant that aggregates distinct predictions across models within a performance band (i.e., a coloured segment) by sorting them for each instance (i.e., a column) separately may instead be preferred due to its improved readability – we use this approach in Figure [fig:experiments:fairnessprofile]. Ideally, each column ought to be in a single – light or dark – shade, which indicates a consistent, thus fair, prediction output by models in a single level or throughout different levels of predictive performance.\n[t]0.96 \n \n[t]0.96 \n[t]0.96 \nThe remaining instrument in our toolkit – shown in Figure [fig:ambdics] – is a visual depiction of two multiplicity metrics: ambiguity and discrepancy . As explained earlier, each execution of an OpenML workflow is run on 10-fold cross validation, which yields 10 distinct pairs of training and test data sets together with the same number of the corresponding models. By calculating these metrics separately for each fold-specific test data (using its respective model), treating them as the fairness validation set, we get a more detailed view of cross-model fairness stability. *Ambiguity* quantifies the (percentage) proportion of instances treated unfairly among models with a given level of predictive performance from a single family. *Discrepancy* measures the (percentage) proportion of instances for which predictions change between any two models with a given level of predictive performance from a single family. We present the former as scatter plot-based trajectories, with flat shapes hovering near 0% being optimal for cross-model fairness under predictive multiplicity – they indicate consistent classification of individuals. We depict the latter as violin plots, showing the overall distribution as well as the minimum and maximum number of instances treated unfairly when switching between two arbitrary models with a given predictive performance level – short violins capture more stable, hence cross-model fair, treatment of individuals. Both figures are accompanied by a bar plot illustrating (on a logarithmic scale) the number of distinct models, i.e., workflow executions, across all utility bands to help better assess the trustworthiness and reliability of these results. Prior to analysing cross-model fairness under predictive multiplicity, we inspect the influence of using non-zero tolerance *δ* (Definition [def:multiplicitydelta]), i.e., relaxing the performance bands, on the quantity of unique utility levels (computed with accuracy). We investigate this phenomenon by rounding predictive performance *ε* to the 3rd (*ε* ≃ 10− 3) and 2nd (*ε* ≃ 10− 2) decimal place, with the results reported in Table [tab:datadetails]. While applying these tolerance values has no effect on Credit Approval and German Credit – given that their performance validation sets have only up to 100 instances – it affects Adult with its 4,885 data points. This observation suggests that for large validation sets we may need to relax the performance bands to get digestible and meaningful results – as we will see later when analysing our findings – however, such an approach makes it easier to identify alternative models and challenge cross-model fairness. In general, this procedure is likely to degrade our notion of fairness as it combines predictions from across strict, i.e., non-relaxed, performance bands, creating a more diverse sample that possibly disagrees on a larger subset of data points.\n[t]\nrrrrr & & \n German Credit & 100 & 26 & 26 & 26 \nAdult & 4,885 & 185 & 57 & 9 \nEquipped with the right insights and tools, we are in a position to assess cross-model fairness under utility-based predictive multiplicity for the selected data sets. Figures [fig:experiments:stabilityprofile] and [fig:experiments:fairnessprofile] show their *stability* and *fairness profiles*; Figure [fig:ambdics] depicts the *ambiguity* and *discrepancy* metrics of model multiplicity. As explained earlier, the former two display results only for the 1st fold of each workflow execution, whereas the latter spans all of the folds. Given that our analysis produces a large number of (accuracy-based) performance bands, the stability and fairness profiles show only a collection of top-performing models. First, we focus on the stability profile of each selected data set shown in Figure [fig:experiments:stabilityprofile]. While for the Credit Approval and German Credit data sets (Figures [fig:experiments:stabilityprofile:creditaproval] and [fig:experiments:stabilityprofile:germancredit]) the high-utility classifiers appear cross-model fair – the stack corresponding to the most accurate models (left-most, depicted in blue) has only one segment – this ceases to hold for classifiers with lower accuracy. In contrast, the predictive workflow used for the Adult data set (Figure [fig:experiments:stabilityprofile:adult]) generates a large collection of models whose accuracy differs beyond the second decimal point – multiple stacks composed of at most 3 segments of width 1 – highlighting potential cross-model fairness issues. One contributing factor may be significantly larger predictive performance and fairness validation sets: 4,885 instances as compared to 69 and 100 for the other two data sets; however, as we proceed with our investigation we see that this size discrepancy is not entirely to blame. Next, we move on to the summary variant of the fairness profiles generated for the selected data sets – shown in Figure [fig:experiments:fairnessprofile] – to paint a more accurate picture of their cross-model fairness. Recall that consistent shading within each column, i.e., for individual data points, both within a single and across the colour-coded performance bands, conveys a stable, thus cross-model fair, prediction. A vertical bar spanning a single performance band that changes its saturation in the middle signifies the maximum disagreement between the models of this utility. While the profiles for the Credit Approval and German Credit data sets (Figures [fig:experiments:fairnessprofile:creditaproval] and [fig:experiments:fairnessprofile:germancredit]) appear relatively consistent, the one for Adult (Figures [fig:experiments:fairnessprofile:adult]) is more jittery (note that for legibility reasons the plot only shows a subset of 250 instances out of 586 individuals who are treated unfairly, with the entire data set composed of 4,885 points). To further investigate cross-model fairness under predictive multiplicity we analyse ambiguity and discrepancy of our predictive workflows; the results are shown in Figure [fig:ambdics] and corroborate the insights gathered thus far. Note that for the Credit Approval and German Credit data sets (Figures [fig:ambdics:creditapproval] and [fig:ambdics:germancredit]) we work with strict model multiplicity F*ε* (Definition [def:multiplicity]), whereas Adult (Figure [fig:ambdics:adult2]) is investigated with relaxed performance bands F*ε* ≃ *δ* (Definition [def:multiplicitydelta]), which in this case is achieved by rounding utility to the second decimal point (*ε* ≃ 10− 2). We apply this procedure since the stability (Figure [fig:experiments:stabilityprofile:adult]) and fairness (Figure [fig:experiments:fairnessprofile:adult]) profiles for this particular data set suggest that ambiguity and discrepancy may otherwise be uninformative. Based on the former we can see an overwhelming number of unique prediction vectors – visible as stacked segments of width 1 – in each colour-coded performance band, signifying cross-model unfair treatment of many instances; the latter plot paints a similar picture. We validate these observations by calculating the number of performance bands as well as computing ambiguity without and with rounding to the third (*ε* ≃ 10− 3) and second (*ε* ≃ 10− 2) decimal points – refer to Table [tab:datadetails] and compare Figures [fig:ambdics:adult2] and [fig:adultamb].\n[t].42 \n[t]0.42 \n[t]0.32 \n[t]0.32 \n \n[t]0.32 \n[t]0.32 \n[t]0.32 \nFor all the data sets, ambiguity raises quite sharply as we move away from an optimal model, i.e., the top performance band, reaching between 40 and 90% at its peak, which draws attention to a surprisingly high degree of cross-model unfairness for these classifiers. Discrepancy shows that while in many cases it is possible to switch between two (performance-wise) equivalent models without affecting individual predictions, a non-negligible proportion of instances – captured by tall violin plots – is likely to be treated differently. More generally, the predictive workflows used with the Credit Approval and German Credit data sets seem to behave consistently without the need of relaxing performance bands, whereas the one used with Adult only provides discernible patterns when utility is rounded to the third (*ε* ≃ 10− 3) or second (*ε* ≃ 10− 2) decimal point. We suspect that in the latter case the culprits are volatility and high expressiveness of the underlying predictive model – decision tree-based AdaBoost – however this intuition cannot be confirmed without further analysis. Notably, relaxing performance bands merges distinct levels of utility, thereby introducing more disagreements on the level of individual predictions. This phenomenon can be observed in the ambiguity range being stretched from 0–25% without rounding the predictive performance to 0–35% and 0–50% when it is rounded to the third and second decimal places respectively (see Figures [fig:ambdics:adult2] and [fig:adultamb] for reference).\nFinally, we analyse the utility – measured with accuracy – of the fair-by-design ensemble models *f*⋆ (Definition [def:fairmodel]) constructed for every performance band of each selected data set. In addition to strict model multiplicity F*ε* (Definition [def:multiplicity]), we investigate relaxed performance bands F*ε* ≃ *δ* (Definition [def:multiplicitydelta]) exclusively for Adult, rounding accuracy to the third (*ε* ≃ 10− 3) and second (*ε* ≃ 10− 2) decimal points. The results – presented in Figure [fig:perfectmodel] – show a nearly universal drop in predictive performance when dealing with *f*⋆. An interesting exception is German Credit – Figure [fig:perfectmodel:germancredit] – for which the proportion of the favourable class is 70% (based on the ground truth labels). This class imbalance improves the accuracy of individually fair models for performance bands (x-axis) below 70%, whereas after this mark the utility drops as expected. The insights provided by our comprehensive inspection toolkit lead to a few important observations. When a classifier is sub-optimal in terms of its predictive performance – which we may not necessarily know at the time – its cross-model unfairness is especially concerning. In our experiments this can be seen in the stability profiles (Figure [fig:experiments:stabilityprofile]), where the height and width of the pyramids rapidly increase as we move towards lower utility bands; the same behaviour is reflected in the sharp raise in ambiguity as we move away from an optimal model (Figure [fig:ambdics]). Plotting the latter metric across *all* the utility bands allows us to pick up another interesting aspect of fairness in view of model multiplicity: incremental improvements of predictive performance may lack measurable influence on cross-model fairness (especially for suboptimal models). This phenomenon is captured by the initial rate of change in ambiguity and the wide span of the worst (highest) value of this metric across numerous performance bands. While specific instances affected by cross-model unfairness may change as we improve the predictive performance of a model, metrics measuring this property may remain relatively unchanged until we cross a certain utility threshold – a phenomenon that may be misleading for model developers. Another observation is the prevalence of cross-model unfairness for large fairness validation sets, which can be seen across our inspection toolkit for the Adult data set. While this is not necessarily a universal characteristic, such a behaviour highlights the importance of considering the difficulty of classifying a particular data set given the capabilities and expressiveness of the selected family of models. More specifically, we should inspect whether certain individuals are more prone than others to be subject to cross-model unfairness either within a subset of performance bands or across all of them – discrepancy plots (Figure [fig:ambdics]) communicate this phenomenon on a high level and fairness profiles (Figure [fig:experiments:fairnessprofile]) allow to zoom in and see a more fine-grain perspective. Identifying the affected data points and generalising them into representative cohorts can help to devise decision heuristic to combat cross-model unfairness; their development can additionally account for domain-specific properties such as the anticipated class imbalance and the selected evaluation metric to deliver better predictive performance than a naïve operationalisation of our notion of cross-model fairness, such as through the fair-by-design ensemble model. The following section offers a more theoretical discussion of cross-model fairness and outlines its broader implications.\nTowards Cross-model Fairness of Individual Predictions[sec:multiplicity]\n========================================================================\nOur experiments show the prevalence and severity of cross-model unfairness in view of utility-based predictive multiplicity for real-life machine learning workflows, which prompts us to explore this problem on a more fundamental level. Such an investigation is especially important given that the naïve operationalisation of the proposed notion of fairness brings about various challenges, in particular a possible drop in predictive performance that curtails the utility of the underlying data-driven modelling effort. This phenomenon is acutely visible for the fair-by-design ensemble model given by Definition [def:fairmodel] whose example is constructed in Figure [fig:lineramodelmultiplicity1merged], where *f*⋆ misclassifies two instances whereas the base models *f* ∈ F*ε* make just one mistake each. Among others, we therefore should pay special attention to the composition of the selected performance and fairness validation sets since they are respectively a key to model equivalence and identification of unfair behaviour. In particular, their representativeness, density and relative distribution should be carefully considered and potentially expanded or augmented over time. In addition to the modelled data space, we ought to scrutinise inherent characteristics of the classifiers themselves. Overall stability of the model family, which may depend on the stochasticity or greediness of the corresponding training procedure, as well as proximity of instances to class boundaries – for example, conveyed by prediction uncertainty and model (over- or under-) confidence – and behaviour of a classifier in sparse data regions all play a role in the volatility of automated decision-making. The inherent expressiveness and flexibility of the chosen model, and more broadly any data modelling workflow that can be built upon it by incorporating new steps such as data pre-processing or feature engineering, can also be problematic. In particular, it may lead to *pathological* instantiations of the fair-by-design ensemble model where *every* individual is offered the most favourable outcome, which we discuss in the next paragraph. All of these observations create opportunities (predominantly on technical grounds) for individuals adversely affected by an automated decision to easily challenge cross-model fairness of this process in view of utility-based predictive multiplicity. One conceivable attack on our notion of cross-model fairness can come from *under-specification* of the selected family of classifiers F or any predictive workflow built around it. Expressive classifiers may be flexible enough to single out individual instances and assign an arbitrary prediction to them, for example, due to their inherent complexity or parameterisation scope. Such freedom can adversely influence cross-model fairness by permitting any two instances to swap predictions – regardless of their placement in the data space – while maintaining the desired level of performance *ε*, i.e., operating within the designated utility-based model multiplicity class F*ε*. This can be easily achieved for classifiers with considerable parameterisation scope such as deep neural networks, but it is also relevant to simpler models, e.g., *k*-nearest neighbours, when they are misconfigured as shown in Figure [fig:neighboursmodelmultiplicity1].\n[t]0.485 \nOne red instance is misclassified.[fig:neighboursmodelmultiplicity1:2]\n[t]0.485 \nOne blue instance is misclassified.[fig:neighboursmodelmultiplicity1:1]\nFlexibility of a model *f* from a given family F can be defined through a proxy such as complexity Ω(*f*) or Vapnik–Chervonenkis dimension , and imposed as an additional constraint. The precise specification of such a metric may be unique to each model family; for example, the number of non-zero parameters for linear models, the highest coefficient degree for polynomial classifiers and the depth, width or number of instances per leaf for decision trees. More broadly, expressiveness, hence flexibility, of models and workflows built upon them may be controlled by ensuring diversity of training data, fixing a lower bound on confidence of each decision, abstaining from predictions, restricting model parameterisation, enforcing regularisation such as pruning for trees and LASSO for linear models, or limiting overfitting via alternative mechanisms. Otherwise, with a budget of errors given by the required level of predictive performance, an excessive number of people could abuse this notion and claim unfair treatment. The aforementioned strategies tasked with constraining the model space can be complemented by (use case-specific) operational requirements. For example, instead of a single performance metric employed to determine model equivalence, their hierarchy can be implemented, measuring accuracy first, and followed by precision and recall to address any ties. More broadly, such an approach could be applied directly to the confusion matrix of a classifier by sequentially imposing restrictions on its individual entries. Arguably, one could optimise exclusively for *recall* and *specificity* to maximise the number of favourable decisions; however, any such course of action is just a stopgap as it is likely to lack a solid foundation rooted in the domain-specific aspects of the modelled problem. A more meaningful heuristic should therefore rely on well-defined properties of the utilised model – for example, monotonicity of a particular feature with respect to the prediction – which strategy is consistent with some definitions of AI and ML explainability . A particularly promising solution that marries technical properties and relevant aspects of the underlying data domain is imposing constraints on the model form . Such an approach ensures a strong notion of (ante-hoc) interpretability and address the problems posed by cross-model fairness when faced with predictive multiplicity, which are especially pronounced for flexible model classes. This solution is particularly promising given that both of these desiderata are of paramount importance for high stakes domains, and constrained model forms can help us to fulfil both of them.\nLacking a (close to) *unique* classifier – obtained by imposing various desiderata to narrow down the scope of the chosen family of predictive models, thus making any alternative difficult or impossible to find – we may need to instead rely on the fair-by-design classifier *f*⋆ outlined by Definition [def:fairmodel]. Treating each individual with the best available model in this fashion may however degrade the overall predictive performance for the task at hand, making the fair application of data-driven decisions on a par with less effective classifiers to begin with. For example, this phenomenon can be observed for the fair-by-design model *f*⋆ shown in Figure [fig:lineramodelmultiplicity1merged]. Notably, any model family F*ε* subject to *population-* or *validation-based* multiplicity, i.e., with meaningful alternatives, is bound to suffer from disputable spaces. Depending on the density of data in these regions and their size, the fair-by-design model *f*⋆ will offer *no worse recall* but *no better specificity* than any individual model *f* ∈ F*ε* as stated by Proposition [col:fairmodel]. This change in predictive performance is especially prominent (reaching its limit) when the chosen family of models F is expressive enough to single out individual data points, in which case the cross-model fair-by-design classifier *f*⋆ will assign the most favourable outcome to every instance – 0% specificity and 100% recall – as given by Proposition [col:fairmodelexpressive].\n[col:fairmodel] **Specificity** *m**s* of a cross-model individually fair-by-design classifier *f*⋆ under utility-based predictive multiplicity F*ε* is no better than that of any other classifier *f* ∈ F*ε*: \n∀ *f* ∈ F*ε* *m**s*(*f*⋆(X), Y) ≤ *m**s*(*f*(X), Y), \n where (X, Y) is the labelled data space. Additionally, **recall** *m**r* of *f*⋆ is no worse than that of any *f* ∈ F*ε*: \n∀ *f* ∈ F*ε* *m**r*(*f*⋆(X), Y) ≥ *m**r*(*f*(X), Y). \n[col:fairmodelexpressive] When a family of classifiers F*ε* under utility-based predictive multiplicity is expressive enough to assign a selected class *c* ∈ Y to any data point *x* ∈ X, i.e., \n∀ *x* ∈ X ∃ *f* ∈ F*ε* s.t. *f*(*x*) = *c*, \n the corresponding cross-model individually fair-by-design classifier *f*⋆ achieves 0% **specificity** (*m**s*) and 100% **recall** (*m**r*): \n*m**s*(*f*⋆(X), Y) = 0% *m**r*(*f*⋆(X), Y) = 100%, \n in which case every point is assigned the most favourable outcome.\nThe results above, specifically Proposition [col:fairmodel], can be interpreted geometrically as expanding the space predicted with the more favourable outcome (red in our case) to the entirety of disputable regions – see Figure [fig:lineramodelmultiplicity1merged] for an example. It follows directly from observing individual predictions under *f*⋆.\n* Each data point from the preferred class (red) predicted as such (true positive) will not be affected, but misclassified positive instances (false negatives) may be corrected by *f*⋆ – a possible improvement in **recall**.\n* Each data point from the undesirable class (blue) predicted as such (true negative) may be (mis)classified as positive, but misclassified negative instances (false positives) will not be affected by *f*⋆ – a possible drop in **specificity**.\nFurthermore, the fair-by-design classifier *f*⋆ for models *f* ∈ F*ε* from a family that is flexible enough to predict any individual with an arbitrary class will assign the preferred output to all data, from which Proposition [col:fairmodelexpressive] follows. These observations alone should encourage the developers of AI and ML systems to meaningfully constrain the model form and strive for high utility computed on a comprehensive and representative validation set to minimise the scope of any disputable regions. Given the benefits of finding a classifier *f* ∈ F*ε* that is favourable for a specific individual, the concept of cross-model fairness under utility-based predictive multiplicity can be framed as an adversarial challenge. In such a setting people disparately affected by an automated decision can confront the owner of the underlying predictive model by building an equivalent classifier (performance-wise) that offers them the desired outcome instead. These “adversaries” attempt to identify a predictor *f*′ from within the employed family of models F*ε* – ensuring that it complies with all of the restrictions imposed by F*ε*, including the predetermined level of predictive performance *ε* – that assigns a selected individual the most favourable class. The party responsible for building and deploying the challenged model, on the other hand, ought to minimise the possible number of such claims by reducing the size of any disputable regions. This tug-of-war is bound to have a positive effect on the predictive model in question by iteratively improving its accountability, robustness and overall quality. Two possible realisations of this process are through *collective action* or by establishing dedicated *oversight bodies*. The former can accumulate and track cross-model unfairness though strategies such as crowdsourcing, the success stories of which are the Ad Observatory[3](#fn3) and Gender Shades projects; the latter can be given unrestricted access to all the required components and attempt to build alternative models in pursuit of uncovering adversely affected individuals. Reporting requirements are another possibility that found success in different areas of AI and ML, e.g., Model Cards . Facilitating this back-and-forth process may require releasing (a subset of) relevant training data as well as performance and fairness validation sets (in addition to the specification of the utilised model family F) – either exclusively to oversight bodies or to the broader public – which may be problematic due to inherent trade secrets. While distributing training data cannot be easily sidestepped, predictive performance of a model may be assessed without access to evaluation data as the classifier itself can instead be submitted for testing, akin to how ML competitions are run. Notably, having a comprehensive performance validation set that faithfully represents all of the individual cases is advantageous for the model creators as it restricts the number of admissible classifiers. Lastly, publishing a fairness validation set may also be beneficial to the owner of the model under investigation, encouraging a broader community to identify unfair predictions – and, more generally, disputable spaces – as well as engendering trust in the deployed model itself. This decoupling of the two – predictive performance validation data and fairness validation set – may therefore be desirable, especially that the latter *does not need to be labelled* since as it stands we are only interested in disparate treatment of these instances across equivalent classifiers *f* ∈ F*ε* to assess cross-model individual fairness. Just as predictive robustness is an important desideratum of a data-driven model for AI and ML engineers, cross-model fairness ought to be of concern to people being affected by automated decision-making. Guaranteeing this property engenders trust in model-makers who in such a scenario have to make the best effort to offer each individual an undisputable outcome (from a technical perspective). Additional reassurance can be had by giving people the benefit of the doubt whenever they fall into a disputable region; this can either be the most favourable prediction when operationalising our fair-by-design ensemble model or a bespoke decision heuristic chosen based on the context and application domain. Notably, such a scenario is a specific instance of favouring the user in the face of uncertainty, quantified here through predictive multiplicity. This approach is in line with augmenting classification with a reject or abstain option to account for scenarios where a model lacks confidence to predict an instance. The definition of cross-model fairness relies heavily on the selected family of data-driven classifiers and any modelling restrictions imposed upon it. This has wide-reaching implications: the fairness of the selected model is not only contingent on its own predictive behaviour but also on the behaviour of other classifiers within the same utility-based multiplicity regime, regardless of whether the model-maker is aware of their existence. While this is a relatively strong requirement that imposes most of the burden on the creators of data-driven models, it is intended to instil a high level of confidence in people subject to algorithmic decision-making, who are relatively powerless in comparison to the organisations collecting the data and building the models. It incentivises the model-makers to impose constraints on the model form as well as continuously monitor and revise their modelling pipelines – which are especially important for high stakes domains – and forgo relying purely on data-driven methods (which tend to exhibit high stochasticity) and ad hoc selection of the final model since such practices are likely to broaden the scope of the underlying model family. As each individual prediction may necessitate an explanation or justification depending on its operational context, or even require facilitating an opportunity for contestation and recourse, cross-model fairness frees people from having to pursue the aspect of this process linked to classification non-uniqueness. Additionally, the difficulty in satisfying this notion of fairness is likely to promote designing human-in-the-loop predictive systems or operating them in the decision support mode instead, thus encouraging responsible adoption of data-driven automation. The challenges of implementing cross-model fairness – such as having to precisely define the admissible family of models and searching through it to identify any disputable spaces – may nonetheless only warrant all this effort when the stakes are high enough.\nConclusion and Future Work[sec:conclusionfuture]\n================================================\nIn this paper we formalised the notion of cross-model fairness that arises due to utility-based model multiplicity – a scenario in which a number of classifiers assign different labels to selected data points despite being considered equivalent based on their predictive power. Such a situation is pertinent to distinct types of models, diverse parameterisation of the same model, or classifiers that result from multiple training runs of a given model when this procedure is, for example, greedy or stochastic. Additionally, high-dimensional and sparse data may contribute to this phenomenon given the curse of dimensionality (i.e., everything being far away from each other). In particular, we built these concepts around a user-specified *family of predictive models*, taking into consideration its *expressiveness*, which in the extreme may lead to awarding the most favourable prediction to every individual. We then defined *two meaningful notions of model multiplicity*: one determined by the entire data space and another measured with respect to a designated *fairness validation data set*. We generalised the former into *disputable spaces* – regions where at least two models disagree on a prediction – which may not necessarily be identified even with a comprehensive fairness validation set. Next, we showed how to combine (performance-wise) equivalent models to present each individual with the best possible decision. After establishing these foundations, we reported results of a comprehensive, large-scale, empirical study of cross-model fairness on three real-life data sets popular in the literature using top-performing classification workflows published in the OpenML repository. Such an experimental setup captures a realistic diversity of models that arises naturally in the machine learning workflow instead of being artificially concocted specifically to study model multiplicity, which ensures credibility of our findings. To help identify cross-model unfairness stemming from predictive multiplicity we adapted two relevant metrics – *discrepancy* and *ambiguity* – and introduced a bespoke visualisation toolkit. Our findings highlight the prevalence and severity of this phenomenon, illustrating the importance of considering utility-based model multiplicity as a new dimension of algorithmic fairness. We also demonstrated that employing the fair-by-design ensemble model, which grants each individual the most favourable outcome, may adversely affect the overall predictive power of the classification task at hand, especially for highly expressive models. As an alternative, we discussed imposing restrictions on the model space to limit the number of viable alternatives, arguing for enforcing (operational) constraints that are meaningful to each individual modelling problem, e.g., prediction monotonicity with respect to a chosen attribute, which is also recognised as a strategy for introducing (ante-hoc) interpretability whereby predictions are aligned with human values and can be easily justified. Otherwise, being able to determine disputable spaces – “grey areas” or “edge cases” from a classification standpoint – can allow to abstain from making a decision or engage a human expert in the process, thus partially alleviate the prevalence of cross-model unfair predictions. In future work, we will investigate numerical metrics and analytical tools to assess cross-model unfairness in a broader context, extending the notion beyond crisp binary classification. For example, we will study some of the techniques discussed briefly in Section [sec:relatedwork]. We will also look into strategies for integrating ground truth labels into calculation of cross-model fairness to capture the quality of each prediction. Moreover, we will investigate developing methods to derive bounds on the most and least favourable treatment of each individual from within a given data set for a selection of predictive models. This should facilitate systematic identification of disputable and stable regions, possibly leading to a new optimisation objective – minimise the former or maximise the latter – for training cross-model fair, robust and effective predictors. Finally, we will explore the lower limit of utility for a fair-by-design classifier that always uses the most beneficial model from their collection determined by a given level of predictive performance.\nThis research was supported by the Estonian Research Council (projects number PUT1458 and PRG1604); the Dora Plus programme, sponsored by the European Regional Development Fund and Estonian government; and the ARC Centre of Excellence for Automated Decision-Making and Society, funded by the Australian Government through the Australian Research Council (project number CE200100005).\n---\n1. <https://github.com/So-Cool/CrossModelFairness>[↩](#fnref1)\n2. <https://github.com/So-Cool/CrossModelFairness>[↩](#fnref2)\n3. <https://adobservatory.org>[↩](#fnref3)\n"},"data_id":{"kind":"string","value":"arxiv_0184767"}}},{"rowIdx":46196,"cells":{"content":{"kind":"string","value":"Chandra observations of outflows from PSR J1509–5850\n====================================================\nPSR J1509–5850 is a middle-aged pulsar with the period *P* ≈ 89 ms, spin-down power $\\dot{E}=5.1\\times 10^{35}$ erg s− 1, at a distance of about 3.8 kpc. We report on deep *Chandra X-ray Observatory* observations of this pulsar and its pulsar wind nebula (PWN). In addition to the previously detected tail extending up to 7ʹ southwest from the pulsar (the southern outflow), the deep images reveal a similarly long, faint diffuse emission stretched toward the north (the northern outflow) and the fine structure of the compact nebula (CN) in the pulsar vicinity. The CN is resolved into two lateral tails and one axial tail pointing southwest (a morphology remarkably similar to that of the Geminga PWN), which supports the assumption that the pulsar moves towards the northeast. The luminosities of the southern and northern outflows are about 1 × 1033 and 4 × 1032 erg s− 1, respectively. The spectra extracted from four regions of the southern outflow do not show any softening with increasing distance from the pulsar. The lack of synchrotron cooling suggests a high flow speed or in-situ acceleration of particles. The spectra extracted from two regions of the northern outflow show a hint of softening with distance from the pulsar, which may indicate slower particle propagation. We speculate that the northern outflow is associated with particle leakage from the bow shock apex into the ISM, while the southern outflow represents the tail of the shocked pulsar wind behind the moving pulsar. We estimate the physical parameters of the observed outflows and compare the J1509–5850 PWN with PWNe of other supersonically moving pulsars.\nINTRODUCTION\n============\nRotation-powered pulsars are known to produce relativistic magnetized winds responsible for a substantial fraction of the pulsar’s spin-down energy loss. High angular resolution observations with the *Chandra X-ray Observatory* (*CXO*) have shown that the winds are intrinsically anisotropic (both polar and equatorial outflow components are often seen, with the latter dominating in most cases; see Kargaltsev & Pavlov 2008). The outflow morphology is also influenced by the interaction with the ambient medium, which can introduce further anisotropy. The motion effects become particularly pronounced when the pulsar speed exceeds the sound speed in the ambient medium leading to the formation of a bow shock. In this case, the outflow from the pulsar is expected to acquire a cometary shape with an extended tail behind the pulsar. Some examples of such outflows are the pulsar wind nebulae (PWNe) associated with PSRs B0355+54 (McGowan et al. 2006), J0357–3205 (De Luca et al. 2010), and J1741–2054 (Auchettl et al. 2015). Moreover, there has been growing evidence that PWNe around high-speed pulsars (such as PSR B2224+65, IGR J11014–6103) exhibit puzzling elongated features in addition to cometary tails (Hui & Becker 2007; Pavan et al. 2014, 2015). To understand the origin of the diverse morphologies of pulsar outflows and their interaction with ambient medium, deep X-ray observations with high spatial resolution are particularly valuable.\nPSR J1509–5850 (J1509 hereafter) is a relatively young pulsar with spin-down age $\\tau\\_{\\rm sd}=154$ kyr and energy loss rate $\\dot{E}=5.1\\times10^{35}$ erg s− 1 (other parameters are listed in Table [tbl-parameters]). An X-ray nebula associated with J1509 was discovered in a 40 ks *CXO* observation (Kargaltsev et al. 2008; K+08 hereafter). It was found to consist of a compact nebula (CN) in the pulsar vicinity and a 6 pc long elongated structure interpreted as a tail behind the pulsar moving in the northeast direction. The PWN’s non-thermal spectra were fit with an absorbed ($N\\_{\\rm H}=2.1^{+0.7}\\_{-0.5}\\times 10^{22}$ cm− 2) power-law (PL) model with photon indices $\\Gamma\\_{\\rm CN}=1.8\\pm0.3$ and $\\Gamma\\_{\\rm tail}=2.4\\pm0.2$ (here and everywhere below the uncertainties are given at 68% confidence unless specified otherwise). The luminosity of the tail was found to be 1 × 1033 erg s$^{-1} \\sim 2\\times 10^{-3} \\dot{E}$ in the 0.5–8 keV band (at the assumed distance of 3.8 kpc; see Table 1). Observations with the Australia Telescope Compact Array (ATCA) detected the tail in the radio and revealed a high degree of linear polarization whose orientation suggested a helical magnetic field (Ng et al. 2010). Additionally, a faint H*α* halo around J1509 and a bow shock cavity were detected (Brownsberger & Romani 2014).\nThe X-ray spectrum extracted from an *r* = 0.9ʺ aperture around the pulsar position was fitted with an absorbed PL model with a best-fit $\\Gamma\\_{\\rm PSR}\\approx 2.2$. Since no X-ray pulsations could be detected (because the 3.2 s frame time was much longer than the pulsar period), the unresolved point-like emission could include a compact PWN contribution. The pulsed emission was detected in *γ*-rays by the *Fermi* Large Area Telescope (LAT) up to energies of 3 GeV, with a PL spectrum (Γ*γ* = 1.36 ± 0.23) and high *γ*-ray efficiency, $\\eta\\_\\gamma \\equiv L\\_\\gamma/\\dot{E} = 0.34\\ (d/3.8 {\\rm\\ kpc})^2$, higher than those of 75% of *γ*-ray pulsars (see Weltevrede et al. 2010 and Abdo et al. 2013).\nIn this paper we report on the results of a series of four deep observations of J1509 and its PWN. In Section 2 we describe the observations and data reduction. In Section 3 we present the images and spectral fit results. In Section 4 we discuss our findings, and in Section 5 we present our conclusions.\nlc R.A. (J2000.0) & 15 09 27.13 \nDecl. (J2000.0) & –58 50 56.1 \nEpoch of position (MJD) & 51463 \nGalactic longitude (deg) & 319.97 \nGalactic latitude (deg) & –0.62 \nSpin period, *P* (ms) & 88.9 \nPeriod derivative, $\\dot{P}$ (10− 14) & 0.92 \nDispersion measure, DM (pc cm− 3) & 137.7 \nDistance[1](#fn1), *d* (kpc) & 3.8 \nDistance from the Galactic plane, *z* (kpc) & 0.04 \nSurface magnetic field, *B**s* (1012 G) & 0.91 \nSpin-down power, $\\dot{E}$ (1035 erg s− 1) & 5.1 \nSpin-down age, $\\tau\\_{\\rm sd} = P/(2\\dot{P})$ (kyr) & 154 \nOBSERVATIONS & DATA REDUCTION\n=============================\nWe conducted four ≈ 90 ks observations of J1509 over the course of 15 months, from 2013 June 14 (MJD 56457) to 2014 September 23 (MJD 56923), totaling 374 ks (see Table [tbl-obs]). The target was placed on the ACIS-I detector, operated in Very Faint mode with a 3.2 s time resolution. We used the Interactive Analysis of Observations (CIAO) software package (ver. 4.6) and the *CXO* Calibration Data Base (CALDB, ver. 4.6.3) for data processing, and restricted the energy range to 0.5–8 keV for images and spectral analysis. We created exposure maps for each observation and an exposure-corrected image from the merged observation data using the CIAO routine merge\\_obs. CIAO’s wavdetect tool (a Mexican-hat wavelet source detection routine; Freeman et al. 2002) was used to detect field point sources, which were used for image alignment and then subtracted for diffuse emission analysis. We used the positions of 12 nearby stationary point sources in conjunction with the CIAO tool wcs\\_match[2](#fn2) to correct slight discrepancies in the astrometry of the event files. Spectra were extracted using CIAO’s specextract tool and analyzed using XSPEC (ver. 12.8.2). In all spectral fits we used the XSPEC wabs model with absorption cross sections from Morrison & McCammon (1983). To study the morphology of the emission in the immediate vicinity of the pulsar, we simulate the Point Spread Function (PSF) using ChaRT (Ray Trace) ver. 2 and MARX ver. 5.1.\nWe also analyzed two *XMM-Newton* observations (ObsIDs 0500630101 and 0500630301) carried out on 2008 January 29 and 2008 March 1. The EPIC (European Photon Imaging Camera) observations of PSR J1509–5850 employed the pn, MOS1, and MOS2 cameras in full frame imaging mode with the medium filter. We used the *XMM-Newton* Science Analysis Software (SAS, ver. 14.0) for the reprocessing of the observations and data reduction. After removal of background flares, the effective exposure times are 45ks (pn), 73ks (MOS1), and 73ks (MOS2). Because of the high background, we only use *XMM-Newton* data to analyze the pulsar spectrum (which, however, is more contaminated by the CN contribution than the *Chandra* spectrum due to the lower spatial resolution of *XMM-Newton*). To minimize the contamination from the CN and the background, we use circular extraction regions with radii of 10ʺ (pn) and 8ʺ (MOS1&2) centered on the pulsar (brightness profile peak). These extraction regions correspond to ≈ 50% of the encircled energy (Figures 6 and 7 in Section 3.2.1.1 of the *XMM-Newton* User Handbook[3](#fn3)). We use an *r* = 45ʺ circular background region close to the pulsar (on the same chip) but away from the CN. The resulting net source counts fraction is ≈ 90% for all instruments.\ncccccc 14523 & 2013 Jun 14 & 93.87 & 26 & 2 & 303∘ \n14524 & 2013 Sep 15 & 93.87 & 17 & 3 & 237∘ \n14525 & 2014 May 9 & 92.01 & 17 & 3 & 2∘ \n14526 & 2014 Sep 23 & 93.86 & 17 & 3 & 232∘ \nRESULTS\n=======\nPWN Morphology\n--------------\nThe merged image produced from four ACIS-I observations (see Figure [image-raw]) reveals a bright CN around the pulsar (514 net source counts) and two fainter large-scale elongated structures. In addition to the extended tail (hereafter, southern outflow; 3530 net counts) previously reported by K+08, the much deeper image shows, for the first time, another fainter structure (hereafter, the northern outflow; 1,760 net counts) apparently originating from the pulsar/CN area and visible up to 7ʹ (or 7.7 pc at *d* = 3.8 kpc) north of the pulsar. Below we consider the individual PWN elements in detail.\n### Small-scale Structure: Compact Nebula\nThe inset in Figure [image-raw] shows the CN structure. The peak of the X-ray brightness distribution (R.A.=$15^{\\rm h}09^{\\rm m}17\\fs136$, Decl.=$-58^{\\circ}50'56\\farcs19$) coincides (within $0\\farcs1$) with the radio position of the pulsar (R.A.=$15^{\\rm h}09^{\\rm m}27\\fs13(3)$, Decl.=$-58^{\\circ}50'56\\farcs1(5)''$ at MJD 51463, i.e., about 14 years before our X-ray observations). The centroiding uncertainty of the X-ray position, $\\approx0\\farcs04$ (calculated by celldetect), is much smaller than the typical uncertainty of the *Chandra* absolute astrometry ($\\approx0\\farcs4$ at 68% confidence[4](#fn4)). This puts an upper limit of $\\sim 0\\farcs5$ on the pulsar position shift during 14 years, corresponding to an upper limit ∼ 36 mas yr− 1 on proper motion.\nThe PSF modeling and subtraction does not reveal any additional structure within the core region. We also attempted PSF deconvolution using the CIAO tool arestore, but it did not produce anything new. The rest of the CN structure bears much resemblance to the Geminga PWN. Therefore, following Pavlov et al. (2006), we will call the three distinct features of the CN the lateral tails and the axial tail[5](#fn5). The lateral tails are noticeably longer ( ≈ 7ʺ − 10ʺ, or 0.13 − 0.18 pc at *d* = 3.8 kpc) than the axial tail ( ≈ 3ʺ, or 0.05 pc at *d* = 3.8 kpc).\nA wind from a fast-moving pulsar is expected to produce a termination shock with a bow-like (parabolic) structure ahead of the pulsar. However, we see no extended emission ahead of the point-like source at the leading tip of the CN (see inset in Figure 1). To constrain the stand-off distance *δ**s* of the possible unresolved shock apex ahead of the pulsar, we assume that the shape of the lateral tails represent the shock surface, fit the shape with a parabola, and estimate *δ**s* as the distance between the vertex of the fitted parabola and the position of the brightness peak. This approach gave us only upper limits on the stand-off distance, $\\delta\\_s \\lesssim 0\\farcs3$–10, depending on the region chosen for fitting. These fits, however, allowed us to estimate the radius of curvature at the parabola’s vertex, *δ**c* ∼ 1ʺ–2ʺ.\n[image-raw]\nExamination of the CN images from individual observations suggests variability associated with the lateral tails. This variability might be caused by the motion of blobs moving along the tails (see Figure [image-blobs-jets][6](#fn6)). To quantify the changes in the positions of the blobs (shown with green and yellow ellipses in Figure [image-blobs-jets]), we measure their positions (relative to the pulsar) in the images from four separate observations and plot them as a function of time in Figure [image-blobs-plot]. The positions are measured by running CIAO’s tool celldetect[7](#fn7). It is apparent from these plots that the green data points can be approximately connected by straight lines corresponding to steady motion. If we fit a steady motion model to the green data points (omitting the others from the fitting procedure), the resulting blob speeds are $v\\_{\\rm b,w} = (0.32 \\pm 0.06)c$ for the western lateral tail and $v\\_{\\rm b,e} = (0.20 \\pm 0.05)c$ for the eastern lateral tail.\nThis result should be taken with some caution because the significance of the individual blobs in the individual observations is low, ∼ 1.5*σ* − 2*σ*, when using a local background (i.e. within the lateral outflows) as calculated by celldetect). However, it increases to 3*σ*–4*σ* when the external (i.e., outside the CN) background region is used. Moreover, under the assumption of steady motion, the same blob is seen in multiple observations and hence the combined blob significances increase to 3*σ* for the local background choice, and to 7*σ*–8*σ* for the external background choice.\nWe have also compared the merged image of the CN with the older observation (ObsID 3515, ACIS-S, 2003 Feb 9). That observation was significantly shorter (40 ks), and the ACIS-S3 background contribution was much higher than that of the ACIS-I, making it difficult to resolve the individual components of the CN (see Figure 4 in K+08).\n[image-blobs-jets]\n[image-blobs-plot]\n[image-mid-scale]\n### Large-scale Structure\nFigure [image-mid-scale] shows the transition from the CN to the extended outflow south of the pulsar. One can see the lateral tails fading and apparently merging at ≈ 27ʺ from the pulsar. Beyond this distance the flow expands in the transverse direction and gives rise to the fainter *southern outflow* seen in Figure [image-large-scale] that presents the merged image of the large-scale diffuse emission with field point sources subtracted and exposure-map correction applied[8](#fn8). Unexpectedly, the same image clearly reveals a similarly shaped emission north-northwest of the pulsar, although this *northern outflow* is not discernible in the pulsar vicinity (Figure [image-mid-scale]).\nThe shapes of both large-scale structures, extending up to 6ʹ − 7ʹ from the pulsar, can be crudely described as two wedges with ≈ 17∘ opening angles and tips at (or near) the pulsar position. The southern outflow is approximately aligned with the CN symmetry axis, while the northern outflow is inclined to the CN symmetry axis by ≈ 33∘. Although the sizes and shapes of these outflows are similar, the distributions of surface brightness within the two structures are markedly different. The averaged (over the transverse direction) surface brightness of the southern outflow (see Figure [image-radio] for the profile extraction regions) appears to decrease almost linearly with distance from the pulsar (see Figure [figure-tail-brightness]), while the averaged surface brightness of the northern outflow does not appear to change significantly over its length (shown in Figure [figure-headlight-brightness]). Moreover, a substantially brighter region (hereafter referred to as the “spine”) is clearly seen within the southern outflow up to ≈ 4ʹ from the pulsar, while no such a feature is seen in the northern outflow.\n[image-large-scale]\nFigure [image-radio] also shows the extended radio emission reported by Ng et al. (2010), which stretches along the southern outflow. It becomes wider than the X-ray outflow beyond ≈ 3ʹ from the pulsar and then narrows down again beyond ≈ 5ʹ. The radio outflow is seen up to 10ʹ from of the pulsar, whereas the visible southern X-ray outflow becomes comparable with the background at about 7ʹ from the pulsar. No radio emission is seen from the northern outflow region in the same ATCA observation.\n[image-radio]\n[figure-tail-brightness]\n[figure-headlight-brightness]\nSpatially-resolved Spectra\n--------------------------\nWe measure the CN spectra in three regions shown in the Figure [image-raw] inset: (1) the bright pulsar/head region (within the $1\\farcs7$ radius white circle around the pulsar), which contains an unresolved CN “head” in addition to the pulsar, (2) the lateral tails (cyan polygons), and (3) the axial tail (the green polygon). For the southern outflow, we define the following regions (shown in Figure [image-large-scale]): the entire southern outflow (combination of green polygons), the spine (magenta polygon), and the southern outflow excluding the spine. We also divide the entire southern outflow region into 4 segments, with segment 1 being the closest to the pulsar and segment 4 being the farthest. For the northern outflow, we define two regions: the proximal and distal halves (cyan polygons). We fit all spectra with an absorbed PL model (unless specified otherwise) and list the fitting parameters Table [spatially-resolved-spectra]. For all regions, the spectra from different observations are fitted simultaneously, with the all parameters being tied together. Fitting the spine of the southern outflow yields an absorbing column density $N\\_{\\rm H}= (1.66\\pm0.19)\\times 10^{22} $ cm− 2 and a photon index Γ = 2.11 ± 0.07, with reduced chi-squared *χ**ν*2 = 1.01 for *ν* = 73 degrees of freedom (d.o.f.). Since the spectrum from this region fits the PL model well, and has a relatively small background contribution and a good statistic, we fix $N\\_{\\rm H}$ at the best-fit value obtained for this region while fitting the spectra from other regions which have poorer statistics[9](#fn9).\nFor the CN regions, we obtained very similar PL slopes for the pulsar/head and the lateral tails (Γ = 1.90 ± 0.12 and 1.80 ± 0.13, respectively) while the axial tail appears to have a harder spectrum (Γ = 1.43 ± 0.18). It is plausible that a substantial fraction of the pulsar/head emission comes from the neutron star and hence it may have a thermal component. Therefore, we attempted to fit the pulsar/head spectrum with a PL + blackbody (BB) model (with a fixed BB radius of 12 km). This resulted in a slightly better fit, with Γ = 1.77 ± 0.13, *k**T* = 103 ± 4 eV, and *χ**ν*2 = 0.77 for 53 d.o.f. However, the best-fit BB+PL model is preferred over the PL model at a 93% (1.8*σ*) confidence, according to the F-test (i.e., the BB component is not required by the data). We find an upper limit on the neutron star surface temperature by gradually increasing the temperature of the BB component (and re-fitting the PL component at each new temperature chosen) until the model becomes inconsistent with the data at a 3*σ* confidence level. This yields an upper limit *k**T* = 115 eV (or *T* = 1.3 × 106 K).\nAlthough the quality of the fits is good for all the CN regions (see Table [spatially-resolved-spectra]), the fits of the pulsar/head spectrum (see Figure [image-psr-spectra]) shows some systematic residuals centered at 3.5 keV (between 3.3–3.6 keV). To examine plausibility of the alleged spectral feature, we jointly fit the *XMM-Newton* spectra from two observations (see Figure [image-xmm-psr-spectra]) extracted from circular regions (*r* = 10ʺ and 8ʺ for the pn and MOS detectors; see Section 2) centered on the pulsar with the absorbed PL model and $N\\_{\\rm H}$ fixed at the same value as for the ACIS spectral fits. This gave a similar spectral slope, Γ = 1.9 ± 0.2, but a hint of an emission feature at ≈ 3.5 keV is seen in the pn spectrum from only one observation while the pn spectrum from the other observation shows a hint of absorption, which means that the feature is likely a statistical fluctuation. The MOS spectra do not have enough counts to look for spectral features.\n[image-psr-spectra]\n[image-xmm-psr-spectra]\n[image-tail-seg-1-spectra]\nThe spectrum of the spine (Γ = 2.11 ± 0.07) appears somewhat softer than that of the rest of the southern outflow (Γ = 1.78 ± 0.09). For the southern outflow segments 1–4, we obtain $\\Gamma\\_{\\rm S1}=1.99\\pm0.12$, $\\Gamma\\_{\\rm S2}=2.28\\pm0.16$, $\\Gamma\\_{\\rm S3}=2.06\\pm0.10$, and $\\Gamma\\_{\\rm S4}=1.82\\pm0.12$. We do not see any systematic softening of the spectrum along the southern emission despite its large extent; rather there is a hint of hardening from segment 2 to 4. Almost all fits are of good quality (the poorest quality fit, *χ**ν*2 = 1.26 for 46 d.o.f., being for segment 1, its spectrum is shown in Figure [image-tail-seg-1-spectra]).\nThe slope of the spectrum for the entire northern outflow (Γ = 1.81 ± 0.10) is similar to the spectrum of the southern outflow excluding the spine (Γ = 1.78 ± 0.09) although, unlike the southern outflow, the northern outflow spectrum shows a hint of softening with increasing distance from the pulsar (Γ changes from 1.50 ± 0.20 to 1.98 ± 0.11). This may explain the somewhat poorer quality of the PL fit to the spectrum of the entire northern outflow (see Table [spatially-resolved-spectra]). The photon indices for all regions as a function of distance from the pulsar are shown in Figure [image-gamma-fn-dist].\n[image-gamma-fn-dist]\nlrccccccc[H] Pulsar/head & 8.96 & 70.9 ± 2.8 & 1.90 ± 0.12 & 1.1 ± 0.14 & 0.82 (54) & 8.69 ± 0.50 & 2.54− 0.15+ 0.12 \nLateral tails & 88.1 & 5.84 ± 0.27 & 1.80 ± 0.13 & 0.85 ± 0.12 & 1.01 (35) & 7.50 ± 0.48 & 2.33− 0.15+ 0.12 \nAxial tail & 23.7 & 6.45 ± 0.54 & 1.43 ± 0.18 & 0.17 ± 0.04 & 790 (2046) & 2.16 ± 0.19 & 0.81− 0.10+ 0.07 \nSouthern outflow (entire) & 36778 & 0.096 ± 0.005 & 1.88 ± 0.06 & 11.9 ± 0.81 & 0.90 (105) & 98.7 ± 3.0 & 29.3 ± − 1.1+ 1.0 \nSpine & 4799 & 0.304 ± 0.011 & 2.11 ± 0.07 & 5.4 ± 0.40 & 1.01 (73) & 38.1 ± 1.4 & 9.71− 0.43+ 0.36 \nS. outflow (excl. spine) & 31979 & 0.066 ± 0.005 & 1.78 ± 0.09 & 6.8 ± 0.067 & 1.09 (83) & 61.1 ± 2.5 & 19.2− 1.1+ 1.0 \nS. outflow seg. 1 & 1805 & 0.434 ± 0.021 & 1.99 ± 0.12 & 1.6 ± 0.22 & 1.26 (46) & 12.7 ± 0.8 & 3.53− 0.21+ 0.20 \nS. outflow seg. 2 & 2224 & 0.250 ± 0.017 & 2.28 ± 0.16 & 2.2 ± 0.34 & 1.06 (35) & 13.9 ± 1.2 & 3.14− 0.33+ 0.23 \nS. outflow seg. 3 & 9463 & 0.183 ± 0.008 & 2.06 ± 0.10 & 4.6 ± 0.49 & 0.71 (42) & 33.8 ± 1.8 & 8.92− 0.45+ 0.59 \nS. outflow seg. 4 & 23162 & 0.082 ± 0.005 & 1.82 ± 0.12 & 3.6 ± 0.49 & 0.95 (73) & 31.6 ± 1.6 & 9.72− 0.65+ 0.5 \nN. outflow (entire) & 21714 & 0.081 ± 0.005 & 1.81 ± 0.10 & 4.8 ± 0.55 & 1.14 (60) & 42.0 ± 2.0 & 13.0− 0.50+ 0.63 \nN. outflow (proximal) & 5960 & 0.080 ± 0.009 & 1.50 ± 0.20 & 0.83 ± 0.20 & 0.95 (47) & 9.74 ± 0.83 & 3.53− 0.47+ 0.35 \nN. emission (distal) & 15754 & 0.082 ± 0.006 & 1.98 ± 0.11 & 4.3 ± 0.52 & 1.09 (67) & 33.1 ± 1.8 & 9.25− 0.63+ 0.56 \nDISCUSSION\n==========\nIn addition to the previously detected CN and southern outflow (K+08), the deep *Chandra* ACIS-I observations have revealed the previously unresolved fine structure of the CN. The overall shape of the CN and the southern outflow supports the earlier suggestion that they are due to the shocked wind of the pulsar moving in the north-northeast direction with a supersonic speed. The deeper image has also revealed a previously undetected faint structure, the northern outflow, extending ≈ 7ʹ to the north from the pulsar position. Below we consider possible interpretations for the different components of the J1509 PWN and compare them to those seen in other PWNe.\nThe Spatial and Spectral Structure of the PWN\n---------------------------------------------\n### Compact Nebula\nThe structure of the J1509 CN is very similar to that of the Geminga PWN (Pavlov et al. 2006, 2010; see Figure [bowshocks]), both in shape (one axial and two lateral tails) and in (projected) physical size ( ∼ 8 × 1017 cm for the lateral tails, and ∼ 2 × 1017 cm for the axial tail, for $d\\_{\\rm Gem} = 250$ pc and $d\\_{\\rm J1509}=3.8$ kpc). It is reasonable to expect that the physical sizes of compact nebulae with similar structures are proportional to $(\\dot{E}/\\rho v^2)^{1/2}$, where *ρ* is the ambient density, and *v* is the pulsar velocity. Assuming also similar angles *i* between the line of sight and the pulsar motion direction, and taking into account that $\\dot{E}\\_{\\rm J1509} \\simeq 15\\, \\dot{E}\\_{\\rm Gem}$, we can crudely estimate the speed of the J1509 pulsar:\n \n$$v\\_{\\rm J1509}\\sim790 \\left( \\frac{v\\_{\\rm Gem}}{200~{\\rm km~s}^{-1}} \\right) \\left(\\frac{\\rho\\_{\\rm Gem}}{\\rho\\_{\\rm J1509}}\\right)^{1/2}~{\\rm km~s}^{-1}.\n\\label{J1509velocity}$$ \nHowever, it is likely that Geminga lies within a local high temperature bubble, with *n* < 10− 2 cm− 3; in contrast, J1509 is in a high density, partly neutral medium (namely the H*α* halo; see Brownsberger & Romani 2014) with *n* > 1 cm− 3. An upper limit on the pulsar velocity follows from the lack of any significant shift of the pulsar position over ≈ 14 years, $\\delta\\lesssim 0\\farcs5$ (see Section 3.1.1), which gives $v\\_{\\rm J1509}\\,\\sin i< 640 (\\delta/0.5'') (d/3.8\\,{\\rm kpc})$ km s− 1. A possible lower limit on the pulsar velocity can be estimated from the lack of resolved extended emission ahead of the pulsar. If there is a (quasi-)isotropic component of the pulsar wind, it should produce a bullet-like termination shock with the apex (bullet head) distance $r\\_s = (\\dot{E}/4\\pi c\\rho v^2)^{1/2}$. For sin*i* = 1, the upper limit on the angular apex distance of $1\\farcs0$ (see Section 3.1.1) implies $r\\_s <5.7\\times10^{16}(d/3.8~{\\rm kpc})$ cm, $p\\_{\\rm amb} = \\rho\\_{\\rm J1509}v\\_{\\rm J1509}^{2}>4.2\\times10^{-10}(d/3.8~{\\rm kpc})^{-2}$ dyn cm− 2, and $v\\_{\\rm J1509}\\gtrsim160 n^{-1/2}(d/3.8~{\\rm kpc})^{-1}$ km s− 1, where *n* = *ρ*/*m**p* is the ISM number density.\nDespite the morphological similarities, there are also notable differences between the two PWNe. The spectra of the lateral tails are much softer in the J1509 CN than in the Geminga PWN (Γ = 1.8 ± 0.1 vs. Γ = 0.7–1.1, respectively), while the lateral tails’ luminosity, $L\\_{\\rm 0.5-8\\,keV} = (7.5\\pm 0.5)\\times 10^{31} (d/3.8\\,{\\rm kpc})^2$ erg s− 1 for J1509, is a factor of ∼ 200 higher[10](#fn10). The higher luminosity could be attributed to the larger $\\dot{E}$, but the radiative X-ray efficiency for J1509’s lateral tails is also significantly higher: $\\eta\\_{\\rm 0.5-8\\,keV} \\equiv L\\_{\\rm 0.5-8\\,keV}/\\dot{E}\\simeq1.5\\times10^{-4}(d/3.8\\,{\\rm kpc})^2$ for J1509 vs. $1.2\\times 10^{-5} (d/0.25\\,{\\rm kpc})^2$ for Geminga. Another difference is that there is the presence of faint diffuse emission between the lateral tails (beyond the extent of the shorter axial tail) in the J1509 CN (see the inset in Figure [image-raw]), while such emission is not detected in the Geminga PWN (Posselt et al., in prep.; Pavlov et al. 2010).\nNumerical simulations of the synchrotron brightness distributions from PWNe of supersonically moving pulsars with isotropic wind ejection (Bucciantini et al. 2005) predict filled structures with the brightness reaching its maximum ahead of the moving pulsar and gradually decreasing behind the pulsar (see Figures 4 and 5 in Bucciantini et al. 2005). These simulated synchrotron brightness maps do not resemble the observed structures of the CNe around J1509 and Geminga. One reason could be that the magnetic field structure is substantially different from that assumed in the simulations. Bucciantini et al. (2005) mention that the magnetic field may be compressed in narrow layers near the contact discontinuity surface, which, in principle, could produce limb-brightened structures, depending on the amount of Doppler boosting (caused by a high bulk flow velocity) and the magnetic field geometry. Another reason could be that the assumption of isotropic wind injection used in the simulations is not valid for real pulsars. For instance, one could interpret the lateral tails as jets or an equatorial outflow bent back by the ram pressure. However, at least the jet interpretation looks unrealistic as it requires stretching some of the parameters to their extremities. Indeed, a jet produced by a moving pulsar bends because the jet flow acquires a momentum from the ambient medium; it becomes comparable to the original jet flow momentum over the bending length scale (curvature radius)[11](#fn11) $l\\_b\\sim\\xi\\_j\\dot{E}/(2 c r\\_j \\rho v^2 \\sin\\alpha)$, where *ξ**j* < 1 is the fraction of the spin-down power $\\dot{E}$ that goes into the jet, *r**j* is the jet radius, and *α* is the angle between the initial jet direction and the pulsar’s velocity. For J1509, we obtain $l\\_b < 3.6 \\times10^{17} \\xi\\_j (r\\_j/5.7\\times10^{16} {\\rm cm})^{-1} (\\sin \\alpha)^{-1} (d/3.8\\ {\\rm kpc})^2 $ cm. Therefore, for a jet with the angular width *δ**j* ∼ 1ʺ, a significant bending on a scale of the observed curvature radius *δ**c* ∼ 1ʺ–2ʺ (see Section 3.1.1) implies that most of the spin-down power goes into the two jets. However, observations of PWNe with both jet and equatorial components suggest *ξ**j* ≪ 1 (e.g., Kargaltsev & Pavlov 2008; Pavlov et al. 2003). Therefore, the jet interpretation of the lateral tails would imply that in some pulsars jets can be more powerful than equatorial outflows. More extensive modeling and simulations are needed to see whether it is possible to reproduce the observed CN morphologies for other assumptions on the angular distribution of the pulsar wind injection.\n[bowshocks]\nWe also note that the morphologies of the Geminga and J1509 CNe are markedly different from those of other bow-shock PWNe. In particular, others exhibit a CN with a local minimum in brightness in the pulsar vicinity (i.e., behind the pulsar; PWNe of PSRs J0357+3205 and J1101–6101), or exhibit a filled CN morphology (PWNe of PSRs J1747–2958, J1741–2054, and B0355+54). The differences, however, are not limited to X-ray morphologies only. For instance, the radio counterparts of the PWNe associated with PSRs J1509–5850 (Ng et al. 2010), J1101–6101 (Pavan et al. 2014), and J1747–2958 (the Mouse; Yusef-Zadeh & Gaensler 2005), the radio brightness distributions are also quite different. While the radio brightness in the Mouse is maximal near the pulsar (similar to X-rays; Gaensler et al. 2004), in the other two PWNe the radio brightness is low near the pulsar and becomes higher further away (see Figure [image-radio]). We also note that the X-ray morphologies of most of the other CNe in Figure [bowshocks] do not bear much resemblance to the simulated images, except perhaps for the Mouse and J1741–2054. Ng et al. (2010) mentioned an open cluster, Ruprecht 112 (C1453–623; Alter et al. 1970), about 4∘ southwest of the current pulsar position, as a possible birth place of the pulsar. At the distance of 3.8 kpc and the pulsar’s spin-down age of 150 kyr, the required pulsar velocity would be ∼ 1700 km s− 1, much higher than the upper limit of 640 km s− 1 on the transverse velocity inferred from the lack of detectable change of the pulsar position in 14 years. Based on the comparison of the older X-ray image with the shape of the termination shock “bullet” seen in numerical simulations, K+08 suggested that the pulsar velocity should be ∼ 300 km s− 1, which is still within the allowed velocity range (see above). However, since the deeper X-ray images reveal substantial differences between the CN structure and the simulations, this velocity estimate should be taken with caution.\n### Large-scale PWN Structure\nThe most puzzling new feature of the J1509 PWN is the large-scale northern outflow seen up to ∼ 7ʹ almost ahead of the pulsar. One could assume that the northern outflow is a powerful relativistic jet piercing the bow shock near the pulsar. The lack of a counter-jet could be explained by Doppler boosting, but there are other problems with the jet interpretation. According to the estimates in Section 4.1.1, the bending length for a jet is very small, even at *ξ**j* ∼ 1 and sin*α* ∼ 0.5, compared to the length of the northern outflow that shows no bending. It is also difficult to explain the relative faintness of the northern outflow closer to the pulsar (but outside the CN) in the jet scenario. Finally, the X-ray radiation efficiency of the northern outflow, *η*0.5 − 8 *k**e**V* ∼ 0.8 × 10− 3, substantially exceeds the efficiencies of jets of younger PWNe with clear torus-jet morphology. Therefore, we discard the jet interpretation for the northern outflow.\nSimilar arguments have led Bandiera (2008) to conclude that the extended X-ray feature accompanying the Guitar nebula (which also exhibits a large-scale outflow misaligned with the pulsar motion direction; see Figure [image-guitar] and Johnson & Wang 2010) cannot be interpreted as a hydrodynamical jet. Following Bandiera (2008), we consider a scenario in which particles with the highest energies leak from the termination shock apex into the ISM, and travel along the ordered ISM magnetic field. Bandiera (2008) suggested that the length of the Guitar PWN feature, ∼ 1.4 pc, is on the order of the radiation length (i.e., the distance over which particles lose a substantial fraction of their energy to synchrotron radiation). However, this assumption implies a propagation speed of only a few thousands km s− 1, instead of the speed of light, as expected for particles with large (bulk) Lorentz factors. Bandiera (2008) argued that the turbulent component of the magnetic field would provide a scattering mechanism and slow down particle propagation (which can then be considered as diffusion) along the ISM magnetic field to the required non-relativistic speed. The observed hint of softening of the X-ray spectrum in J1509’s northern outflow can be considered as supporting this scenario. A problem with this scenario is that in all three cases with prominent misaligned (with respect to pulsar motion) X-ray features (the Guitar, J1509, and J1101–6101), the features appear to be one-sided or show very faint “counter-features”, which means that there is a preferred direction for the escape of the high-energy particles. The cause of this asymmetry is currently unclear. On the other hand, since the particles escaping from the termination shock apex into the ISM must be the most energetic ones (those with large gyration radii; see Bandiera 2008), the lack of radio synchrotron emission can be naturally explained by the deficit of lower energy particles.\nThe southern outflow is aligned with the CN symmetry axis and clearly represents a continuation of the CN (see Figure [image-mid-scale]). Therefore, we can assume that at least the brighter part of the southern outflow can be interpreted as a tail filled with the shocked pulsar wind confined by the ram pressure. The lack of evidence for synchrotron burn-off in the X-ray spectra suggests a high flow speed and/or a low magnetic field. Alternatively, there could be some “reheating” due to in-situ conversion of magnetic field energy into particle energy, e.g., via turbulent processes and accompanying reconnection, which might explain the hint of spectral hardening at large distances from the pulsar (see Figure [image-gamma-fn-dist].) Indeed, Ng et al. (2010) reported the presence of substantial turbulent magnetic field components in the tail using radio polarimetry data.\nThere are, however, some puzzling aspects associated with the southern outflow. One is that the radio emission brightens with distance from the pulsar and becomes broader than the X-ray emission beyond ≈ 3ʹ (see Figure [image-radio]). Such behavior could possibly be caused by entrainment of ambient matter (see also Morlino et al. 2015), which decreases the flow speed, increases the transverse size of the tail, and leads to adiabatic cooling of the outflowing matter. This tentative explanation remains to be checked with detailed modeling of entrainment effects.\nAnother puzzling aspect is the similarity of the wedge-like shapes of the fainter part of the southern outflow (i.e., the southern outflow excluding the spine) and the northern outflow. Given the qualitative differences in the above interpretations of the northern and southern outflows, these similarities could be attributed to merely a chance coincidence. Alternatively, the structural and spectral similarities could be explained if both the northern emission and the southern emission (excluding the spine) are caused by very energetic particles that leaked from the termination shock apex in opposite directions and diffused along the curved ISM magnetic field (the curvature of the magnetic field is required to explain the different orientations of the “wedges”). In this scenario, the spine (whose spectrum is slightly softer than that of the rest of the southern outflow) is still interpreted as the pulsar tail, coincidentally projected onto the sky within the southern outflow. This interpretation may look somewhat artificial, but it cannot be firmly excluded with the currently available data.\nThe spectra of the large-scale outflows measured in several regions allow one to crudely estimate magnetic fields in these regions. For a PL synchrotron spectrum with photon index Γ, the magnetic field at a given magnetization parameter *σ* = *w**B*/*w**e* (ratio of energy densities of magnetic field and relativistic electrons) depends on the ratio of the luminosity *L*(*ν**m*, *ν**M*), measured in the *ν**m* < *ν* < *ν**M* frequency range, to the radiating volume *V*. A simple generalization of Equation (7.14) in Pacholczyk (1970)[12](#fn12) to the case of arbitrary *σ* gives\n \n$$B=\\left[\\frac{L(\\nu\\_m,\\nu\\_M)\\sigma}{\\mathcal{A} V} \\frac{\\Gamma-2}{\\Gamma-1.5}\\frac{\\nu\\_1^{1.5-\\Gamma}-\\nu\\_2^{1.5-\\Gamma}}{\\nu\\_m^{2-\\Gamma}-\\nu\\_M^{2-\\Gamma}} \\right]^{2/7}.\n\\label{synch\\_magn\\_field}$$ \nIn this equation, *ν*1 and *ν*2 are the characteristic synchrotron frequencies ($\\nu\\_{\\rm syn}\\simeq eB\\gamma^2/2\\pi mc$) corresponding to the boundary energies (*γ*1*m**e**c*2 and *γ*2*m**e**c*2) of the electron spectrum (*d**N**e*/*d**γ* ∝ *γ*− *p* ∝ *γ*− 2Γ + 1; *γ*1 < *γ* < *γ*2), and A = 21/2*e*7/2/18*π*1/2*m**e*5/2*c*9/2. The photon indices and 0.5–8 keV luminosities have been measured from the spectral fits (see Table [spatially-resolved-spectra]). We estimate the radiating volumes from the images assuming conical shapes of the outflows. Since *L* ∝ *d*2 and *V* ∝ *d*3, the magnetic field estimated from Equation ([synchmagnfield]) weakly depends on the assumed distance, *B* ∝ *d*− 2/7. The boundary frequencies *ν*1 and *ν*2 are unknown. The exact value of *ν*2 is not really important as long as (*ν*1/*ν*2)Γ − 1.5 ≪ 1, so we choose a plausible (but arbitrary) value *h**ν*2 = 100 keV. The unknown frequency *ν*1 is the main source of uncertainty of the magnetic field at a given *σ*, for the measured spectral slopes. For the southern outflow, we can at least estimate a lower limit on *ν*1 from the requirement that extrapolation of the X-ray PL spectrum to lower frequencies does not exceed the harder radio spectrum ($\\Gamma\\_{\\rm rad}=1.26\\pm 0.04$; Ng et al. 2010), but even the limit, $\\nu\\_1\\gtrsim 10^{10}$–1012 Hz, is not very certain because of the uncertainty of the X-ray photon indices (and because the slope of the radio spectrum was measured in a broader region). Since the northern outflow was not detected in the radio, we can only guess that *ν*1 is higher there than in the southern outflow (which would be consistent with our hypothesis that the northern outflow emission is due to leaked particles with very high energies). Given these uncertainties, we estimate the magnetic fields assuming *ν*1 = 100 GHz for all the regions. If future multiwavelength observations yield different estimates of *ν*1, the magnetic field could be scaled (approximately) as *B* ∝ *ν*1(3 − 2Γ)/7 (if (*ν*1/*ν*2)Γ − 1.5 ≪ 1).\nThe equipartition magnetic fields, $B\\_{\\rm eq}=B\\sigma^{-2/7}$, for different parts of the J1509 southern and northern outflows are given in Table [b-fields], for the above-described choice of boundary frequencies and *d* = 3.8 kpc. Although the actual magnetic field strengths can differ from those in Table [b-fields] by a factor of a few, we can at least hope that their relative values reflect the spatial dependence of the magnetic field in the emitting regions. Surprisingly, the magnetic field in the southern outflow shows a hint of non-monotonic dependence on distance from the pulsar: the field grows from segment 1 to segment 2, and then decreases outwards (in concert with the photon index Γ and slope *p* of electron spectrum). The reason for such behavior is unclear, and we cannot exclude that it is simply due to a variation of unknown parameters (e.g., *ν*1 and/or *σ* along the outflow).\nSince the southern outflow was detected in radio (Ng et al. 2010), the radio spectrum ($\\Gamma\\_{\\rm rad}\\approx 1.26$) can also be used for magnetic field estimates. If the radio and X-ray spectra are generated in different regions (e.g., the radio emission comes from a “sheath” around an X-ray emitting “spine” of the outflow), the magnetic field in the radio-emitting region should be estimated solely from the radio spectrum, which gives $B\\_{\\rm rad} \\simeq 13\\, \\sigma\\_{\\rm rad}^{2/7}$ *μ*G for $L\\_{\\rm rad}(\\nu\\_m,\\nu\\_M) = 1.5\\times 10^{32}$ erg s− 1, $V\\_{\\rm rad} = 5\\times 10^{56}$ cm3, *ν*1 = *ν**m* = 10 MHz, *ν*2 = *ν**M* = 100 GHz ($\\sigma\\_{\\rm rad}$ is the magnetization in this region). This value of $B\\_{\\rm rad}$ is close to those determined from the X-ray spectra in segments 3 and 4 (assuming $\\sigma\\_{\\rm rad} \\sim \\sigma$). If the radio spectrum in the X-ray detected part of the southern outflow is generated in the same region as the X-ray emission, then the mutiwavelength (radio to X-rays) spectrum can be considered as a broken PL. The magnetic field from such a spectrum could be estimated for a given magnetization $\\sigma = (\\sigma\\_{\\rm rad}^{-1} + \\sigma\\_X^{-1})^{-1}$, where $\\sigma\\_{\\rm rad}$ and *σ**X* (each of them is greater than *σ*) are magnetizations corresponding to the energy densities of particles responsible for radiation below and above the break frequency, respectively. In this case, the equipartition field would be slightly higher than those estimated separately from the X-ray and radio spectra, but the difference would be within the uncertainties.\nAccording to Table [b-fields], the difference of the equipartition fields in the two segments of the northern outflow is within the $B\\_{\\rm eq}$ uncertainties caused by the uncertainties of the spectral slopes. The characteristic equipartition field, $B\\_{\\rm eq}\\sim 5$–10 *μ*G, is lower than typical values for the southern outflow[13](#fn13). However, if the X-ray emission from the northern outflow is indeed due to particles of very high energy leaked from the termination shock apex, then *σ* can be very different from unity, and *B* very different from $B\\_{\\rm eq}$.\nThe magnetic field strengths can be used to estimate the Lorentz factors, gyration radii, and characteristic synchrotron cooling times of electrons radiating synchrotron photons with energy *E*:\n \n$$\\gamma\\sim 10^{8}(E/1\\,{\\rm keV})^{1/2}(B/20\\,\\mu{\\rm G})^{-1/2},$$ \n \n$$r\\_g\\sim 10^{14}(E/1\\,{\\rm keV})^{1/2}(B/20\\,\\mu{\\rm G})^{-3/2}~{\\rm cm},$$ \nand\n \n$$t\\_{\\rm syn}\\sim 400 (E/1\\,{\\rm keV})^{-1/2}(B/20\\,\\mu{\\rm G})^{-3/2}~{\\rm yr}\\,.$$ \nThe spatially-resolved X-ray spectra of the outflows allow one to crudely estimate (or constrain) characteristic flow velocities. For instance, if the apparent spectral softening in the northern outflow is real, the flow velocity can be estimated as $v\\_{\\rm flow} \\sim l/\\tau\\_{\\rm syn}\\sim 3000 (l/6\\,{\\rm pc}) (B/7\\,\\mu{\\rm G})^{3/2}$ km s− 1, where *l* is the spatial scale of the alleged softening. This is, of course, just an order-of-magnitude estimate, but the fact that $v\\_{\\rm flow} \\ll c$ is consistent with the scenario by Bandiera (2008), which suggests that propagation of the leaked high-energy particles is slowed down by diffusion in a turbulent magnetic field (see above).\nKnowing the equipartition magnetic field and the flow velocity, we can estimate the energy density,\n \n$$\\begin{aligned}\nw & = & w\\_B + w\\_e = [B\\_{\\rm eq}^2/(8\\pi)](\\sigma^{4/7}+\\sigma^{-3/7})^2 \\\\\n\\nonumber\n & \\sim & 2\\times 10^{-12} (B\\_{\\rm eq}/7\\,\\mu{\\rm G})^2 (\\sigma^{4/7}+\\sigma^{-3/7})^2\\,\\,{\\rm erg}\\,{\\rm cm}^{-3}, \n\\label{energy\\_density\\_north}\\end{aligned}$$ \nand the energy transfer rate ( ∼ energy injection rate),\n \n$$\\begin{aligned}\n\\dot{E}\\_{\\rm North} & \\sim & (w+p) S v\\_{\\rm flow} \n\\label{energy\\_injection\\_north}\\\\\n\\nonumber\n& \\sim & 4\\times 10^{33} (B\\_{\\rm eq}/7\\,\\mu{\\rm G})^2 (S/3\\times 10^{36}\\, {\\rm cm}^2)\\\\\n\\nonumber\n&\\times & (v\\_{\\rm flow}/3000\\,{\\rm km}\\,{\\rm s}^{-1}) (\\sigma^{4/7}+(2/3)\\sigma^{-3/7})^2\\,\\,{\\rm erg}\\,{\\rm s}^{-1},\\end{aligned}$$ \nfor the northern outflow, **where *w* + *p* is the enthalpy density, *p* = *w**B* + *w**e*/3 is the pressure of the relativistic particles and magnetic field, and *S* is the characteristic cross-sectional area.** As expected, $\\dot{E}\\_{\\rm North}$ is a small fraction of the pulsar’s spin-down power ($\\dot{E}\\_{\\rm North}\\sim 10^{-2}\\dot{E}$ for *σ* ∼ 1). On the other hand, $\\dot{E}\\_{\\rm North}$ exceeds the 0.5–8 keV luminosity of the northern outflow (by a factor of 7 at the parameters chosen and *σ* ∼ 1, but the value of this factor strongly depends on the poorly known outflow parameters).\nIn the southern outflow we see no spectral softening with increasing distance from the pulsar (i.e., no synchrotron cooling). If there is no in-situ particle acceleration in the outflow, the lack of cooling sets a lower limit on the flow velocity of a few thousand km s− 1 (K+08), well above the upper limit on the pulsar velocity. The value of the limiting speed is somewhat uncertain because of the magnetic field uncertainties. Moreover, the hint of spectral hardening with increasing distance from the pulsar (see Figure [image-gamma-fn-dist]) suggests that particle acceleration may occur in the southern outflow (e.g., due to magnetic field reconnection), which would invalidate the velocity estimates based on passive cooling. An upper limit of a few tens of thousands km s− 1 on the characteristic flow velocity can be estimated from the condition $\\dot{E}\\_{\\rm South}< \\dot{E}$, where $\\dot{E}\\_{\\rm South}\\sim (w+p) S v\\_{\\rm flow}$, similar to Equation (7).\nlcccc[H] S. outflow seg. 1 & 2.98 ± 0.24 &1.3 & 2.6 & 36− 12+ 23 \nS. outflow seg. 2 & 3.56 ± 0.32 & 1.4 & 7.4 & 89− 42+ 16 \nS. outflow seg. 3 & 3.12 ± 0.20 & 3.4 & 42 & 29− 9+ 14 \nS. outflow seg. 4 & 2.64 ± 0.24 & 3.2 & 125 & 9− 3+ 4 \nN. outflow (proximal) & 2.00 ± 0.40 & 1.0 & 9.1 & 5− 1+ 4 \nN. outflow (distal) & 2.96 ± 0.22 & 3.3 & 76 & 10− 2+ 4 \nAs we mentioned above, the J1509 and Guitar PWNe are not the only ones which exhibit puzzling asymmetric structures misaligned with their apparent pulsar motion directions. Two other examples are the outflows associated with PSR J1101–6101 (the Lighthouse Nebula; Pavan et al. 2014) and PSR J2055+2539 (Marelli et al. 2015). The former one shows a particularly remarkable structure. The deep 300 ks ACIS image of the Lighthouse Nebula shown in Figure [image-lighthouse] reveals a long ( ∼ 11 pc) jet-like outflow that originates near the pulsar, bends back in the direction opposite to that of the pulsars’ proper motion, and then sharply turns in a direction nearly orthogonal to the proper motion direction (see also Pavan et al. 2015). This outflow can possibly be interpreted in the framework of the Bandiera (2008) scenario, similar to Guitar and J1509 misaligned outflows. **In this scenario, the external (ISM) magnetic field lines are illuminated by the escaping wind particles traveling along the distorted magnetic field (cf. magnetic “draping”; Lyutikov 2006 and Dursi & Pfrommer 2008).** Additionally, the PWN of PSR B0355+54 also shows hints of a misaligned outflow (see Figure [bowshocks]), although in this case it is rather faint and is seen on both sides of the bow shock (perhaps due to the lower speed of the pulsar). An in-depth discussion of this PWN will be given in a separate paper (Klingler et al., in prep.).\n[image-guitar]\n[image-lighthouse]\nSummary\n=======\nThe most important findings from our deep ACIS-I observations of J1509 can be summarized as follows.\nWe were able to resolve the fine structure of the CN in the immediate pulsar vicinity. The CN consists of two lateral tails, apparently emerging from the pulsar and resolved up to 0.2 pc behind the pulsar, and a short, 0.05 pc, axial tail along the CN symmetry axis. The CN structure and size are remarkably similar to those of the Geminga, PWN but different from several other bow shock PWNe. Such PWN morphology challenges the existing numerical simulations of synchrotron emission from the PWNe of supersonically moving pulsars. The morphological differences from other PWNe could be attributed to different initial wind magnetizations or, more likely, to anisotropic wind ejection, which affects the geometry of the magnetic field downstream of the termination shock and the PWN morphology.\nWe found that the CN narrows beyond 0.2–0.3 pc, reaching the minimum width (where the lateral tails apparently merge) at about 0.5 pc behind the pulsar. Beyond this distance the flow expands and forms the diverging 7 pc long southern outflow, discovered by K+08 who interpreted it as a tail of shocked pulsar wind confined by the ram pressure of the ambient medium. The PL spectra of the southern outflow do not show softening (synchrotron cooling) with increasing distance from the pulsar, which is expected for such long tails. The lack of softening (with a hint of hardening) may suggest that particle acceleration occurs within the tail.\nWe also discovered a previously undetected wedge-like northern outflow, seen up to about 6 pc, with the wedge axis inclined at an angle of about 33∘ from the pulsar’s proper motion direction, inferred from the CN symmetry axis. This outflow is possibly formed by energetic pulsar wind particles escaping from the apex of the bow shock and diffusing along the ISM magnetic field. The equipartion magnetic field in this outflow is about a few *μ*G, the flow speed is a few thousand km s− 1, and the energy injection rate is $\\gtrsim 10^{33}$ erg s− 1. The discovery of the northern outflow puts J1509 into the (so far) small group of supersonically moving pulsars with misaligned outflows (i.e., outflows inclined to the pulsar trajectory at substantial angles). To better understand the nature of the misaligned outflows, further deep multiwavelength observations with high angular resolution are needed.\n*Facilities:*\nWe would like to thank Maxim Lyutikov, Andrei Bykov, Giovanni Morlino, Marina Romanova, and John Hewitt for the very insightful discussions, as well as our referee, Roger Romani, for the helpful suggestions. Support for this work was provided by the National Aeronautics and Space Administration through *Chandra* Award Number G03-14082 issued by the *Chandra* X-ray Observatory Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of the National Aeronautics Space Administration under contract NAS8-03060. The work was also partly supported by NASA grant NNX08AD71G.\n \nAbdo, A. A., Ajello, M., Allafort, A., et al. 2013,, 208, 17\nAuchettl, K., Slane, P., Romani, R. W., et al. 2015,, 802, 68\nBandiera, R. 2008,, 490, L3\nBrownsberger, S., & Romani, R. W. 2014,, 784, 154\nBucciantini, N., Amato, E., & Del Zanna, L. 2005,, 434, 189\nDe Luca, A., Marelli, M., Mignani, R. P., et al. 2011,, 733, 104\nDursi, L. J., & Pfrommer, C. 2008,, 677, 993\nFreeman, P. E., Kashyap, V., Rosner, R., & Lamb, D. Q. 2002,, 138, 185\nGaensler, B. M., van der Swaluw, E., Camilo, F., et al. 2004,, 616, 383\nHui, C. Y., & Becker, W. 2007,, 467, 1209\nJohnson, S. P., & Wang, Q. D. 2010,, 408, 1216\nKargaltsev, O., & Pavlov, G. G. 2008, 40 Years of Pulsars: Millisecond Pulsars, Magnetars and More, 983, 171\nKargaltsev, O., Misanovic, Z., Pavlov, G. G., Wong, J. A., & Garmire, G. P. 2008,, 684, 542\nKramer, M., Bell, J. F., Manchester, R. N., et al. 2003,, 342, 1299\nLyutikov, M. 2006,, 373, 73\nMarelli, M., Pizzocaro, D., De Luca, A., et al. 2015, arXiv:1509.05833\nMcGowan, K. E., Vestrand, W. T., Kennea, J. A., et al. 2006,, 647, 1300\nMorlino, G., Lyutikov, M., & Vorster, M. 2015,, 454, 3886\nMorrison, R., & McCammon, D. 1983,, 270, 119\nNg, C.-Y., Gaensler, B. M., Chatterjee, S., & Johnston, S. 2010,, 712, 596\nPacholczyk, A. G. 1970, Series of Books in Astronomy and Astrophysics, San Francisco: Freeman, 1970,\nPavan, L., Bordas, P., Pühlhofer, G., et al. 2014,, 562, A122\nPavan, L., Pühlhofer, G., Bordas, P., et al. 2015, arXiv:1511.01944\nPavlov, G. G., Teter, M. A., Kargaltsev, O., & Sanwal, D. 2003,, 591, 1157\nPavlov, G. G., Sanwal, D., & Zavlin, V. E. 2006,, 643, 1146\nPavlov, G. G., Bhattacharyya, S., & Zavlin, V. E. 2010,, 715, 66\nTaylor, J. H., & Cordes, J. M. 1993,, 411, 674\nTomsick, J. A., Bodaghee, A., Rodriguez, J., et al. 2012,, 750, L39\nWeltevrede, P., Abdo, A. A., Ackermann, M., et al. 2010,, 708, 1426\nYusef-Zadeh, F., & Gaensler, B. M. 2005, Advances in Space Research, 35, 1129\n---\n1. The distance estimate is based on the dispersion measure and the Galactic electron density distribution the model by Taylor & Cordes (1993). The model by Cordes & Lazio (2002) places the pulsar at a distance of 2.6 kpc. However, the smaller distance implies a lower $N\\_{\\rm H}$ than observed, and, based on the J1509’s Galactic coordinates, would place the pulsar in between the Sagittarius and Scutum-Centaurus arms, whereas the distance of 3.8 kpc places J1509 in the middle of the Scutum-Centaurus arm.[↩](#fnref1)\n2. http://cxc.harvard.edu/ciao/ahelp/wcs\\_match.html[↩](#fnref2)\n3. http://xmm.esac.esa.int/external/xmm\\_user\\_support/ \ndocumentation/uhb\\_2.1/node17.html[↩](#fnref3)\n4. http://cxc.harvard.edu/cal/ASPECT/celmon/[↩](#fnref4)\n5. The naming convention used here does not imply any physical interpretation (see Section 4 for discussion)[↩](#fnref5)\n6. See also the movie at [http://home.gwu.edu/ ∼ kargaltsev/ \nj1509movie.mov](http://home.gwu.edu/~kargaltsev/j1509movie.mov))[↩](#fnref6)\n7. The tool also calculates the uncertainties of the blob positions, but they are likely to be underestimated in the presence of fainter emission within the lateral tails (typical 1*σ* values range between 01–02). Therefore, we conservatively take the actual uncertainties to be 05 for all blobs and use these in the fits.[↩](#fnref7)\n8. To create this image, we followed the procedure described at http://cxc.harvard.edu/ciao/threads/merge\\_all/[↩](#fnref8)\n9. We have also obtained the same value of *N**H* by fitting the spectra from the lateral tails and the pulsar/head region.[↩](#fnref9)\n10. The photon indices and luminosity of Geminga’s lateral tails are based on recent deep *Chandra* ACIS observations (Posselt et al., in prep.)[↩](#fnref10)\n11. This estimate neglects possible entrainment of the ambient medium into the jet. **However, if *ξ**j* ≪ 1, it is likely that the jets are inside the contact discontinuity, in which case the deflection is a result of the post-shock sweep back, and not from ballistic impact from the unperturbed ISM.**[↩](#fnref11)\n12. Equation (7.14) gives the magnetic field for *σ* = 3/4 (which minimizes the total energy density *w**B* + *w**e*) and assumes *ν**m* = *ν*1 and *ν**M* = *ν*2.[↩](#fnref12)\n13. It would be even lower if we assume a *ν*1 value higher than 100 GHz for the northern outflow.[↩](#fnref13)\n\nChandra observations of outflows from PSR J1509–5850\n====================================================\nPSR J1509–5850 is a middle-aged pulsar with the period *P* ≈ 89 ms, spin-down power $\\dot{E}=5.1\\times 10^{35}$ erg s− 1, at a distance of about 3.8 kpc. We report on deep *Chandra X-ray Observatory* observations of this pulsar and its pulsar wind nebula (PWN). In addition to the previously detected tail extending up to 7ʹ southwest from the pulsar (the southern outflow), the deep images reveal a similarly long, faint diffuse emission stretched toward the north (the northern outflow) and the fine structure of the compact nebula (CN) in the pulsar vicinity. The CN is resolved into two lateral tails and one axial tail pointing southwest (a morphology remarkably similar to that of the Geminga PWN), which supports the assumption that the pulsar moves towards the northeast. The luminosities of the southern and northern outflows are about 1 × 1033 and 4 × 1032 erg s− 1, respectively. The spectra extracted from four regions of the southern outflow do not show any softening with increasing distance from the pulsar. The lack of synchrotron cooling suggests a high flow speed or in-situ acceleration of particles. The spectra extracted from two regions of the northern outflow show a hint of softening with distance from the pulsar, which may indicate slower particle propagation. We speculate that the northern outflow is associated with particle leakage from the bow shock apex into the ISM, while the southern outflow represents the tail of the shocked pulsar wind behind the moving pulsar. We estimate the physical parameters of the observed outflows and compare the J1509–5850 PWN with PWNe of other supersonically moving pulsars.\nINTRODUCTION\n============\nRotation-powered pulsars are known to produce relativistic magnetized winds responsible for a substantial fraction of the pulsar’s spin-down energy loss. High angular resolution observations with the *Chandra X-ray Observatory* (*CXO*) have shown that the winds are intrinsically anisotropic (both polar and equatorial outflow components are often seen, with the latter dominating in most cases; see Kargaltsev & Pavlov 2008). The outflow morphology is also influenced by the interaction with the ambient medium, which can introduce further anisotropy. The motion effects become particularly pronounced when the pulsar speed exceeds the sound speed in the ambient medium leading to the formation of a bow shock. In this case, the outflow from the pulsar is expected to acquire a cometary shape with an extended tail behind the pulsar. Some examples of such outflows are the pulsar wind nebulae (PWNe) associated with PSRs B0355+54 (McGowan et al. 2006), J0357–3205 (De Luca et al. 2010), and J1741–2054 (Auchettl et al. 2015). Moreover, there has been growing evidence that PWNe around high-speed pulsars (such as PSR B2224+65, IGR J11014–6103) exhibit puzzling elongated features in addition to cometary tails (Hui & Becker 2007; Pavan et al. 2014, 2015). To understand the origin of the diverse morphologies of pulsar outflows and their interaction with ambient medium, deep X-ray observations with high spatial resolution are particularly valuable.\nPSR J1509–5850 (J1509 hereafter) is a relatively young pulsar with spin-down age $\\tau\\_{\\rm sd}=154$ kyr and energy loss rate $\\dot{E}=5.1\\times10^{35}$ erg s− 1 (other parameters are listed in Table [tbl-parameters]). An X-ray nebula associated with J1509 was discovered in a 40 ks *CXO* observation (Kargaltsev et al. 2008; K+08 hereafter). It was found to consist of a compact nebula (CN) in the pulsar vicinity and a 6 pc long elongated structure interpreted as a tail behind the pulsar moving in the northeast direction. The PWN’s non-thermal spectra were fit with an absorbed ($N\\_{\\rm H}=2.1^{+0.7}\\_{-0.5}\\times 10^{22}$ cm− 2) power-law (PL) model with photon indices $\\Gamma\\_{\\rm CN}=1.8\\pm0.3$ and $\\Gamma\\_{\\rm tail}=2.4\\pm0.2$ (here and everywhere below the uncertainties are given at 68% confidence unless specified otherwise). The luminosity of the tail was found to be 1 × 1033 erg s$^{-1} \\sim 2\\times 10^{-3} \\dot{E}$ in the 0.5–8 keV band (at the assumed distance of 3.8 kpc; see Table 1). Observations with the Australia Telescope Compact Array (ATCA) detected the tail in the radio and revealed a high degree of linear polarization whose orientation suggested a helical magnetic field (Ng et al. 2010). Additionally, a faint H*α* halo around J1509 and a bow shock cavity were detected (Brownsberger & Romani 2014).\nThe X-ray spectrum extracted from an *r* = 0.9ʺ aperture around the pulsar position was fitted with an absorbed PL model with a best-fit $\\Gamma\\_{\\rm PSR}\\approx 2.2$. Since no X-ray pulsations could be detected (because the 3.2 s frame time was much longer than the pulsar period), the unresolved point-like emission could include a compact PWN contribution. The pulsed emission was detected in *γ*-rays by the *Fermi* Large Area Telescope (LAT) up to energies of 3 GeV, with a PL spectrum (Γ*γ* = 1.36 ± 0.23) and high *γ*-ray efficiency, $\\eta\\_\\gamma \\equiv L\\_\\gamma/\\dot{E} = 0.34\\ (d/3.8 {\\rm\\ kpc})^2$, higher than those of 75% of *γ*-ray pulsars (see Weltevrede et al. 2010 and Abdo et al. 2013).\nIn this paper we report on the results of a series of four deep observations of J1509 and its PWN. In Section 2 we describe the observations and data reduction. In Section 3 we present the images and spectral fit results. In Section 4 we discuss our findings, and in Section 5 we present our conclusions.\nlc R.A. (J2000.0) & 15 09 27.13 \nDecl. (J2000.0) & –58 50 56.1 \nEpoch of position (MJD) & 51463 \nGalactic longitude (deg) & 319.97 \nGalactic latitude (deg) & –0.62 \nSpin period, *P* (ms) & 88.9 \nPeriod derivative, $\\dot{P}$ (10− 14) & 0.92 \nDispersion measure, DM (pc cm− 3) & 137.7 \nDistance[1](#fn1), *d* (kpc) & 3.8 \nDistance from the Galactic plane, *z* (kpc) & 0.04 \nSurface magnetic field, *B**s* (1012 G) & 0.91 \nSpin-down power, $\\dot{E}$ (1035 erg s− 1) & 5.1 \nSpin-down age, $\\tau\\_{\\rm sd} = P/(2\\dot{P})$ (kyr) & 154 \nOBSERVATIONS & DATA REDUCTION\n=============================\nWe conducted four ≈ 90 ks observations of J1509 over the course of 15 months, from 2013 June 14 (MJD 56457) to 2014 September 23 (MJD 56923), totaling 374 ks (see Table [tbl-obs]). The target was placed on the ACIS-I detector, operated in Very Faint mode with a 3.2 s time resolution. We used the Interactive Analysis of Observations (CIAO) software package (ver. 4.6) and the *CXO* Calibration Data Base (CALDB, ver. 4.6.3) for data processing, and restricted the energy range to 0.5–8 keV for images and spectral analysis. We created exposure maps for each observation and an exposure-corrected image from the merged observation data using the CIAO routine merge\\_obs. CIAO’s wavdetect tool (a Mexican-hat wavelet source detection routine; Freeman et al. 2002) was used to detect field point sources, which were used for image alignment and then subtracted for diffuse emission analysis. We used the positions of 12 nearby stationary point sources in conjunction with the CIAO tool wcs\\_match[2](#fn2) to correct slight discrepancies in the astrometry of the event files. Spectra were extracted using CIAO’s specextract tool and analyzed using XSPEC (ver. 12.8.2). In all spectral fits we used the XSPEC wabs model with absorption cross sections from Morrison & McCammon (1983). To study the morphology of the emission in the immediate vicinity of the pulsar, we simulate the Point Spread Function (PSF) using ChaRT (Ray Trace) ver. 2 and MARX ver. 5.1.\nWe also analyzed two *XMM-Newton* observations (ObsIDs 0500630101 and 0500630301) carried out on 2008 January 29 and 2008 March 1. The EPIC (European Photon Imaging Camera) observations of PSR J1509–5850 employed the pn, MOS1, and MOS2 cameras in full frame imaging mode with the medium filter. We used the *XMM-Newton* Science Analysis Software (SAS, ver. 14.0) for the reprocessing of the observations and data reduction. After removal of background flares, the effective exposure times are 45ks (pn), 73ks (MOS1), and 73ks (MOS2). Because of the high background, we only use *XMM-Newton* data to analyze the pulsar spectrum (which, however, is more contaminated by the CN contribution than the *Chandra* spectrum due to the lower spatial resolution of *XMM-Newton*). To minimize the contamination from the CN and the background, we use circular extraction regions with radii of 10ʺ (pn) and 8ʺ (MOS1&2) centered on the pulsar (brightness profile peak). These extraction regions correspond to ≈ 50% of the encircled energy (Figures 6 and 7 in Section 3.2.1.1 of the *XMM-Newton* User Handbook[3](#fn3)). We use an *r* = 45ʺ circular background region close to the pulsar (on the same chip) but away from the CN. The resulting net source counts fraction is ≈ 90% for all instruments.\ncccccc 14523 & 2013 Jun 14 & 93.87 & 26 & 2 & 303∘ \n14524 & 2013 Sep 15 & 93.87 & 17 & 3 & 237∘ \n14525 & 2014 May 9 & 92.01 & 17 & 3 & 2∘ \n14526 & 2014 Sep 23 & 93.86 & 17 & 3 & 232∘ \nRESULTS\n=======\nPWN Morphology\n--------------\nThe merged image produced from four ACIS-I observations (see Figure [image-raw]) reveals a bright CN around the pulsar (514 net source counts) and two fainter large-scale elongated structures. In addition to the extended tail (hereafter, southern outflow; 3530 net counts) previously reported by K+08, the much deeper image shows, for the first time, another fainter structure (hereafter, the northern outflow; 1,760 net counts) apparently originating from the pulsar/CN area and visible up to 7ʹ (or 7.7 pc at *d* = 3.8 kpc) north of the pulsar. Below we consider the individual PWN elements in detail.\n### Small-scale Structure: Compact Nebula\nThe inset in Figure [image-raw] shows the CN structure. The peak of the X-ray brightness distribution (R.A.=$15^{\\rm h}09^{\\rm m}17\\fs136$, Decl.=$-58^{\\circ}50'56\\farcs19$) coincides (within $0\\farcs1$) with the radio position of the pulsar (R.A.=$15^{\\rm h}09^{\\rm m}27\\fs13(3)$, Decl.=$-58^{\\circ}50'56\\farcs1(5)''$ at MJD 51463, i.e., about 14 years before our X-ray observations). The centroiding uncertainty of the X-ray position, $\\approx0\\farcs04$ (calculated by celldetect), is much smaller than the typical uncertainty of the *Chandra* absolute astrometry ($\\approx0\\farcs4$ at 68% confidence[4](#fn4)). This puts an upper limit of $\\sim 0\\farcs5$ on the pulsar position shift during 14 years, corresponding to an upper limit ∼ 36 mas yr− 1 on proper motion.\nThe PSF modeling and subtraction does not reveal any additional structure within the core region. We also attempted PSF deconvolution using the CIAO tool arestore, but it did not produce anything new. The rest of the CN structure bears much resemblance to the Geminga PWN. Therefore, following Pavlov et al. (2006), we will call the three distinct features of the CN the lateral tails and the axial tail[5](#fn5). The lateral tails are noticeably longer ( ≈ 7ʺ − 10ʺ, or 0.13 − 0.18 pc at *d* = 3.8 kpc) than the axial tail ( ≈ 3ʺ, or 0.05 pc at *d* = 3.8 kpc).\nA wind from a fast-moving pulsar is expected to produce a termination shock with a bow-like (parabolic) structure ahead of the pulsar. However, we see no extended emission ahead of the point-like source at the leading tip of the CN (see inset in Figure 1). To constrain the stand-off distance *δ**s* of the possible unresolved shock apex ahead of the pulsar, we assume that the shape of the lateral tails represent the shock surface, fit the shape with a parabola, and estimate *δ**s* as the distance between the vertex of the fitted parabola and the position of the brightness peak. This approach gave us only upper limits on the stand-off distance, $\\delta\\_s \\lesssim 0\\farcs3$–10, depending on the region chosen for fitting. These fits, however, allowed us to estimate the radius of curvature at the parabola’s vertex, *δ**c* ∼ 1ʺ–2ʺ.\n[image-raw]\nExamination of the CN images from individual observations suggests variability associated with the lateral tails. This variability might be caused by the motion of blobs moving along the tails (see Figure [image-blobs-jets][6](#fn6)). To quantify the changes in the positions of the blobs (shown with green and yellow ellipses in Figure [image-blobs-jets]), we measure their positions (relative to the pulsar) in the images from four separate observations and plot them as a function of time in Figure [image-blobs-plot]. The positions are measured by running CIAO’s tool celldetect[7](#fn7). It is apparent from these plots that the green data points can be approximately connected by straight lines corresponding to steady motion. If we fit a steady motion model to the green data points (omitting the others from the fitting procedure), the resulting blob speeds are $v\\_{\\rm b,w} = (0.32 \\pm 0.06)c$ for the western lateral tail and $v\\_{\\rm b,e} = (0.20 \\pm 0.05)c$ for the eastern lateral tail.\nThis result should be taken with some caution because the significance of the individual blobs in the individual observations is low, ∼ 1.5*σ* − 2*σ*, when using a local background (i.e. within the lateral outflows) as calculated by celldetect). However, it increases to 3*σ*–4*σ* when the external (i.e., outside the CN) background region is used. Moreover, under the assumption of steady motion, the same blob is seen in multiple observations and hence the combined blob significances increase to 3*σ* for the local background choice, and to 7*σ*–8*σ* for the external background choice.\nWe have also compared the merged image of the CN with the older observation (ObsID 3515, ACIS-S, 2003 Feb 9). That observation was significantly shorter (40 ks), and the ACIS-S3 background contribution was much higher than that of the ACIS-I, making it difficult to resolve the individual components of the CN (see Figure 4 in K+08).\n[image-blobs-jets]\n[image-blobs-plot]\n[image-mid-scale]\n### Large-scale Structure\nFigure [image-mid-scale] shows the transition from the CN to the extended outflow south of the pulsar. One can see the lateral tails fading and apparently merging at ≈ 27ʺ from the pulsar. Beyond this distance the flow expands in the transverse direction and gives rise to the fainter *southern outflow* seen in Figure [image-large-scale] that presents the merged image of the large-scale diffuse emission with field point sources subtracted and exposure-map correction applied[8](#fn8). Unexpectedly, the same image clearly reveals a similarly shaped emission north-northwest of the pulsar, although this *northern outflow* is not discernible in the pulsar vicinity (Figure [image-mid-scale]).\nThe shapes of both large-scale structures, extending up to 6ʹ − 7ʹ from the pulsar, can be crudely described as two wedges with ≈ 17∘ opening angles and tips at (or near) the pulsar position. The southern outflow is approximately aligned with the CN symmetry axis, while the northern outflow is inclined to the CN symmetry axis by ≈ 33∘. Although the sizes and shapes of these outflows are similar, the distributions of surface brightness within the two structures are markedly different. The averaged (over the transverse direction) surface brightness of the southern outflow (see Figure [image-radio] for the profile extraction regions) appears to decrease almost linearly with distance from the pulsar (see Figure [figure-tail-brightness]), while the averaged surface brightness of the northern outflow does not appear to change significantly over its length (shown in Figure [figure-headlight-brightness]). Moreover, a substantially brighter region (hereafter referred to as the “spine”) is clearly seen within the southern outflow up to ≈ 4ʹ from the pulsar, while no such a feature is seen in the northern outflow.\n[image-large-scale]\nFigure [image-radio] also shows the extended radio emission reported by Ng et al. (2010), which stretches along the southern outflow. It becomes wider than the X-ray outflow beyond ≈ 3ʹ from the pulsar and then narrows down again beyond ≈ 5ʹ. The radio outflow is seen up to 10ʹ from of the pulsar, whereas the visible southern X-ray outflow becomes comparable with the background at about 7ʹ from the pulsar. No radio emission is seen from the northern outflow region in the same ATCA observation.\n[image-radio]\n[figure-tail-brightness]\n[figure-headlight-brightness]\nSpatially-resolved Spectra\n--------------------------\nWe measure the CN spectra in three regions shown in the Figure [image-raw] inset: (1) the bright pulsar/head region (within the $1\\farcs7$ radius white circle around the pulsar), which contains an unresolved CN “head” in addition to the pulsar, (2) the lateral tails (cyan polygons), and (3) the axial tail (the green polygon). For the southern outflow, we define the following regions (shown in Figure [image-large-scale]): the entire southern outflow (combination of green polygons), the spine (magenta polygon), and the southern outflow excluding the spine. We also divide the entire southern outflow region into 4 segments, with segment 1 being the closest to the pulsar and segment 4 being the farthest. For the northern outflow, we define two regions: the proximal and distal halves (cyan polygons). We fit all spectra with an absorbed PL model (unless specified otherwise) and list the fitting parameters Table [spatially-resolved-spectra]. For all regions, the spectra from different observations are fitted simultaneously, with the all parameters being tied together. Fitting the spine of the southern outflow yields an absorbing column density $N\\_{\\rm H}= (1.66\\pm0.19)\\times 10^{22} $ cm− 2 and a photon index Γ = 2.11 ± 0.07, with reduced chi-squared *χ**ν*2 = 1.01 for *ν* = 73 degrees of freedom (d.o.f.). Since the spectrum from this region fits the PL model well, and has a relatively small background contribution and a good statistic, we fix $N\\_{\\rm H}$ at the best-fit value obtained for this region while fitting the spectra from other regions which have poorer statistics[9](#fn9).\nFor the CN regions, we obtained very similar PL slopes for the pulsar/head and the lateral tails (Γ = 1.90 ± 0.12 and 1.80 ± 0.13, respectively) while the axial tail appears to have a harder spectrum (Γ = 1.43 ± 0.18). It is plausible that a substantial fraction of the pulsar/head emission comes from the neutron star and hence it may have a thermal component. Therefore, we attempted to fit the pulsar/head spectrum with a PL + blackbody (BB) model (with a fixed BB radius of 12 km). This resulted in a slightly better fit, with Γ = 1.77 ± 0.13, *k**T* = 103 ± 4 eV, and *χ**ν*2 = 0.77 for 53 d.o.f. However, the best-fit BB+PL model is preferred over the PL model at a 93% (1.8*σ*) confidence, according to the F-test (i.e., the BB component is not required by the data). We find an upper limit on the neutron star surface temperature by gradually increasing the temperature of the BB component (and re-fitting the PL component at each new temperature chosen) until the model becomes inconsistent with the data at a 3*σ* confidence level. This yields an upper limit *k**T* = 115 eV (or *T* = 1.3 × 106 K).\nAlthough the quality of the fits is good for all the CN regions (see Table [spatially-resolved-spectra]), the fits of the pulsar/head spectrum (see Figure [image-psr-spectra]) shows some systematic residuals centered at 3.5 keV (between 3.3–3.6 keV). To examine plausibility of the alleged spectral feature, we jointly fit the *XMM-Newton* spectra from two observations (see Figure [image-xmm-psr-spectra]) extracted from circular regions (*r* = 10ʺ and 8ʺ for the pn and MOS detectors; see Section 2) centered on the pulsar with the absorbed PL model and $N\\_{\\rm H}$ fixed at the same value as for the ACIS spectral fits. This gave a similar spectral slope, Γ = 1.9 ± 0.2, but a hint of an emission feature at ≈ 3.5 keV is seen in the pn spectrum from only one observation while the pn spectrum from the other observation shows a hint of absorption, which means that the feature is likely a statistical fluctuation. The MOS spectra do not have enough counts to look for spectral features.\n[image-psr-spectra]\n[image-xmm-psr-spectra]\n[image-tail-seg-1-spectra]\nThe spectrum of the spine (Γ = 2.11 ± 0.07) appears somewhat softer than that of the rest of the southern outflow (Γ = 1.78 ± 0.09). For the southern outflow segments 1–4, we obtain $\\Gamma\\_{\\rm S1}=1.99\\pm0.12$, $\\Gamma\\_{\\rm S2}=2.28\\pm0.16$, $\\Gamma\\_{\\rm S3}=2.06\\pm0.10$, and $\\Gamma\\_{\\rm S4}=1.82\\pm0.12$. We do not see any systematic softening of the spectrum along the southern emission despite its large extent; rather there is a hint of hardening from segment 2 to 4. Almost all fits are of good quality (the poorest quality fit, *χ**ν*2 = 1.26 for 46 d.o.f., being for segment 1, its spectrum is shown in Figure [image-tail-seg-1-spectra]).\nThe slope of the spectrum for the entire northern outflow (Γ = 1.81 ± 0.10) is similar to the spectrum of the southern outflow excluding the spine (Γ = 1.78 ± 0.09) although, unlike the southern outflow, the northern outflow spectrum shows a hint of softening with increasing distance from the pulsar (Γ changes from 1.50 ± 0.20 to 1.98 ± 0.11). This may explain the somewhat poorer quality of the PL fit to the spectrum of the entire northern outflow (see Table [spatially-resolved-spectra]). The photon indices for all regions as a function of distance from the pulsar are shown in Figure [image-gamma-fn-dist].\n[image-gamma-fn-dist]\nlrccccccc[H] Pulsar/head & 8.96 & 70.9 ± 2.8 & 1.90 ± 0.12 & 1.1 ± 0.14 & 0.82 (54) & 8.69 ± 0.50 & 2.54− 0.15+ 0.12 \nLateral tails & 88.1 & 5.84 ± 0.27 & 1.80 ± 0.13 & 0.85 ± 0.12 & 1.01 (35) & 7.50 ± 0.48 & 2.33− 0.15+ 0.12 \nAxial tail & 23.7 & 6.45 ± 0.54 & 1.43 ± 0.18 & 0.17 ± 0.04 & 790 (2046) & 2.16 ± 0.19 & 0.81− 0.10+ 0.07 \nSouthern outflow (entire) & 36778 & 0.096 ± 0.005 & 1.88 ± 0.06 & 11.9 ± 0.81 & 0.90 (105) & 98.7 ± 3.0 & 29.3 ± − 1.1+ 1.0 \nSpine & 4799 & 0.304 ± 0.011 & 2.11 ± 0.07 & 5.4 ± 0.40 & 1.01 (73) & 38.1 ± 1.4 & 9.71− 0.43+ 0.36 \nS. outflow (excl. spine) & 31979 & 0.066 ± 0.005 & 1.78 ± 0.09 & 6.8 ± 0.067 & 1.09 (83) & 61.1 ± 2.5 & 19.2− 1.1+ 1.0 \nS. outflow seg. 1 & 1805 & 0.434 ± 0.021 & 1.99 ± 0.12 & 1.6 ± 0.22 & 1.26 (46) & 12.7 ± 0.8 & 3.53− 0.21+ 0.20 \nS. outflow seg. 2 & 2224 & 0.250 ± 0.017 & 2.28 ± 0.16 & 2.2 ± 0.34 & 1.06 (35) & 13.9 ± 1.2 & 3.14− 0.33+ 0.23 \nS. outflow seg. 3 & 9463 & 0.183 ± 0.008 & 2.06 ± 0.10 & 4.6 ± 0.49 & 0.71 (42) & 33.8 ± 1.8 & 8.92− 0.45+ 0.59 \nS. outflow seg. 4 & 23162 & 0.082 ± 0.005 & 1.82 ± 0.12 & 3.6 ± 0.49 & 0.95 (73) & 31.6 ± 1.6 & 9.72− 0.65+ 0.5 \nN. outflow (entire) & 21714 & 0.081 ± 0.005 & 1.81 ± 0.10 & 4.8 ± 0.55 & 1.14 (60) & 42.0 ± 2.0 & 13.0− 0.50+ 0.63 \nN. outflow (proximal) & 5960 & 0.080 ± 0.009 & 1.50 ± 0.20 & 0.83 ± 0.20 & 0.95 (47) & 9.74 ± 0.83 & 3.53− 0.47+ 0.35 \nN. emission (distal) & 15754 & 0.082 ± 0.006 & 1.98 ± 0.11 & 4.3 ± 0.52 & 1.09 (67) & 33.1 ± 1.8 & 9.25− 0.63+ 0.56 \nDISCUSSION\n==========\nIn addition to the previously detected CN and southern outflow (K+08), the deep *Chandra* ACIS-I observations have revealed the previously unresolved fine structure of the CN. The overall shape of the CN and the southern outflow supports the earlier suggestion that they are due to the shocked wind of the pulsar moving in the north-northeast direction with a supersonic speed. The deeper image has also revealed a previously undetected faint structure, the northern outflow, extending ≈ 7ʹ to the north from the pulsar position. Below we consider possible interpretations for the different components of the J1509 PWN and compare them to those seen in other PWNe.\nThe Spatial and Spectral Structure of the PWN\n---------------------------------------------\n### Compact Nebula\nThe structure of the J1509 CN is very similar to that of the Geminga PWN (Pavlov et al. 2006, 2010; see Figure [bowshocks]), both in shape (one axial and two lateral tails) and in (projected) physical size ( ∼ 8 × 1017 cm for the lateral tails, and ∼ 2 × 1017 cm for the axial tail, for $d\\_{\\rm Gem} = 250$ pc and $d\\_{\\rm J1509}=3.8$ kpc). It is reasonable to expect that the physical sizes of compact nebulae with similar structures are proportional to $(\\dot{E}/\\rho v^2)^{1/2}$, where *ρ* is the ambient density, and *v* is the pulsar velocity. Assuming also similar angles *i* between the line of sight and the pulsar motion direction, and taking into account that $\\dot{E}\\_{\\rm J1509} \\simeq 15\\, \\dot{E}\\_{\\rm Gem}$, we can crudely estimate the speed of the J1509 pulsar:\n \n$$v\\_{\\rm J1509}\\sim790 \\left( \\frac{v\\_{\\rm Gem}}{200~{\\rm km~s}^{-1}} \\right) \\left(\\frac{\\rho\\_{\\rm Gem}}{\\rho\\_{\\rm J1509}}\\right)^{1/2}~{\\rm km~s}^{-1}.\n\\label{J1509velocity}$$ \nHowever, it is likely that Geminga lies within a local high temperature bubble, with *n* < 10− 2 cm− 3; in contrast, J1509 is in a high density, partly neutral medium (namely the H*α* halo; see Brownsberger & Romani 2014) with *n* > 1 cm− 3. An upper limit on the pulsar velocity follows from the lack of any significant shift of the pulsar position over ≈ 14 years, $\\delta\\lesssim 0\\farcs5$ (see Section 3.1.1), which gives $v\\_{\\rm J1509}\\,\\sin i< 640 (\\delta/0.5'') (d/3.8\\,{\\rm kpc})$ km s− 1. A possible lower limit on the pulsar velocity can be estimated from the lack of resolved extended emission ahead of the pulsar. If there is a (quasi-)isotropic component of the pulsar wind, it should produce a bullet-like termination shock with the apex (bullet head) distance $r\\_s = (\\dot{E}/4\\pi c\\rho v^2)^{1/2}$. For sin*i* = 1, the upper limit on the angular apex distance of $1\\farcs0$ (see Section 3.1.1) implies $r\\_s <5.7\\times10^{16}(d/3.8~{\\rm kpc})$ cm, $p\\_{\\rm amb} = \\rho\\_{\\rm J1509}v\\_{\\rm J1509}^{2}>4.2\\times10^{-10}(d/3.8~{\\rm kpc})^{-2}$ dyn cm− 2, and $v\\_{\\rm J1509}\\gtrsim160 n^{-1/2}(d/3.8~{\\rm kpc})^{-1}$ km s− 1, where *n* = *ρ*/*m**p* is the ISM number density.\nDespite the morphological similarities, there are also notable differences between the two PWNe. The spectra of the lateral tails are much softer in the J1509 CN than in the Geminga PWN (Γ = 1.8 ± 0.1 vs. Γ = 0.7–1.1, respectively), while the lateral tails’ luminosity, $L\\_{\\rm 0.5-8\\,keV} = (7.5\\pm 0.5)\\times 10^{31} (d/3.8\\,{\\rm kpc})^2$ erg s− 1 for J1509, is a factor of ∼ 200 higher[10](#fn10). The higher luminosity could be attributed to the larger $\\dot{E}$, but the radiative X-ray efficiency for J1509’s lateral tails is also significantly higher: $\\eta\\_{\\rm 0.5-8\\,keV} \\equiv L\\_{\\rm 0.5-8\\,keV}/\\dot{E}\\simeq1.5\\times10^{-4}(d/3.8\\,{\\rm kpc})^2$ for J1509 vs. $1.2\\times 10^{-5} (d/0.25\\,{\\rm kpc})^2$ for Geminga. Another difference is that there is the presence of faint diffuse emission between the lateral tails (beyond the extent of the shorter axial tail) in the J1509 CN (see the inset in Figure [image-raw]), while such emission is not detected in the Geminga PWN (Posselt et al., in prep.; Pavlov et al. 2010).\nNumerical simulations of the synchrotron brightness distributions from PWNe of supersonically moving pulsars with isotropic wind ejection (Bucciantini et al. 2005) predict filled structures with the brightness reaching its maximum ahead of the moving pulsar and gradually decreasing behind the pulsar (see Figures 4 and 5 in Bucciantini et al. 2005). These simulated synchrotron brightness maps do not resemble the observed structures of the CNe around J1509 and Geminga. One reason could be that the magnetic field structure is substantially different from that assumed in the simulations. Bucciantini et al. (2005) mention that the magnetic field may be compressed in narrow layers near the contact discontinuity surface, which, in principle, could produce limb-brightened structures, depending on the amount of Doppler boosting (caused by a high bulk flow velocity) and the magnetic field geometry. Another reason could be that the assumption of isotropic wind injection used in the simulations is not valid for real pulsars. For instance, one could interpret the lateral tails as jets or an equatorial outflow bent back by the ram pressure. However, at least the jet interpretation looks unrealistic as it requires stretching some of the parameters to their extremities. Indeed, a jet produced by a moving pulsar bends because the jet flow acquires a momentum from the ambient medium; it becomes comparable to the original jet flow momentum over the bending length scale (curvature radius)[11](#fn11) $l\\_b\\sim\\xi\\_j\\dot{E}/(2 c r\\_j \\rho v^2 \\sin\\alpha)$, where *ξ**j* < 1 is the fraction of the spin-down power $\\dot{E}$ that goes into the jet, *r**j* is the jet radius, and *α* is the angle between the initial jet direction and the pulsar’s velocity. For J1509, we obtain $l\\_b < 3.6 \\times10^{17} \\xi\\_j (r\\_j/5.7\\times10^{16} {\\rm cm})^{-1} (\\sin \\alpha)^{-1} (d/3.8\\ {\\rm kpc})^2 $ cm. Therefore, for a jet with the angular width *δ**j* ∼ 1ʺ, a significant bending on a scale of the observed curvature radius *δ**c* ∼ 1ʺ–2ʺ (see Section 3.1.1) implies that most of the spin-down power goes into the two jets. However, observations of PWNe with both jet and equatorial components suggest *ξ**j* ≪ 1 (e.g., Kargaltsev & Pavlov 2008; Pavlov et al. 2003). Therefore, the jet interpretation of the lateral tails would imply that in some pulsars jets can be more powerful than equatorial outflows. More extensive modeling and simulations are needed to see whether it is possible to reproduce the observed CN morphologies for other assumptions on the angular distribution of the pulsar wind injection.\n[bowshocks]\nWe also note that the morphologies of the Geminga and J1509 CNe are markedly different from those of other bow-shock PWNe. In particular, others exhibit a CN with a local minimum in brightness in the pulsar vicinity (i.e., behind the pulsar; PWNe of PSRs J0357+3205 and J1101–6101), or exhibit a filled CN morphology (PWNe of PSRs J1747–2958, J1741–2054, and B0355+54). The differences, however, are not limited to X-ray morphologies only. For instance, the radio counterparts of the PWNe associated with PSRs J1509–5850 (Ng et al. 2010), J1101–6101 (Pavan et al. 2014), and J1747–2958 (the Mouse; Yusef-Zadeh & Gaensler 2005), the radio brightness distributions are also quite different. While the radio brightness in the Mouse is maximal near the pulsar (similar to X-rays; Gaensler et al. 2004), in the other two PWNe the radio brightness is low near the pulsar and becomes higher further away (see Figure [image-radio]). We also note that the X-ray morphologies of most of the other CNe in Figure [bowshocks] do not bear much resemblance to the simulated images, except perhaps for the Mouse and J1741–2054. Ng et al. (2010) mentioned an open cluster, Ruprecht 112 (C1453–623; Alter et al. 1970), about 4∘ southwest of the current pulsar position, as a possible birth place of the pulsar. At the distance of 3.8 kpc and the pulsar’s spin-down age of 150 kyr, the required pulsar velocity would be ∼ 1700 km s− 1, much higher than the upper limit of 640 km s− 1 on the transverse velocity inferred from the lack of detectable change of the pulsar position in 14 years. Based on the comparison of the older X-ray image with the shape of the termination shock “bullet” seen in numerical simulations, K+08 suggested that the pulsar velocity should be ∼ 300 km s− 1, which is still within the allowed velocity range (see above). However, since the deeper X-ray images reveal substantial differences between the CN structure and the simulations, this velocity estimate should be taken with caution.\n### Large-scale PWN Structure\nThe most puzzling new feature of the J1509 PWN is the large-scale northern outflow seen up to ∼ 7ʹ almost ahead of the pulsar. One could assume that the northern outflow is a powerful relativistic jet piercing the bow shock near the pulsar. The lack of a counter-jet could be explained by Doppler boosting, but there are other problems with the jet interpretation. According to the estimates in Section 4.1.1, the bending length for a jet is very small, even at *ξ**j* ∼ 1 and sin*α* ∼ 0.5, compared to the length of the northern outflow that shows no bending. It is also difficult to explain the relative faintness of the northern outflow closer to the pulsar (but outside the CN) in the jet scenario. Finally, the X-ray radiation efficiency of the northern outflow, *η*0.5 − 8 *k**e**V* ∼ 0.8 × 10− 3, substantially exceeds the efficiencies of jets of younger PWNe with clear torus-jet morphology. Therefore, we discard the jet interpretation for the northern outflow.\nSimilar arguments have led Bandiera (2008) to conclude that the extended X-ray feature accompanying the Guitar nebula (which also exhibits a large-scale outflow misaligned with the pulsar motion direction; see Figure [image-guitar] and Johnson & Wang 2010) cannot be interpreted as a hydrodynamical jet. Following Bandiera (2008), we consider a scenario in which particles with the highest energies leak from the termination shock apex into the ISM, and travel along the ordered ISM magnetic field. Bandiera (2008) suggested that the length of the Guitar PWN feature, ∼ 1.4 pc, is on the order of the radiation length (i.e., the distance over which particles lose a substantial fraction of their energy to synchrotron radiation). However, this assumption implies a propagation speed of only a few thousands km s− 1, instead of the speed of light, as expected for particles with large (bulk) Lorentz factors. Bandiera (2008) argued that the turbulent component of the magnetic field would provide a scattering mechanism and slow down particle propagation (which can then be considered as diffusion) along the ISM magnetic field to the required non-relativistic speed. The observed hint of softening of the X-ray spectrum in J1509’s northern outflow can be considered as supporting this scenario. A problem with this scenario is that in all three cases with prominent misaligned (with respect to pulsar motion) X-ray features (the Guitar, J1509, and J1101–6101), the features appear to be one-sided or show very faint “counter-features”, which means that there is a preferred direction for the escape of the high-energy particles. The cause of this asymmetry is currently unclear. On the other hand, since the particles escaping from the termination shock apex into the ISM must be the most energetic ones (those with large gyration radii; see Bandiera 2008), the lack of radio synchrotron emission can be naturally explained by the deficit of lower energy particles.\nThe southern outflow is aligned with the CN symmetry axis and clearly represents a continuation of the CN (see Figure [image-mid-scale]). Therefore, we can assume that at least the brighter part of the southern outflow can be interpreted as a tail filled with the shocked pulsar wind confined by the ram pressure. The lack of evidence for synchrotron burn-off in the X-ray spectra suggests a high flow speed and/or a low magnetic field. Alternatively, there could be some “reheating” due to in-situ conversion of magnetic field energy into particle energy, e.g., via turbulent processes and accompanying reconnection, which might explain the hint of spectral hardening at large distances from the pulsar (see Figure [image-gamma-fn-dist].) Indeed, Ng et al. (2010) reported the presence of substantial turbulent magnetic field components in the tail using radio polarimetry data.\nThere are, however, some puzzling aspects associated with the southern outflow. One is that the radio emission brightens with distance from the pulsar and becomes broader than the X-ray emission beyond ≈ 3ʹ (see Figure [image-radio]). Such behavior could possibly be caused by entrainment of ambient matter (see also Morlino et al. 2015), which decreases the flow speed, increases the transverse size of the tail, and leads to adiabatic cooling of the outflowing matter. This tentative explanation remains to be checked with detailed modeling of entrainment effects.\nAnother puzzling aspect is the similarity of the wedge-like shapes of the fainter part of the southern outflow (i.e., the southern outflow excluding the spine) and the northern outflow. Given the qualitative differences in the above interpretations of the northern and southern outflows, these similarities could be attributed to merely a chance coincidence. Alternatively, the structural and spectral similarities could be explained if both the northern emission and the southern emission (excluding the spine) are caused by very energetic particles that leaked from the termination shock apex in opposite directions and diffused along the curved ISM magnetic field (the curvature of the magnetic field is required to explain the different orientations of the “wedges”). In this scenario, the spine (whose spectrum is slightly softer than that of the rest of the southern outflow) is still interpreted as the pulsar tail, coincidentally projected onto the sky within the southern outflow. This interpretation may look somewhat artificial, but it cannot be firmly excluded with the currently available data.\nThe spectra of the large-scale outflows measured in several regions allow one to crudely estimate magnetic fields in these regions. For a PL synchrotron spectrum with photon index Γ, the magnetic field at a given magnetization parameter *σ* = *w**B*/*w**e* (ratio of energy densities of magnetic field and relativistic electrons) depends on the ratio of the luminosity *L*(*ν**m*, *ν**M*), measured in the *ν**m* < *ν* < *ν**M* frequency range, to the radiating volume *V*. A simple generalization of Equation (7.14) in Pacholczyk (1970)[12](#fn12) to the case of arbitrary *σ* gives\n \n$$B=\\left[\\frac{L(\\nu\\_m,\\nu\\_M)\\sigma}{\\mathcal{A} V} \\frac{\\Gamma-2}{\\Gamma-1.5}\\frac{\\nu\\_1^{1.5-\\Gamma}-\\nu\\_2^{1.5-\\Gamma}}{\\nu\\_m^{2-\\Gamma}-\\nu\\_M^{2-\\Gamma}} \\right]^{2/7}.\n\\label{synch\\_magn\\_field}$$ \nIn this equation, *ν*1 and *ν*2 are the characteristic synchrotron frequencies ($\\nu\\_{\\rm syn}\\simeq eB\\gamma^2/2\\pi mc$) corresponding to the boundary energies (*γ*1*m**e**c*2 and *γ*2*m**e**c*2) of the electron spectrum (*d**N**e*/*d**γ* ∝ *γ*− *p* ∝ *γ*− 2Γ + 1; *γ*1 < *γ* < *γ*2), and A = 21/2*e*7/2/18*π*1/2*m**e*5/2*c*9/2. The photon indices and 0.5–8 keV luminosities have been measured from the spectral fits (see Table [spatially-resolved-spectra]). We estimate the radiating volumes from the images assuming conical shapes of the outflows. Since *L* ∝ *d*2 and *V* ∝ *d*3, the magnetic field estimated from Equation ([synchmagnfield]) weakly depends on the assumed distance, *B* ∝ *d*− 2/7. The boundary frequencies *ν*1 and *ν*2 are unknown. The exact value of *ν*2 is not really important as long as (*ν*1/*ν*2)Γ − 1.5 ≪ 1, so we choose a plausible (but arbitrary) value *h**ν*2 = 100 keV. The unknown frequency *ν*1 is the main source of uncertainty of the magnetic field at a given *σ*, for the measured spectral slopes. For the southern outflow, we can at least estimate a lower limit on *ν*1 from the requirement that extrapolation of the X-ray PL spectrum to lower frequencies does not exceed the harder radio spectrum ($\\Gamma\\_{\\rm rad}=1.26\\pm 0.04$; Ng et al. 2010), but even the limit, $\\nu\\_1\\gtrsim 10^{10}$–1012 Hz, is not very certain because of the uncertainty of the X-ray photon indices (and because the slope of the radio spectrum was measured in a broader region). Since the northern outflow was not detected in the radio, we can only guess that *ν*1 is higher there than in the southern outflow (which would be consistent with our hypothesis that the northern outflow emission is due to leaked particles with very high energies). Given these uncertainties, we estimate the magnetic fields assuming *ν*1 = 100 GHz for all the regions. If future multiwavelength observations yield different estimates of *ν*1, the magnetic field could be scaled (approximately) as *B* ∝ *ν*1(3 − 2Γ)/7 (if (*ν*1/*ν*2)Γ − 1.5 ≪ 1).\nThe equipartition magnetic fields, $B\\_{\\rm eq}=B\\sigma^{-2/7}$, for different parts of the J1509 southern and northern outflows are given in Table [b-fields], for the above-described choice of boundary frequencies and *d* = 3.8 kpc. Although the actual magnetic field strengths can differ from those in Table [b-fields] by a factor of a few, we can at least hope that their relative values reflect the spatial dependence of the magnetic field in the emitting regions. Surprisingly, the magnetic field in the southern outflow shows a hint of non-monotonic dependence on distance from the pulsar: the field grows from segment 1 to segment 2, and then decreases outwards (in concert with the photon index Γ and slope *p* of electron spectrum). The reason for such behavior is unclear, and we cannot exclude that it is simply due to a variation of unknown parameters (e.g., *ν*1 and/or *σ* along the outflow).\nSince the southern outflow was detected in radio (Ng et al. 2010), the radio spectrum ($\\Gamma\\_{\\rm rad}\\approx 1.26$) can also be used for magnetic field estimates. If the radio and X-ray spectra are generated in different regions (e.g., the radio emission comes from a “sheath” around an X-ray emitting “spine” of the outflow), the magnetic field in the radio-emitting region should be estimated solely from the radio spectrum, which gives $B\\_{\\rm rad} \\simeq 13\\, \\sigma\\_{\\rm rad}^{2/7}$ *μ*G for $L\\_{\\rm rad}(\\nu\\_m,\\nu\\_M) = 1.5\\times 10^{32}$ erg s− 1, $V\\_{\\rm rad} = 5\\times 10^{56}$ cm3, *ν*1 = *ν**m* = 10 MHz, *ν*2 = *ν**M* = 100 GHz ($\\sigma\\_{\\rm rad}$ is the magnetization in this region). This value of $B\\_{\\rm rad}$ is close to those determined from the X-ray spectra in segments 3 and 4 (assuming $\\sigma\\_{\\rm rad} \\sim \\sigma$). If the radio spectrum in the X-ray detected part of the southern outflow is generated in the same region as the X-ray emission, then the mutiwavelength (radio to X-rays) spectrum can be considered as a broken PL. The magnetic field from such a spectrum could be estimated for a given magnetization $\\sigma = (\\sigma\\_{\\rm rad}^{-1} + \\sigma\\_X^{-1})^{-1}$, where $\\sigma\\_{\\rm rad}$ and *σ**X* (each of them is greater than *σ*) are magnetizations corresponding to the energy densities of particles responsible for radiation below and above the break frequency, respectively. In this case, the equipartition field would be slightly higher than those estimated separately from the X-ray and radio spectra, but the difference would be within the uncertainties.\nAccording to Table [b-fields], the difference of the equipartition fields in the two segments of the northern outflow is within the $B\\_{\\rm eq}$ uncertainties caused by the uncertainties of the spectral slopes. The characteristic equipartition field, $B\\_{\\rm eq}\\sim 5$–10 *μ*G, is lower than typical values for the southern outflow[13](#fn13). However, if the X-ray emission from the northern outflow is indeed due to particles of very high energy leaked from the termination shock apex, then *σ* can be very different from unity, and *B* very different from $B\\_{\\rm eq}$.\nThe magnetic field strengths can be used to estimate the Lorentz factors, gyration radii, and characteristic synchrotron cooling times of electrons radiating synchrotron photons with energy *E*:\n \n$$\\gamma\\sim 10^{8}(E/1\\,{\\rm keV})^{1/2}(B/20\\,\\mu{\\rm G})^{-1/2},$$ \n \n$$r\\_g\\sim 10^{14}(E/1\\,{\\rm keV})^{1/2}(B/20\\,\\mu{\\rm G})^{-3/2}~{\\rm cm},$$ \nand\n \n$$t\\_{\\rm syn}\\sim 400 (E/1\\,{\\rm keV})^{-1/2}(B/20\\,\\mu{\\rm G})^{-3/2}~{\\rm yr}\\,.$$ \nThe spatially-resolved X-ray spectra of the outflows allow one to crudely estimate (or constrain) characteristic flow velocities. For instance, if the apparent spectral softening in the northern outflow is real, the flow velocity can be estimated as $v\\_{\\rm flow} \\sim l/\\tau\\_{\\rm syn}\\sim 3000 (l/6\\,{\\rm pc}) (B/7\\,\\mu{\\rm G})^{3/2}$ km s− 1, where *l* is the spatial scale of the alleged softening. This is, of course, just an order-of-magnitude estimate, but the fact that $v\\_{\\rm flow} \\ll c$ is consistent with the scenario by Bandiera (2008), which suggests that propagation of the leaked high-energy particles is slowed down by diffusion in a turbulent magnetic field (see above).\nKnowing the equipartition magnetic field and the flow velocity, we can estimate the energy density,\n \n$$\\begin{aligned}\nw & = & w\\_B + w\\_e = [B\\_{\\rm eq}^2/(8\\pi)](\\sigma^{4/7}+\\sigma^{-3/7})^2 \\\\\n\\nonumber\n & \\sim & 2\\times 10^{-12} (B\\_{\\rm eq}/7\\,\\mu{\\rm G})^2 (\\sigma^{4/7}+\\sigma^{-3/7})^2\\,\\,{\\rm erg}\\,{\\rm cm}^{-3}, \n\\label{energy\\_density\\_north}\\end{aligned}$$ \nand the energy transfer rate ( ∼ energy injection rate),\n \n$$\\begin{aligned}\n\\dot{E}\\_{\\rm North} & \\sim & (w+p) S v\\_{\\rm flow} \n\\label{energy\\_injection\\_north}\\\\\n\\nonumber\n& \\sim & 4\\times 10^{33} (B\\_{\\rm eq}/7\\,\\mu{\\rm G})^2 (S/3\\times 10^{36}\\, {\\rm cm}^2)\\\\\n\\nonumber\n&\\times & (v\\_{\\rm flow}/3000\\,{\\rm km}\\,{\\rm s}^{-1}) (\\sigma^{4/7}+(2/3)\\sigma^{-3/7})^2\\,\\,{\\rm erg}\\,{\\rm s}^{-1},\\end{aligned}$$ \nfor the northern outflow, **where *w* + *p* is the enthalpy density, *p* = *w**B* + *w**e*/3 is the pressure of the relativistic particles and magnetic field, and *S* is the characteristic cross-sectional area.** As expected, $\\dot{E}\\_{\\rm North}$ is a small fraction of the pulsar’s spin-down power ($\\dot{E}\\_{\\rm North}\\sim 10^{-2}\\dot{E}$ for *σ* ∼ 1). On the other hand, $\\dot{E}\\_{\\rm North}$ exceeds the 0.5–8 keV luminosity of the northern outflow (by a factor of 7 at the parameters chosen and *σ* ∼ 1, but the value of this factor strongly depends on the poorly known outflow parameters).\nIn the southern outflow we see no spectral softening with increasing distance from the pulsar (i.e., no synchrotron cooling). If there is no in-situ particle acceleration in the outflow, the lack of cooling sets a lower limit on the flow velocity of a few thousand km s− 1 (K+08), well above the upper limit on the pulsar velocity. The value of the limiting speed is somewhat uncertain because of the magnetic field uncertainties. Moreover, the hint of spectral hardening with increasing distance from the pulsar (see Figure [image-gamma-fn-dist]) suggests that particle acceleration may occur in the southern outflow (e.g., due to magnetic field reconnection), which would invalidate the velocity estimates based on passive cooling. An upper limit of a few tens of thousands km s− 1 on the characteristic flow velocity can be estimated from the condition $\\dot{E}\\_{\\rm South}< \\dot{E}$, where $\\dot{E}\\_{\\rm South}\\sim (w+p) S v\\_{\\rm flow}$, similar to Equation (7).\nlcccc[H] S. outflow seg. 1 & 2.98 ± 0.24 &1.3 & 2.6 & 36− 12+ 23 \nS. outflow seg. 2 & 3.56 ± 0.32 & 1.4 & 7.4 & 89− 42+ 16 \nS. outflow seg. 3 & 3.12 ± 0.20 & 3.4 & 42 & 29− 9+ 14 \nS. outflow seg. 4 & 2.64 ± 0.24 & 3.2 & 125 & 9− 3+ 4 \nN. outflow (proximal) & 2.00 ± 0.40 & 1.0 & 9.1 & 5− 1+ 4 \nN. outflow (distal) & 2.96 ± 0.22 & 3.3 & 76 & 10− 2+ 4 \nAs we mentioned above, the J1509 and Guitar PWNe are not the only ones which exhibit puzzling asymmetric structures misaligned with their apparent pulsar motion directions. Two other examples are the outflows associated with PSR J1101–6101 (the Lighthouse Nebula; Pavan et al. 2014) and PSR J2055+2539 (Marelli et al. 2015). The former one shows a particularly remarkable structure. The deep 300 ks ACIS image of the Lighthouse Nebula shown in Figure [image-lighthouse] reveals a long ( ∼ 11 pc) jet-like outflow that originates near the pulsar, bends back in the direction opposite to that of the pulsars’ proper motion, and then sharply turns in a direction nearly orthogonal to the proper motion direction (see also Pavan et al. 2015). This outflow can possibly be interpreted in the framework of the Bandiera (2008) scenario, similar to Guitar and J1509 misaligned outflows. **In this scenario, the external (ISM) magnetic field lines are illuminated by the escaping wind particles traveling along the distorted magnetic field (cf. magnetic “draping”; Lyutikov 2006 and Dursi & Pfrommer 2008).** Additionally, the PWN of PSR B0355+54 also shows hints of a misaligned outflow (see Figure [bowshocks]), although in this case it is rather faint and is seen on both sides of the bow shock (perhaps due to the lower speed of the pulsar). An in-depth discussion of this PWN will be given in a separate paper (Klingler et al., in prep.).\n[image-guitar]\n[image-lighthouse]\nSummary\n=======\nThe most important findings from our deep ACIS-I observations of J1509 can be summarized as follows.\nWe were able to resolve the fine structure of the CN in the immediate pulsar vicinity. The CN consists of two lateral tails, apparently emerging from the pulsar and resolved up to 0.2 pc behind the pulsar, and a short, 0.05 pc, axial tail along the CN symmetry axis. The CN structure and size are remarkably similar to those of the Geminga, PWN but different from several other bow shock PWNe. Such PWN morphology challenges the existing numerical simulations of synchrotron emission from the PWNe of supersonically moving pulsars. The morphological differences from other PWNe could be attributed to different initial wind magnetizations or, more likely, to anisotropic wind ejection, which affects the geometry of the magnetic field downstream of the termination shock and the PWN morphology.\nWe found that the CN narrows beyond 0.2–0.3 pc, reaching the minimum width (where the lateral tails apparently merge) at about 0.5 pc behind the pulsar. Beyond this distance the flow expands and forms the diverging 7 pc long southern outflow, discovered by K+08 who interpreted it as a tail of shocked pulsar wind confined by the ram pressure of the ambient medium. The PL spectra of the southern outflow do not show softening (synchrotron cooling) with increasing distance from the pulsar, which is expected for such long tails. The lack of softening (with a hint of hardening) may suggest that particle acceleration occurs within the tail.\nWe also discovered a previously undetected wedge-like northern outflow, seen up to about 6 pc, with the wedge axis inclined at an angle of about 33∘ from the pulsar’s proper motion direction, inferred from the CN symmetry axis. This outflow is possibly formed by energetic pulsar wind particles escaping from the apex of the bow shock and diffusing along the ISM magnetic field. The equipartion magnetic field in this outflow is about a few *μ*G, the flow speed is a few thousand km s− 1, and the energy injection rate is $\\gtrsim 10^{33}$ erg s− 1. The discovery of the northern outflow puts J1509 into the (so far) small group of supersonically moving pulsars with misaligned outflows (i.e., outflows inclined to the pulsar trajectory at substantial angles). To better understand the nature of the misaligned outflows, further deep multiwavelength observations with high angular resolution are needed.\n*Facilities:*\nWe would like to thank Maxim Lyutikov, Andrei Bykov, Giovanni Morlino, Marina Romanova, and John Hewitt for the very insightful discussions, as well as our referee, Roger Romani, for the helpful suggestions. Support for this work was provided by the National Aeronautics and Space Administration through *Chandra* Award Number G03-14082 issued by the *Chandra* X-ray Observatory Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of the National Aeronautics Space Administration under contract NAS8-03060. The work was also partly supported by NASA grant NNX08AD71G.\n \nAbdo, A. A., Ajello, M., Allafort, A., et al. 2013,, 208, 17\nAuchettl, K., Slane, P., Romani, R. W., et al. 2015,, 802, 68\nBandiera, R. 2008,, 490, L3\nBrownsberger, S., & Romani, R. W. 2014,, 784, 154\nBucciantini, N., Amato, E., & Del Zanna, L. 2005,, 434, 189\nDe Luca, A., Marelli, M., Mignani, R. P., et al. 2011,, 733, 104\nDursi, L. J., & Pfrommer, C. 2008,, 677, 993\nFreeman, P. E., Kashyap, V., Rosner, R., & Lamb, D. Q. 2002,, 138, 185\nGaensler, B. M., van der Swaluw, E., Camilo, F., et al. 2004,, 616, 383\nHui, C. Y., & Becker, W. 2007,, 467, 1209\nJohnson, S. P., & Wang, Q. D. 2010,, 408, 1216\nKargaltsev, O., & Pavlov, G. G. 2008, 40 Years of Pulsars: Millisecond Pulsars, Magnetars and More, 983, 171\nKargaltsev, O., Misanovic, Z., Pavlov, G. G., Wong, J. A., & Garmire, G. P. 2008,, 684, 542\nKramer, M., Bell, J. F., Manchester, R. N., et al. 2003,, 342, 1299\nLyutikov, M. 2006,, 373, 73\nMarelli, M., Pizzocaro, D., De Luca, A., et al. 2015, arXiv:1509.05833\nMcGowan, K. E., Vestrand, W. T., Kennea, J. A., et al. 2006,, 647, 1300\nMorlino, G., Lyutikov, M., & Vorster, M. 2015,, 454, 3886\nMorrison, R., & McCammon, D. 1983,, 270, 119\nNg, C.-Y., Gaensler, B. M., Chatterjee, S., & Johnston, S. 2010,, 712, 596\nPacholczyk, A. G. 1970, Series of Books in Astronomy and Astrophysics, San Francisco: Freeman, 1970,\nPavan, L., Bordas, P., Pühlhofer, G., et al. 2014,, 562, A122\nPavan, L., Pühlhofer, G., Bordas, P., et al. 2015, arXiv:1511.01944\nPavlov, G. G., Teter, M. A., Kargaltsev, O., & Sanwal, D. 2003,, 591, 1157\nPavlov, G. G., Sanwal, D., & Zavlin, V. E. 2006,, 643, 1146\nPavlov, G. G., Bhattacharyya, S., & Zavlin, V. E. 2010,, 715, 66\nTaylor, J. H., & Cordes, J. M. 1993,, 411, 674\nTomsick, J. A., Bodaghee, A., Rodriguez, J., et al. 2012,, 750, L39\nWeltevrede, P., Abdo, A. A., Ackermann, M., et al. 2010,, 708, 1426\nYusef-Zadeh, F., & Gaensler, B. M. 2005, Advances in Space Research, 35, 1129\n---\n1. The distance estimate is based on the dispersion measure and the Galactic electron density distribution the model by Taylor & Cordes (1993). The model by Cordes & Lazio (2002) places the pulsar at a distance of 2.6 kpc. However, the smaller distance implies a lower $N\\_{\\rm H}$ than observed, and, based on the J1509’s Galactic coordinates, would place the pulsar in between the Sagittarius and Scutum-Centaurus arms, whereas the distance of 3.8 kpc places J1509 in the middle of the Scutum-Centaurus arm.[↩](#fnref1)\n2. http://cxc.harvard.edu/ciao/ahelp/wcs\\_match.html[↩](#fnref2)\n3. http://xmm.esac.esa.int/external/xmm\\_user\\_support/ \ndocumentation/uhb\\_2.1/node17.html[↩](#fnref3)\n4. http://cxc.harvard.edu/cal/ASPECT/celmon/[↩](#fnref4)\n5. The naming convention used here does not imply any physical interpretation (see Section 4 for discussion)[↩](#fnref5)\n6. See also the movie at [http://home.gwu.edu/ ∼ kargaltsev/ \nj1509movie.mov](http"},"data_id":{"kind":"string","value":"arxiv_0184770"}}},{"rowIdx":46197,"cells":{"content":{"kind":"string","value":"Order algebras: a quantitative model of interaction\n===================================================\n### July 7, 2011\nA quantitative model of concurrent interaction is introduced. The basic objects are linear combinations of partial order relations, acted upon by a group of permutations that represents potential non-determinism in synchronisation. This algebraic structure is shown to provide faithful interpretations of finitary process algebras, for an extension of the standard notion of testing semantics, leading to a model that is both denotational (in the sense that the internal workings of processes are ignored) and non-interleaving. Constructions on algebras and their subspaces enjoy a good structure that make them (nearly) a model of differential linear logic, showing that the underlying approach to the representation of non-determinism as linear combinations is the same.\nIntroduction\n============\nThe theory of concurrency has developed several very different models for interactive processes, focusing on different aspects of computation. Among those, process calculi are an appealing framework, because the formal language approach is well suited to modular reasoning, allowing to study sophisticated systems by means of abstract programming primitives for which powerful theoretical tools can be developed. They are also the setting of choice for extending the vast body of results of proof theory to less sequential settings. However, the vast majority of the semantic studies on process calculi like the -calculus have focused on the so-called interleaving operational semantics, which is the basic definition of the dynamic of a process: the interaction of a program with its environment is reduced to possible sequences of transitions, thus considering that parallel composition of program components is merely an abstraction that represents all possible ways of combining several sequential processes into one. In Hoare’s seminal work on Communicating Sequential Processes , this is even an explicit design choice.\nThere is clearly something unsatisfactory in this state of things. Although sophisticated theories have been established for interleaving semantics, most of which are based on various forms of bisimulation, they fundamentally forget the crucial (and obvious) fact that concurrent processes are intended to model situations where some events may occur independently, and event explicitly in parallel. This fact is well known, and the search for non interleaving semantics for process calculi is an active field of research, with fruitful interaction with proof theory and denotational semantics. Recently, the old idea of Winskel’s interpretation of CCS in event structures has been revisited by Crafa, Varacca and Yoshida to provide an actually non-interleaving operational semantics for the -calculus, using extensions of event structures . Event structures are also one of the starting points of extensions of game semantics to non-sequential frameworks, for instance in asynchronous games and concurrent extensions of ludics . In a neighbouring line of research, the recent differential extension of linear logic is known to be expressive enough to represent the dynamics of the -calculus . However the implications of this fact in the search for denotational semantics of the -calculus are still unclear, in particular the quantitative contents of differential linear logic lacks a proper status in concurrency.\nThis paper presents a new semantic framework that addresses this question, following previous work by the author on the search for algebraically pleasant denotational semantics of process calculi. The first step was to introduce in the -calculus an additive structure (a formal sum with zero) that represents pure non-determinism, and this technique proved efficient enough to provide a readiness trace semantics with a complete axiomatization of equivalence for finite terms. The second step presented here further extends the space of processes with arbitrary linear combinations, giving a meaning to these combinations in terms of quantitative testing. This introduction of scalar coefficients was not possible in the interleaving case, because of the combinatorial explosion that arose even when simply composing independent traces; moving to a non-interleaving setting through a quotient by homotopy of executions is the solution to this problem. Growing the space of processes to get more algebraic structure is also motivated by the idea that better structured semantics gives cleaner mathematical foundations for the object of study, in the hope that the obtained theory will be reusable for different purposes and that it will benefit from existing mathematical tools.\n#### Informal description\nAn order algebra is defined on an *arena*, which represents the set of all observable events that may occur in the execution of a process. Basic interaction scenarii, named *plays*, are partial order relations over finite subsets of the arena. We then postulate two principles:\n* Linear combinations are used to represent non-determinism, which, although not the defining feature, is an unavoidable effect in concurrent interaction. Coefficients form the quantitative part of the model, the first thing they represent is how many times a given play may occur in a given situation. They can also represent more subtle things, like under which conditions a given play is relevant. This allows for the representation of features such as probabilistic choice, in which case coefficients will be random variables. In general, coefficients are taken in an arbitrary semiring with some additional properties. This use of linear combinations is a novelty of the differential λ-calculus and subsequent work , although a decomposition of processes as formal linear combinations was first proposed by Boreale and Gadducci , albeit without the quantitative aspect we develop here.\n* The fact that some events may be indistinguishable by the environment of a process, typically different inputs (or outputs) on the same channel, is represented by a group action over the arena. Each element of the group acts as a permutation that represents a possible way of rearranging the events. A comparable approach was used in particular in AJM game semantics to represent the interchangeability of copies in the exponentials of linear logic.\nSome words are borrowed from game semantics, since our objects have similarities with games, but this is not a “game” semantics, at most a degenerate one. In particular, there is no real notion of player and opponent interacting, since there is no polarity that could distinguish them or distinguish inputs and outputs. The term “strategy” does not really apply either since there is no notion of choosing the next move in a given situation. Under these circumstances, calling anything a “game” is kind of far fetched.\n#### Outline\nSection [sec:algebras] defines order algebras from these ideas. Arenas, plays and linear combinations of plays (simply called vectors) are defined, with the two basic operations on vectors: *synchronisation*, which extends the merging of orders to take permutations into account, and *outcome*, which is a scalar that acts as the “result” of a process. Two vectors are equivalent if they are indistinguishable by synchronisation and outcome, and the order algebra is the quotient of the vectors by this equivalence.\nSection [sec:logic] describes constructs involving order algebras and their subspaces. Cartesian and tensor products are described in terms of interaction, and the symmetric algebra is constructed in the framework. This algebra is of particular interest because it represents the basic source of non-determinism in interaction, namely the fact that any number of interchangeable actions may occur at a given synchronisation point.\nSection [sec:processes] shows how order algebras can be used to provide fully abstract models of process calculi, with the example of the -calculus. The crucial ingredient is a quantitative extension of the standard notion of testing, from which the present work stems. Standard forms of testing are obtained as particular choices of the semiring of scalars.\n#### Future work\nOrder algebras as defined and studied in the present work are very finitary in nature, because vectors are finite linear combinations of finite plays. This setting already has an interesting structure, as this paper illustrates, but it is unable to represent any kind of potentially infinitary behaviour. This includes identity functions over types that are not finite dimensional, and as a consequence we do not get a model of differential linear logic. Handling infinity is the natural next step, and for this we need to add topology to the structure, in order to get a sensible notion of convergence. Order algebras will then appear not only as the quotient of combinations of plays by equivalence, but as the separated and completed space generated by plays. In this line of thought, the dual space should play an important role, in order to define duality in the logical sense.\nAnother direction is to exploit the fact that the semiring of scalars is a parameter of the construction. In particular, going from a semiring $\\Scal$ to the semiring of $\\Scal$-valued random variables over a given probabilistic space properly extends the model to a probabilistic one. Similarly, using complex numbers and unitary transformations could provide a way to represent quantum computation in the same framework. Developing these ideas correctly is a line of research by itself, as the question of denotational models for these aspects of computation is known to be a difficult matter.\n#### Related work\nPart of the construction of order algebras is concerned with modelling of features like name binding or creation of fresh names. The topic of proper formal handling of binders in syntax is a vast topic known as *nominal techniques* (see for instance Gabbay’s survey ), and it has been applied in particular to construct operational semantics for process calculi in a generic way . We feel that our approach is orthogonal: arenas present a flattened version of the name structure, in which remains no notion of name creation or binding (or only indirectly); permutations are used only to relate different *occurrences* of names. Moreover, local names, by essence, are absent from order algebras, since our intent is to build a denotational model, in which internal behaviour is forgotten.\nOur work aims in particular at constructing models of interaction that are not interleaving, a featured sometimes referred to as “true concurrency”. This objective, of course, is not new, and the reference model in this respect is that of event structures. A relationship between our framework and event structures can be formulated: using the simplest semiring of coefficients, namely {0, 1} with 1 + 1 = 1 (thus losing any “quantitative” content), linear combinations of plays are simply finite sets of plays. The set of plays interpreting a given process turns out to be exactly the set of configurations of the event structure interpreting this process, forgetting any internal events. We do not develop this correspondence in the present paper, as it is of limited interest in the current state of development of order algebras, however it will certainly be of great interest in the development of the theory, notably when applying it to modelling probabilistic processes, for which event structure semantics has been developed . Besides, the use of symmetry in event structures has been recently identified as a crucial feature; we defer to future work the comparison with our approach based on group actions.\nThe shift from sets of configurations to formal linear combinations in the interpretation of processes has a notable precedent in Boreale and Gadducci’s interpretation of CSP processes as formal power series , building on Rutten’s work relating coinduction and formal power series . Boreale and Gadducci’s work differs from the present paper in two respects. Firstly, their interpretation of the semiring of coefficient is of a different nature: sum and product are seen as internal and external choice respectively, while we interpret them as internal choice and parallel composition without interaction. Secondly, their technical development uses only idempotent semirings (where *x* + *x* = *x* for all *x*), which does not handle quantitative features, and leads inevitably to interleaving semantics (as proved in our setting by Theorem [thm:basis] and remarks in Section [sec:consequences]). Nevertheless, Rutten’s approach to coinduction, and the idea of coinductive definitions by behavioural differential equations is certainly relevant to our work and is a promising source of inspiration for the extension of the present setting to infinitary behaviours.\nOrder algebras\n==============\nArenas and plays\n----------------\nAn order algebra is defined on an *arena*, which represents a fixed set of potential events. The arena is equipped with a permutation group that represents the non-determinism that arises when synchronising events, as described below. Then a play is a partial order relation over a finite subset of the arena.\nAn *arena* *X* is a pair $(\\web{X},\\pgroup[X])$ where $\\web{X}$ is a countable set (the *web* of *X*) and $\\pgroup[X]$ is a subgroup of the group $\\Perm{\\web{X}}$ of permutations of $\\web{X}$. If $\\pgroup[X]$ is trivial, then *X* is called *static* and it is identified with its web.\nThe points in the web are called *events*, rather than moves, since there is no actual notion of players interacting. Permutations represent the fact that there may be several different ways for two processes to synchronise. In process calculus language, permutations can be seen as relating different occurrences of the same action label.\n[ex:csp] When modelling a simple process algebra like CSP over an alphabet *A* (with no value passing), we can use a web like $A×\\Nat$, where $\\Nat$ is the set of natural numbers; (*a*, *i*) is interpreted as the *i*-th copy of *a* (any other infinite set than $\\Nat$ would do: the actual values are irrelevant). The permutation group will consist of all permutations of $A×\\Nat$ that leave the first member unchanged in each pair: different occurrences of a given event can be freely permuted, but obviously they cannot be exchanged for events of a different name.\n[ex:ccs] When modelling a calculus like CCS , the same arena can be used as in CSP, taking for *A* the set of action labels, including polarities, that is $N\\uplus\\set{\\bar{u}}{u∈N}$ if *N* is the set of names.\nThings get more subtle when modelling a calculus with name passing like the *π*-calculus . For the monadic case, the arena will consist of triples (*ε*, *a*, *i*) where *ε* is a polarity (input or output), *a* is a name (either a free name or a name bound by an action) and *i* is an occurrence number. Names bound by different input events will be considered different: in process terms, instead of $u(x).P\\para u(x).Q$, write $u(x\\_1).P[x\\_1/x]\\para u(x\\_2).P[x\\_2/x]$. The considered permutations are those that respect the name structure: if *σ* maps an event *u*(*x*1) to an event *u*(*x*2), then it must map any event involving *x*1 to an event of the same type involving *x*2 instead. Private names, like *a* in $\\new{a}(a.P\\para\\bar{a}.Q)$, will not be represented in arenas, since by definition they cannot be involved in interaction with the environment, unless they are communicated by scope extrusion, as in $\\new{a}\\bar{u}a$, in which case they will be modelled the same way as binding input prefixes. This construction is developed in more detail in Section [sec:processes].\nA *play* over *X* is a pair $s=(\\web{s},\\order{s})$ where $\\web{s}$ is a finite subset of $\\web{X}$ (the support) and $\\order{s}$ is a preorder over $\\web{s}$; the set of plays over *X* is written $\\Plays{X}$. A play *s* is called *consistent* if the relation $\\order{s}$ is a partial order relation ( if it is acyclic).\nThe intuition is that a play represents a possible way a process may act: the support contains the set of all events that will actually occur, the preorder represents scheduling constraints for these events. Consistency means that these constraints are not contradictory, that they do not lead to a deadlock. Synchronisation, defined below, consists in combining constraints from two plays, assuming they have the same events. The primitive definition of plays as pre-orders is a way to make it a total operator by separating it from the consistency condition: two plays can synchronise even if their scheduling constraints are not compatible, but then the result is inconsistent.\n[ex:diagram] We will represent a (consistent) play graphically as the Hasse diagram of its order relation, with each node labelled by the event’s name. By convention, when two events are part of the same orbit under $\\pgroup[X]$, we use the same name with different indices:\n0.7 This represents a play with support {*a*, *b*1, *b*2, *c*, *d*, *e*}, with the order relation such that *a* < *b*1, *a* < *b*2, *b*1 < *d*, *c* < *b*2, *b*2 < *d* and *b*2 < *e*, in an arena that has a permutation that swaps *b*1 and *b*2.\nFor $r,s∈\\Plays{X}$ with $\\web{r}=\\web{s}$, the *synchronisation* of *r* and *s* is the play \n$$r\\sync s := \\bigl( \\web{r}, (\\order{r}\\cup\\order{s})^\\* \\bigr),$$ \n where (⋅)\\* denotes the reflexive transitive closure. Given a finite subset *A* of $\\web{X}$, define the **A*-neutral* play as *e**A* : = (*A*, id*A*) where id*A* is the identity relation.\n[ex:sync] We have the following synchronizations: \n$$\\left(\\!\\tikzinline{\n \\eventnode{(0,0)}{a1}{left}{a\\_1}\n \\eventnode{(-0.5,1)}{b}{left}{b}\n \\eventnode{(0.5,1)}{a2}{right}{a\\_2}\n \\draw (a1)--(b) (a1)--(a2); }\\!\\right)\n \\sync\n \\left(\\!\\tikzinline{\n \\eventnode{(0,0)}{a1}{left}{a\\_1}\n \\eventnode{(1,0)}{b}{right}{b}\n \\eventnode{(1,1)}{a2}{right}{a\\_2}\n \\draw (b)--(a2); }\\!\\right)\n =\n \\left(\\!\\tikzinline{\n \\eventnode{(0,0)}{a1}{left}{a\\_1}\n \\eventnode{(0,1)}{b}{left}{b}\n \\eventnode{(0,2)}{a2}{left}{a\\_2}\n \\draw (a1)--(b)--(a2); }\\right)\n,\\quad\n \\left(\\!\\tikzinline{\n \\eventnode{(0,0)}{a1}{left}{a\\_1}\n \\eventnode{(-0.5,1)}{b}{left}{b}\n \\eventnode{(0.5,1)}{a2}{right}{a\\_2}\n \\draw (a1)--(b) (a1)--(a2); }\\!\\right)\n \\sync\n \\left(\\!\\tikzinline{\n \\eventnode{(0,0)}{a2}{left}{a\\_2}\n \\eventnode{(1,0)}{b}{right}{b}\n \\eventnode{(1,1)}{a1}{right}{a\\_1}\n \\draw (b)--(a1); }\\!\\right)\n =\n \\left(\\!\\tikzinline{\n \\eventnode{(0,0)}{a1}{left}{a\\_1}\n \\eventnode{(1,0)}{b}{right}{b}\n \\eventnode{(1,1)}{a2}{right}{a\\_2}\n \\draw (b) to[bend left] (a1) to[bend left] (b) -- cycle;\n \\draw (b)--(a2); }\\!\\right)\n.$$ \n The second one leads to an inconsistent play, since the union of the order relations is cyclic.\nNote that synchronisation is a very restrictive operator because it requires the event sets to be equal. The possibility of synchronising on some events while keeping the others independent, which is a natural notion, will be defined in Section [sec:partial] using this primitive form of total synchronisation.\nCommutativity of $\\sync$ is immediate from the definition. Associativity is also clear: for $r,s,t∈\\Plays{X}$, $(r\\sync s)\\sync t$ and $r\\sync(s\\sync t)$ are defined if and only if $\\web{r},\\web{s},\\web{t}$ are equal, and in this case we have $\\order{(r\\sync s)\\sync t}=\n(\\order{r}\\cup\\order{s}\\cup\\order{t})^\\*=\\order{r\\sync(s\\sync t)}$. Because of the constraint on supports, there cannot be a neutral element. However, among plays of a given support *A*, the neutral play *e**A* is actually neutral for synchronisation.\nWe now define the action of the permutation group $\\pgroup[X]$ over the set of plays. Since there is usually no ambiguity, we overload the notation for group actions: given $σ∈\\pgroup[X]$, for *x*∈*X* we write *σ**x* for the image of *x*, for *A*⊆*X* we write *σ**A* for the set of images $\\set{σx}{x∈A}$, and similarly for $r∈\\Plays{X}$ we write *σ**r* for the play *r* permuted by *σ*, as defined below.\nLet *X* be an arena. The action of a permutation $σ∈\\pgroup[X]$ on a play $r∈\\Plays{X}$ is defined as \n$$σ r := \\bigl(\n σ \\web{r}, \\; \\set{(σ x, σ y)}{(x,y)∈\\order{r}}\n \\bigr).$$ \n The *orbit* of a play *r* in $\\Plays{X}$ is the set \n$$\\porbit[X]{r} := \\set{σ r}{σ∈\\pgroup[X]}.$$ \nWe refer the reader to some reference textbook (for instance Lang’s *Algebra* ) for details on the standard group-theoretic notions in use here. For reference, given a group *G* acting on a set *X*, the *stabilizer* of a point *x*∈*X* in the action of *G* is, by definition, the subgroup of *G* consisting of all the *σ*∈*G* that leave *x* unchanged, *σ**x* = *x*. The *pointwise* stabilizer of a set *A*⊆*X* is the subgroup of those that leave each point in *A* unchanged, as opposed to the *setwise* stabilizer which includes all permutations that leave the set *A* unchanged as a whole ( $\\set{σx}{x∈A}=A$). The index of a subgroup *H* in a group *G*, written (*H* : *G*), is the number of left cosets of *H* in *G*, that is the cardinal of $\\set{σH}{σ∈G}$. When *H* is a normal subgroup of *G*, the index (*H* : *G*) is the cardinal of the quotient group *G*/*H*.\nLet *X* be an arena and *r* be a play in $\\Plays{X}$. Let $\\pstab[X]{r}$ be the stabilizer of *r* in the action of $\\pgroup[X]$ over $\\Plays{X}$ and let $\\pstab[X]{\\web{r}}$ be the pointwise stabilizer of $\\web{r}$ in $\\pgroup[X]$, then the *multiplicity* of *r* in *X* is the index of $\\pstab[X]{\\web{r}}$ in $\\pstab[X]{r}$: \n$$\\multiplicity[X]{r} := (\\pstab[X]{r}:\\pstab[X]{\\web{r}}).$$ \nHence, the multiplicity of *r* is the number of different ways one can permute *r* into itself. Indeed, the definition as $(\\pstab{r}:\\pstab{\\web{r}})$ exactly means the number of permutations of *r* into itself (elements of $\\pstab{r}$), up to permutations that leave each point of $\\web{r}$ invariant (elements of $\\pstab{\\web{r}}$), in other words $\\multiplicity{r}$ is the order of the group $\\set[1]{\\restr{σ}{\\web{r}}}{σ∈\\pgroup,σ r=r}$, which is always finite since the support $\\web{r}$ is finite.\nUsing the same conventions as in Example [ex:diagram], we have \n$$\\multiplicity{\\tikzinline{\n \\eventnode{(0,0)}{a}{left}{a}\n \\eventnode{(-1,1)}{b1}{left}{b\\_1}\n \\eventnode{(-1,2)}{c1}{left}{c\\_1}\n \\eventnode{(1,1)}{b2}{right}{b\\_2}\n \\eventnode{(1,2)}{c2}{right}{c\\_2}\n \\draw (a)--(b1)--(c1) (a)--(b2)--(c2);\n }} = 2\n \\qquad\\text{and}\\qquad\n \\multiplicity{\\tikzinline{\n \\eventnode{(0,0)}{a}{left}{a}\n \\eventnode{(-1,1)}{b1}{left}{b\\_1}\n \\eventnode{(-1,2)}{c1}{left}{c\\_1}\n \\eventnode{(1,1)}{b2}{right}{b\\_2}\n \\eventnode{(0,3)}{c2}{right}{c\\_2}\n \\draw (a)--(b1)--(c1)--(c2) (a)--(b2)--(c2);\n }} = 1$$ \n Both plays have the same support, there are 4 permutations of this support: *b*1 and *b*2 can be exchanged, idem for *c*1 and *c*2. In the first case if we exchange *b*1 with *b*2 and *c*1 with *c*2, we get the same play (permuting the *b* but not the *c* yields a different play). In the second case, no permutation can yield the same play.\nLinear combinations\n-------------------\nThe set of plays of an arena *X* is independent of the group $\\pgroup$, but our idea is that plays that are permutations of each other should be considered equivalent, since permutations exchange occurrences of indistinguishable actions. In the presence of permutations, however, there are several ways to synchronise two plays, so in order to extend the definition of synchronisation we have to be able to consider combinations of possible plays. For genericity, and because our aim is to get a quantitative account of interaction, we will use linear combinations, with coefficients in an unspecified commutative semiring.\nA *commutative semiring* $\\Scal$ is a tuple $(\\Scal,{+},{⋅},0,1)$ such that $(\\Scal,{+},0)$ and $(\\Scal,{⋅},1)$ are commutative monoids and for all $x,y,z∈\\Scal$ it holds that *x*⋅(*y* + *z*) = *x*⋅*y* + *x*⋅*z* and *x*⋅0 = 0. A *semimodule* over $\\Scal$ is a commutative monoid (*M*, + , 0) with an action $(⋅):\\Scal×M→M$ that commutes with addition on both sides and satisfies *λ*⋅(*μ*⋅*x*) = (*λ*⋅*μ*)⋅*x* for all $λ,μ∈\\Scal$ and *x*∈*M*. A *commutative semialgebra* over $\\Scal$ is a semimodule *M* with a bilinear operation that is associative and commutative.\nTerminology about semirings, semimodules and semialgebras is not standard, in particular some definitions do not require both neutrals. Sometimes, the neutrals are not required to be distinct (they are equal if and only if the semiring is a singleton, but this is a degenerate case that we will not consider). In the above definitions, if all elements of $\\Scal$ have additive inverses, then $\\Scal$ is a (commutative unitary) ring, and the semimodules and semialgebras are actually modules and algebras (indeed, the action of $\\Scal$ imposes the existence of additive inverses in them too). If $\\Scal$ is a field, we get the usual notions of vector space and algebra.\nLet $\\Scal$ be a commutative semiring. The *integers* of $\\Scal$ are the finite sums of 1 including the empty sum 0, the *non-zero integers* are the finite non-empty sums. We call $\\Scal$ *regular* if for every non-zero integer $n\\in\\Scal$, for all $x,y\\in\\Scal$, *n**x* = *n**y* implies *x* = *y*. We call $\\Scal$ *rational* if every non-zero integer has a multiplicative inverse.\nIn particular, regularity applied to *x* = 1 and *y* = 0 imposes that no non-empty sum of 1 can be equal to 0, in other words $\\Scal$ has characteristic zero. The rationality condition means that it is possible to divide by non-zero natural numbers, or in more abstract terms that the considered semiring is a semimodule over the semiring of non-negative rationals. This obviously implies regularity.\nTwo important cases of rational semirings will be considered here. The first case is that of commutative algebras over the field $\\Rat$ of rational numbers, which includes fields of characteristic zero (among which rational, real and complex numbers) and commutative algebras over them. The second case is when addition is idempotent, which includes so-called *tropical* semirings , and the canonical examples are that of min-plus and max-plus semirings. Boolean algebras with disjunction as sum and conjunction as product are another typical example. In this case, all integers except 0 are equal to 1, so they obviously have multiplicative inverses.\nThroughout this paper, unless explicitly stated otherwise, $\\Scal$ is any semiring. Note that the semiring $\\Nat$ of natural numbers and the ring $\\Int$ of integers are regular but not rational. Indeed, when using natural numbers as scalars, some properties of order algebras will be lost, for instance the existence of bases. Hence some statements will explicitly require $\\Scal$ to be regular or rational.\nFor an arbitrary set *X*, a formal linear combination over *X* is a function from *X* to $\\Scal$ that has a value other than 0 on a finite number of points. Formal linear combinations over *X*, with sum and scalar product defined pointwise, form the free $\\Scal$-semimodule over *X* and an element *x*∈*X* is identified with its “characteristic” function $δ\\_x:X→\\Scal$, such that *δ**x*(*x*) = 1 and *δ**x*(*y*) = 0 for all *y*≠*x*. If *X* is finite, then the set of formal linear combinations is the $\\Scal$-semimodule $\\Scal^X$. For an arbitrary subset *A* of a $\\Scal$-semimodule *E*, we denote by $\\Vect[\\Scal]{A}$, or simply $\\Vect{A}$, the submodule of *E* generated by *A*, the smallest submodule of *E* that contains *A*, that is the set of finite linear combinations of elements of *A*.\nLet *X* be an arena. The *preliminary order algebra* $\\PreOrdAlg[\\Scal]{X}$ is the free $\\Scal$-semimodule over $\\Plays{X}$. The *outcome* is the linear form $\\outcome{⋅}$ over $\\PreOrdAlg[\\Scal]{X}$ such that $\\outcome{r}=1$ when *r* is consistent and $\\outcome{r}=0$ otherwise.\nWe usually keep the semiring $\\Scal$ implicit in our notations. Vectors in $\\PreOrdAlg{X}$ are finite linear combination of plays in *X*, they represent the collection of all possible behaviours of a finite process. The coefficients can be understood as the amount of each behaviour that is present in the process. Examples in further sections also illustrate that $\\Scal$ can be chosen to represent conditions on the availability of each behaviour. The outcome represents how relevant each play is, and by the intuition exposed in the previous section, plays with cyclic dependencies cannot happen, so they are considered irrelevant.\nConsider the CSP term $P=a{\\to}(b\\mathbin{\\|}c)\\mid a{\\to}c$ (remember that in CSP ∣ is the choice operator, and ∥ is parallel composition). An interpretation of *P* in a preliminary order algebra containing only *a*, *b*, *c* as events could be \n$$\\tikzformula{ } +\n 2 \\tikzformula{\n \\eventnode{(0,0)}{a}{right}{a}} +\n \\tikzformula{\n \\eventnode{(0,0)}{a}{right}{a}\n \\eventnode{(0,1)}{b}{right}{b}\n \\draw (a)--(b); } +\n 2 \\tikzformula{\n \\eventnode{(0,0)}{a}{right}{a}\n \\eventnode{(0,1)}{c}{right}{c}\n \\draw (a)--(c); } +\n \\tikzformula{\n \\eventnode{(0,0)}{a}{right}{a}\n \\eventnode{(-0.5,1)}{b}{left}{b}\n \\eventnode{(0.5,1)}{c}{right}{c}\n \\draw (a)--(b) (a)--(c); }$$ \n where we have a summand for each partial run of *P*. The coefficient 2 in the second and fourth summands represent the fact that there are two ways to perform only *a*, and two ways to perform *a* then *c*, depending on the choice one has done.\n[def:psync] Let *X* be an arena. Permuted synchronisation in *X* is the bilinear operator $\\psync$ over $\\PreOrdAlg{X}$ such that for all plays $r,s∈\\Plays{X}$, \n$$r \\psync s := \\multiplicity[X]{s}\n \\sum\\_{\\substack{s'∈\\porbit[X]{s} \\\\ \\web[1]{s'}=\\web{r}}} r\\sync s'$$ \nObserve that this sum is always finite. The reason is that each *s*ʹ can be written *σ**s* for some $σ∈\\pgroup[X]$, and $\\web{σs}=\\web{r}$ implies $σ\\web{s}=\\web{r}$. Since *σ**s* is determined by the action of *σ* on $\\web{s}$, there is at most one image of *s* for each bijection between $\\web{s}$ and $\\web{r}$. Since $\\web{r}$ and $\\web{s}$ are finite, the number of such bijections is finite.\n[ex:psync] Considering again the plays in Example [ex:sync], we have \n$$\\tikzformula{\n \\eventnode{(0,0)}{a1}{left}{a\\_1}\n \\eventnode{(-0.5,1)}{b}{left}{b}\n \\eventnode{(0.5,1)}{a2}{right}{a\\_2}\n \\draw (a1)--(b) (a1)--(a2); }\n \\psync\n \\tikzformula{\n \\eventnode{(0,0)}{a2}{left}{a\\_2}\n \\eventnode{(1,0)}{b}{right}{b}\n \\eventnode{(1,1)}{a1}{right}{a\\_1}\n \\draw (b)--(a1); }\n =\n \\tikzformula{\n \\eventnode{(0,0)}{a1}{left}{a\\_1}\n \\eventnode{(0,1)}{b}{left}{b}\n \\eventnode{(0,2)}{a2}{left}{a\\_2}\n \\draw (a1)--(b)--(a2); }\n +\n \\tikzformula{\n \\eventnode{(0,0)}{a1}{left}{a\\_1}\n \\eventnode{(1,0)}{b}{right}{b}\n \\eventnode{(1,1)}{a2}{right}{a\\_2}\n \\draw (b) to[bend left] (a1) to[bend left] (b) -- cycle;\n \\draw (b)--(a2); }\n.$$ \n The first term in the sum corresponds to the identity permutation, the second one exchanges *a*1 and *a*2. Here, all plays involved have multiplicity 1.\nPermuted synchronisation is similar to the parallel composition operator of CSP, in the case of processes defined on the same alphabet: in $r\\psync s$, every event of *r* must be synchronized with some event of the same name in *s*. There is a difference between plays and processes, however, in that in a play, every event must occur, whereas in a process, an action may be cancelled, for lack of a partner action to synchronize with.\nAny partial function $f:\\Plays{X}^n\\to\\Plays{X}$ extends as an *n*-linear operator $\\bar{f}:\\PreOrdAlg{X}^n\\to\\PreOrdAlg{X}$, by setting *f̄*(*r*1, …, *r**n*) = 0 when *f*(*r*1, …, *r**n*) is undefined. This applies in particular to synchronisation, which yields a bilinear operator $\\bar{\\sync}$ over $\\PreOrdAlg{X}$. As a slight abuse of notations, we will write it simply as $\\sync$ when there is no ambiguity.\nUsing this convention, permuted synchronisation can be seen as a generalisation of non-permuted synchronisation, since when the permutation group is trivial, all multiplicities are 1 and all orbits are singletons. Although $\\psync$ is a generalisation of $\\sync$, we still use different notations, since both operators are of interest in a given non-static arena. The non-permuted version will be referred to as *static synchronisation* to avoid confusion.\nLet *X* be an arena. Observational equivalence in $\\PreOrdAlg[\\Scal]{X}$ is defined as $u\\obseqv[X]u'$ when $\\outcome{u\\psync v}=\\outcome{u'\\psync v}$ for all $v∈\\PreOrdAlg[\\Scal]{X}$. The *order algebra* over *X* is $\\OrdAlg[\\Scal]{X}:=\\PreOrdAlg[\\Scal]{X}/{\\obseqv[X]}$.\nThe scalar $\\outcome{u\\psync v}$ is understood as the result of testing a process *u* against a process *v*. It linearly extends the basic case of single plays: $\\outcome{r\\sync s}$ is 1 if *r* and *s* are compatible and 0 otherwise; $\\outcome{r\\psync s}$ is the number of different ways *r* and *s* can be permuted so that they become compatible. Hence the definition: $u\\obseqv v$ if *u* and *v* are indistinguishable by this testing protocol.\nAny inconsistent play is observationally equivalent to 0, since synchronising it with any order yields an inconsistent play. Hence the synchronisation of Example [ex:psync] implies \n$$\\tikzformula{\n \\eventnode{(0,0)}{a1}{left}{a\\_1}\n \\eventnode{(-0.5,1)}{b}{left}{b}\n \\eventnode{(0.5,1)}{a2}{right}{a\\_2}\n \\draw (a1)--(b) (a1)--(a2); }\n \\psync\n \\tikzformula{\n \\eventnode{(0,0)}{a2}{left}{a\\_2}\n \\eventnode{(1,0)}{b}{right}{b}\n \\eventnode{(1,1)}{a1}{right}{a\\_1}\n \\draw (b)--(a1); }\n \\obseqv\n \\tikzformula{\n \\eventnode{(0,0)}{a1}{left}{a\\_1}\n \\eventnode{(0,1)}{b}{left}{b}\n \\eventnode{(0,2)}{a2}{left}{a\\_2}\n \\draw (a1)--(b)--(a2); }\n.$$ \n[lemma:perm-eq] Let *X* be an arena. For all $u∈\\PreOrdAlg{X}$ and $σ∈\\pgroup[X]$, we have $u\\obseqvσ u$.\nSince plays generate the module $\\PreOrdAlg{X}$, clearly $u\\obseqvσ u$ if and only if $\\outcome{u\\psync s}=\\outcome{σ u\\psync s}$ for all play *s*. Since *u* is a finite linear combination of plays, the definition of synchronisation on plays extends as $\\outcome{u\\psync s}\n =\\multiplicity{s}\\sum\\_{s'∈\\porbit{s}}\\outcome{u\\sync s'}$. It is clear that for all *σ*∈*G* we have $σ(u\\sync s')=σ u\\syncσ s'$, moreover outcomes are preserved by permutations, so we have $\\outcome{u\\sync s'}=\\outcome{σu\\syncσs'}$ for all *s*ʹ, hence $\\outcome{u\\psync s}\n =\\multiplicity{s}\\sum\\_{s'∈\\porbit{s}}\\outcome{σ u\\syncσ s'}$. Since *σ* acts as a permutation on the orbit $\\porbit{s}$, *σ**s*ʹ and *s*ʹ range over the same set, so we have $\\outcome{u\\psync s}=\\outcome{σ u\\psync s}$, and finally $u\\obseqvσ u$.\nThe fact that observational equivalence is preserved by linear combinations is immediate from the definition, since synchronisation and outcome are linear. As a consequence, in each orbit $\\porbit{s}$, we can choose a representant $\\repr{s}$ such that each vector in $\\PreOrdAlg{X}$ is equivalent to a linear combination of representants.\n[def:choice] Let *X* be an arena. A *choice of representants* for *X* is a pair of an idempotent map $A\\mapsto\\repr{A}$ over $\\PartsFin{\\web{X}}$ and an idempotent map $r\\mapsto\\repr{r}$ over $\\Plays{X}$ such that for all $r∈\\Plays{X}$ $\\web{\\repr{r}}=\\repr{\\web{r}}$, and for all $r,s∈\\Plays{X}$, $\\repr{r}=\\repr{s}$ if and only if *r* = *σ**s* for some $σ∈\\pgroup[X]$.\nSo a choice of representants picks one play in each orbit under $\\pgroup[X]$ in such a way that representants have the same support if it is possible. There always exists such choices, and in the sequel we assume that each arena comes with a particular choice, written $r\\mapsto\\repr[X]{r}$. The choice function over $\\Plays{X}$ induces a projection in $\\PreOrdAlg{X}$ by linearity, and for all $u∈\\PreOrdAlg{X}$ we have $u\\obseqv\\repr[X]{u}$ by Lemma [lemma:perm-eq].\n[def:saturate] Let *X* be an arena. *Saturation* in *X* is the linear map $\\saturate[X]:\\PreOrdAlg{X}\\to\\PreOrdAlg{X}$ such that for each play *r*, \n$$\\saturate[X] r := \\sum\\_{σ∈\\rpgroup{\\web{r}}} σ r\n \\qquad\\text{where}\\qquad\n \\rpgroup{A} := \\set[2]{\\restr{σ}{A}}{σ∈\\pgroup[X],σ A=A}.$$ \nThe set $\\rpgroup{A}$ is the group of permutations of *A* induced by $\\pgroup$, it is isomorphic to the quotient $\\pstab{\\{A\\}}/\\pstab{A}$ where $\\pstab{\\{A\\}}$ is the setwise stabilizer of *A* in $\\pgroup$ and $\\pstab{A}$ is its pointwise stabilizer (it is easy to check that the latter is a normal subgroup of the former).\nAgain using the conventions of Example [ex:diagram], we have \n$$\\saturate\n \\tikzformula{\n \\eventnode{(0,0)}{a}{left}{a}\n \\eventnode{(-1,1)}{b}{left}{b}\n \\eventnode{(-1,2)}{c1}{left}{c\\_1}\n \\eventnode{(0,1)}{c2}{above}{c\\_2}\n \\eventnode{(1,1)}{c3}{above}{c\\_3}\n \\draw (a)--(b)--(c1) (a)--(c2) (a)--(c3); }\n = 2 \\tikzformula{\n \\eventnode{(0,0)}{a}{left}{a}\n \\eventnode{(-1,1)}{b}{left}{b}\n \\eventnode{(-1,2)}{c1}{left}{c\\_1}\n \\eventnode{(0,1)}{c2}{above}{c\\_2}\n \\eventnode{(1,1)}{c3}{above}{c\\_3}\n \\draw (a)--(b)--(c1) (a)--(c2) (a)--(c3); }\n + 2 \\tikzformula{\n \\eventnode{(0,0)}{a}{left}{a}\n \\eventnode{(-1,1)}{b}{left}{b}\n \\eventnode{(-1,2)}{c1}{left}{c\\_2}\n \\eventnode{(0,1)}{c2}{above}{c\\_1}\n \\eventnode{(1,1)}{c3}{above}{c\\_3}\n \\draw (a)--(b)--(c1) (a)--(c2) (a)--(c3); }\n + 2 \\tikzformula{\n \\eventnode{(0,0)}{a}{left}{a}\n \\eventnode{(-1,1)}{b}{left}{b}\n \\eventnode{(-1,2)}{c1}{left}{c\\_3}\n \\eventnode{(0,1)}{c2}{above}{c\\_1}\n \\eventnode{(1,1)}{c3}{above}{c\\_2}\n \\draw (a)--(b)--(c1) (a)--(c2) (a)--(c3); },$$ \n where the factor 2 comes from the fact that exchanging *c*2 and *c*3 in the original play does not change it. Indeed, the multiplicity of this play is 2.\n[lemma:saturate] Let *X* be an arena. For all $r,s\\in\\Plays{X}$ such that $\\web{r}=\\web{s}$ we have $s\\psync r=s\\mathbin{\\bar{\\sync}}\\saturate r$.\nWe use the notations of Definition [def:saturate]. As *σ* ranges over $\\rpgroup{\\web{r}}$, *σ**r* ranges over all the elements of the orbit of *r* under $\\pgroup$ that have the same support as *r*. Moreover, each play in the orbit is hit a number of times equal to $\\multiplicity{r}$, so for any play *s* we have the expected equality.\nIn particular, this implies the equivalence $u\\psync v\\obseqv\\repr{u}\\mathbin{\\bar{\\sync}}\\saturate\\repr{v}$ for all *u* and *v*. We will use this fact in Proposition [prop:represent] to get a representation of arbitrary order algebras in static ones.\nPermuted synchronisation is compatible with observational equivalence. Up to observational equivalence, it is associative and commutative.\nWe first prove that permuted synchronisation is strictly associative. Consider three plays *r*, *s*, *t*. For all $σ∈\\pgroup[X]$ we have $r\\psyncσ s=r\\psync s$, so we can assume that *r*, *s*, *t* have equal support (if no permutation can let them have the same support, then synchronisation in any order is zero). Then we have $r\\psync(s\\psync t)=r\\sync\\saturate(s\\sync\\saturate t)$ by Lemma [lemma:saturate], hence \n$$r\\psync(s\\psync t)\n = \\sum\\_{σ,\\tau∈\\rpgroup{\\web{r}}} r\\syncσ(s\\sync\\tau t)\n = \\sum\\_{σ,\\tau∈\\rpgroup{\\web{r}}} (r\\syncσ s)\\syncσ\\tau t\n = \\sum\\_{σ,\\tau∈\\rpgroup{\\web{r}}} (r\\syncσ s)\\sync\\tau t\n = (r\\psync s)\\psync t$$ \n using associativity of strict synchronisation and the fact that, for a fixed *σ*, the permutations *σ**τ* and *τ* range over the same set. This extends to all vectors by linearity.\nAssociativity implies compatibility with observational equivalence: for two equivalent vectors $u\\obseqv u'$ and an arbitrary $v∈\\PreOrdAlg{X}$, for all $w∈\\PreOrdAlg{X}$ we have $\\outcome{(u\\psync v)\\psync w}=\\outcome{u\\psync(v\\psync w)}\n =\\outcome{u'\\psync(v\\psync w)}=\\outcome{(u'\\psync v)\\psync w}$ so $u\\psync v\\obseqv u'\\psync v$.\nSince observational equivalence is preserved by permutations, using commutativity of strict synchronisation we have \n$$s\\psync r\n = \\sum\\_{σ∈\\rpgroup{\\web{r}}} s\\syncσ r\n = \\sum\\_{σ∈\\rpgroup{\\web{r}}} σ(σ^{-1}s\\sync r)\n \\obseqv \\sum\\_{σ∈\\rpgroup{\\web{r}}} σ^{-1}s\\sync r\n = \\sum\\_{σ∈\\rpgroup{\\web{r}}} r\\syncσ^{-1}s\n = r\\psync s.$$ \n which proves commutativity of permuted synchronisation up to observational equivalence.\n[lemma:partial-neutral] Let *X* be an arena and let (*u**i*)*i*∈*I* be a finite family of vectors in $\\PreOrdAlg{X}$. There exists a vector *e* and an integer *n* > 0 such that for all *i*∈*I*, $u\\_i\\psync e=n\\,u\\_i$.\nObserve that for all finite subset *A* of *X*, every setwise stabilizer of *A* is a stabilizer of the *A*-neutral play *e**A*, so we have $\\saturate e\\_A=\\multiplicity{e\\_A}e\\_A$, and subsequently for all play $s∈\\Plays{X}$ such that $\\repr{\\web{s}}=\\repr{A}$ we get $s\\psync e\\_A=\\multiplicity{e\\_A}(s\\sync e\\_A)=\\multiplicity{e\\_A}s$. Call *P* the set of all $\\repr{\\web{r}}$ such that the play *r* has a non-zero coefficient in some *u**i*, then *P* is finite since each *u**i* is a finite linear combination of plays. Let *n* be the least common multiple of the $\\multiplicity{e\\_A}$ for *A* in *P*, then the vector \n$$e := \\sum\\_{A∈P} \\frac{n}{\\multiplicity{e\\_A}} e\\_A$$ \n satisfies $e\\psync u\\_i=nu\\_i$ for each *i* by construction. Note that the coefficients $n/\\multiplicity{e\\_A}$ are all natural numbers, by construction.\nIf $\\Scal$ is regular then outcome is preserved by observational equivalence.\nLet $u\\obseqv v$ be a pair of equivalent vectors in $\\PreOrdAlg{X}$. By Lemma [lemma:partial-neutral] there is a vector *e* and an integer *n*≠0 such that $u\\psync e=nu$ and $v\\psync e=nv$, so we have $n\\outcome{u}=\\outcome{u\\psync e}=\\outcome{v\\psync e}=n\\outcome{v}$. By regularity, we can deduce $\\outcome{u}=\\outcome{v}$.\nAs a consequence, the order algebra $\\OrdAlg{X}$, which is defined as the quotient of $\\PreOrdAlg{X}$ by observational equivalence, is a commutative semialgebra over $\\Scal$ with synchronisation $\\psync$ as the product, and outcome $\\outcome{⋅}$ is a linear form over it. The choice of representants for orbits of finite sets and plays induces the following representation property of $\\OrdAlg{X}$ in the static algebra $\\OrdAlg{\\web{X}}$.\n[prop:represent] Let *X* be an arena. Define the linear map $\\psplit[X]:\\PreOrdAlg{X}\\to\\PreOrdAlg{X}$ as \n$$\\psplit[X](u) := \\saturate[X]\\repr[X]{u}.$$ \n For all $u,v∈\\PreOrdAlg{X}$, \n$$u \\obseqv[X] v\n \\qquad\\text{if and only if}\\qquad\n \\psplit[X](u) \\obseqv[{\\web{X}}] \\psplit[X](v).$$ \n Hence $\\psplit[X]$ is an injective map from $\\OrdAlg{X}$ into $\\OrdAlg{\\web{X}}$. For all $u,v∈\\OrdAlg{X}$, \n$$\\psplit[X](u \\psync v) = \\psplit[X](u) \\sync \\psplit[X](v).$$ \nFor compatibility with observational equivalences, first suppose that *u* and *v* are such that $\\saturate\\repr{u}\\obseqv\\_{\\web{X}}\\saturate\\repr{v}$. Consider a play $r∈\\Plays{X}$, then we have $\\outcome{u\\psync r}=\\outcome{\\saturate\\repr{u}\\sync\\repr{r}}\n =\\outcome{\\saturate\\repr{v}\\sync\\repr{r}}=\\outcome{v\\psync r}$ using Lemma [lemma:saturate], so we have $u\\obseqv[X]v$.\nReciprocally suppose $u\\obseqv[X]v$, then by definition for all play $r∈\\Plays{X}$ we have $\\outcome{u\\psync r}=\\outcome{v\\psync r}$. By the remarks above, the outcome $\\outcome{u\\psync r}$ is equal to $\\outcome{r\\psync u}=\\outcome{\\repr{r}\\sync\\saturate\\repr{u}}$, so we have $\\outcome{\\repr{r}\\sync\\saturate\\repr{u}}=\\outcome{\\repr{r}\\sync\\saturate\\repr{v}}$ for all *r*. Let *s* be an arbitrary play in $\\Plays{X}$. Writing *u* as a linear combination ∑*i*∈*I**λ**i**r**i*, we get $\n \\outcome{\\saturate\\repr{u}\\sync s} =\n \\sum\\_{i∈I} λ\\_i \\outcome{\\saturate\\repr{r\\_i}\\sync\\repr{s}}.\n $ If $\\web{s}$ is not a representant subset of $\\web{X}$, then this sum is zero since $\\saturate{r\\_i}$ is a combination of plays whose supports are representant subsets. The same applies to *v* so we have $\\outcome{\\saturate\\repr{u}\\sync s}=\\outcome{\\saturate\\repr{v}\\sync s}=0$. Now suppose that $\\web{s}$ is a representant subset of $\\web{X}$, then the representant $\\repr{s}$ of *s* has the same support as *s* by definition, so there is a permutation *σ* such that $σ s=\\repr{s}$ and $σ\\web{s}=\\web{s}$. For all *i*∈*I*, if $\\web{\\repr{r\\_i}}=\\web{s}$, then by definition of saturation we have $σ\\saturate\\repr{r\\_i}=\\saturate\\repr{r\\_i}$, so we get $\\outcome{\\saturate\\repr{r\\_i}\\sync s}\n =\\outcome{\\saturate\\repr{r\\_i}\\sync\\repr{s}}$. If $\\web{\\repr{r\\_i}}\\neq\\web{s}$, then the equality holds trivially since both sides are 0. By linearity, we can deduce $\\outcome{\\saturate\\repr{u}\\sync s}\n =\\outcome{\\saturate\\repr{u}\\sync\\repr{s}}$, and applying the same reasoning to *v*, from our initial remarks we deduce $\\outcome{\\saturate\\repr{u}\\sync s}=\\outcome{\\saturate\\repr{v}\\sync s}$. Hence we get $\\saturate\\repr{u}\\obseqv\\_{\\web{X}}\\saturate\\repr{v}$.\nFor the commutation property with synchronisation, consider two plays $r,s∈\\Plays{X}$. If the supports $\\web{\\repr{r}}$ and $\\web{\\repr{s}}$ are distinct, then clearly $\\psplit[X](r\\psync s)=\\psplit[X](r)\\sync\\psplit[X](s)=0$. Otherwise, let *A* be this support. If *ρ* is a permutation in $\\pgroup[X]$ such that $ρ r=\\repr{r}$, we have $\\repr{r\\psync s}=\\repr{ρ(r\\psync s)}=\\repr{ρ r\\psync s}\n =\\repr{\\repr{r}\\psync s}$. Moreover, $\\repr{r}\\psync s=\\repr{r}\\sync\\saturate\\repr{s}$ so all terms in $\\repr{r}\\psync s$ have support *A*, and since for all play *t* with $\\web{t}=A$ we have $\\saturate\\repr{t}=\\saturate t$, we get \n$$\\saturate \\repr{r \\psync s}\n = \\saturate (\\repr{r} \\psync s)\n = \\sum\\_{σ∈\\rpgroup{A}} σ(\\repr{r} \\psync s)\n = \\sum\\_{σ∈\\rpgroup{A}} σ\\repr{r} \\psync s\n = \\sum\\_{σ∈\\rpgroup{A}}\n \\sum\\_{\\tau∈\\rpgroup{A}} σ\\repr{r} \\sync \\tau\\repr{s}\n = \\saturate\\repr{r} \\sync \\saturate\\repr{s}$$ \n which concludes the proof.\nThe commutation property could actually be written $\\psplit[X](u \\psync v) = \\psplit[X](u) \\psync \\psplit[X](v)$ since permuted and static synchronisations coincide in the static order algebra $\\OrdAlg{\\web{X}}$, but we keep the notations distinct to stress the fact that the second is static. This establishes an injective morphism of $\\Scal$-semialgebras, however this morphism does not preserve outcomes: for a play *s*, we have $\\outcome{\\psplit s}=\\sharp(\\rpgroup{\\web{s}})\\outcome{s}$; since this factor depends on $\\web{s}$, the outcome of $\\psplit(u)$ is not even proportional to that of *u* in general.\n[prop:units] Let *X* be an arena. Assume $\\Scal$ is rational. Then the $\\Scal$-semialgebra $\\OrdAlg{X}$ has a unit element if and only if the web $\\web{X}$ is finite.\nSuppose $\\web{X}$ is finite, then the set of plays $\\Plays{X}$ is finite, so we can apply Lemma [lemma:partial-neutral] to the whole set $\\Plays{X}$, which provides a vector *e* and a non-zero integer *n* such that $e\\psync s=ns$ for all $s∈\\Plays{X}$; then *e*/*n* is a neutral element for synchronisation. Now suppose that $\\web{X}$ is infinite. Let *u* be an arbitrary vector in $\\PreOrdAlg{X}$. Since *u* is a finite linear combination of plays with finite support, there is an integer *n* such that all non-zero components of *u* are plays with supports of cardinal strictly less than *n*. Let *A* be a subset of $\\web{X}$ of cardinal *n*, then we must have $u\\psync e\\_A=0\\neq e\\_A$. This implies that no finite linear combination of plays can be neutral.\nBases\n-----\nIn this section, we describe the $\\Scal$-semimodule $\\OrdAlg{X}$ by providing a subset of plays whose equivalence classes forms a basis. Linear independence does not have a unique definition for modules over arbitrary semirings , so we state the appropriate definition for our needs, which clearly extends the standard one for vector spaces:\nLet $\\Scal$ be a semiring and *E* a semimodule over $\\Scal$. A family (*u**i*)*i*∈*I* in *E* is linearly independent if, for any two families (*λ**i*)*i*∈*I* and (*μ**i*)*i*∈*I* in $\\Scal$ with finite support, if ∑*i*∈*I**λ**i**u**i* = ∑*i*∈*I**μ**i**u**i* then for all *i*, *λ**i* = *μ**i*. A basis of *E* is a linearly independent generating family.\nWe first concentrate on the case of static order algebras. The first thing we can remark about observational equivalence is that plays of different supports are always independent, since compatibility explicitly requires having the same support, so we have the following decomposition:\n[prop:direct-sum] For a finite static arena *X*, let $\\PreOrdAlgS{X}$ be the submodule of $\\PreOrdAlg{X}$ generated by plays of support *X*. Define the *strict order algebra* over *X* as the submodule $\\OrdAlgS{X}$ of $\\OrdAlg{X}$ made of equivalence classes of elements of $\\PreOrdAlgS{X}$. Then for all static arena *X* we have \n$$\\OrdAlg{X} = \\bigoplus\\_{Y∈\\PartsFin{X}} \\OrdAlgS{Y}.$$ \nClearly $\\PreOrdAlg{X}$ is the direct sum of the $\\PreOrdAlgS{Y}$, since this decomposition amounts to partitioning the basis $\\Plays{X}$ according to the supports *Y* of its elements. As a consequence, $\\OrdAlg{X}$ is the sum of the $\\OrdAlgS{Y}$, and we have to prove that this sum is direct. Consider two vectors $u=\\sum\\_{Y∈\\PartsFin{X}}u\\_Y$ and $v=\\sum\\_{Y∈\\PartsFin{X}}v\\_Y$ such that $u\\obseqv v$ and for all $Y∈\\PartsFin{X}$, $u\\_Y,v\\_Y∈\\PreOrdAlgS{Y}$ (necessarily, only finitely many of the *u**Y* and *v**Y* are not 0). For each $Y∈\\PartsFin{X}$, we have $u\\sync e\\_Y=u\\_Y$ and $v\\sync\n e\\_Y=v\\_Y$, so $u\\_Y\\obseqv v\\_Y$ since $\\sync$ is compatible with $\\obseqv$. As a consequence, the decomposition of a vector in $\\OrdAlg{X}$ on the submodules $\\OrdAlgS{Y}$ is unique.\nWe can thus focus on the study of strict order algebras. These have the definite advantage of being finitely generated, since there are finitely many different binary relations over a given finite set. We will now provide explicit bases for them, depending on the structure of $\\Scal$.\nLet *X* be a finite static arena. Clearly, for all inconsistent plays *r* we have $r\\obseqv0$, so we can consider only consistent plays, plays *r* such that $\\order{r}$ is an order relation. In the following statements, as a slight abuse of notations, a play *r* with $\\web{r}=X$ is identified with its order relation $\\order{r}$, and also with its equivalence class in $\\OrdAlgS{X}$. Let $\\Ord{X}$ be the set of all partial order relations over *X*.\nThe notations < *r*, ≥*r*, > *r* are defined as expected. We denote by $\\conc\\_r$ the incomparability relation: $x\\conc\\_ry$ if and only if neither *x*≤*r**y* nor *y*≤*r**x*. We write $x\\conceq\\_ry$ if *x* = *y* or $x\\conc\\_ry$. If there is no ambiguity, we may omit the subscript *r* in these notations. The notation [*a* < *b*]*X*, for *a*, *b*∈*X*, represents the smallest partial order over *X* for which *a* < *b*, that is $\\identity{X}\\cup\\{(a,b)\\}$. The notation extends to more complicated formulas, for instance [*a* < *b*, *c* < *d*] is the smallest partial order for which *a* < *b* and *c* < *d*. We write $r\\comp s$ to denote that two partial orders *r* and *s* are compatible.\n[prop:totals] Let $\\Totals{X}$ be the set of total orders over *X*, then $\\Totals{X}$ is a linearly independent family in $\\OrdAlgS{X}$.\nWe prove the equivalent statement that two observationally equivalent combinations of total orders are necessarily equal. Let $u=\\sum\\_{t∈\\Totals{X}}λ\\_tt$ and $v=\\sum\\_{t∈\\Totals{X}}\\mu\\_tt$ be two combinations such that $u\\obseqv v$. If *r* and *s* are two distinct total orders over *X*, there exists a pair (*a*, *b*)∈*X*2 such that *a* < *r**b* and *b* < *s**a*, hence *r* and *s* are not compatible, so $\\inter{r}{s}=0$. Besides, it always holds that $\\inter{r}{r}=1$, so for all $t∈\\Totals{X}$, $\\inter{u}{t}=λ\\_t$ and $\\inter{v}{t}=\\mu\\_t$, so $u\\obseqv v$ implies *λ**t* = *μ**t* for all *t*, hence *u* = *v*.\nHowever, in general, $\\Totals{X}$ is not a generating family for $\\OrdAlgS{X}$. The simplest counter-example can be found if *X* has two points. Write *X* = {*a*, *b*}, then $\\Ord{X}$ has three elements: \n$$\\Ord{\\{a,b\\}}=\\bigl\\{[a\\conc b],[a<b],[a>b]\\bigr\\}.$$ \n Then in the canonical basis $([a\\conc b],[a<b],[a>b])$ of $\\PreOrdAlgS{X}$, the matrix of $(u,v)\\mapsto\\inter{u}{v}$ is \n$$\\begin{pmatrix}\n 1 & 1 & 1 \\\\\n 1 & 1 & 0 \\\\\n 1 & 0 & 1\n \\end{pmatrix}$$ \n If $\\Scal$ is the field of reals, for instance, then this matrix is invertible, which means that the three orders are linearly independent, hence $\\OrdAlgS{X}$ is isomorphic to $\\Scal^{\\Ord{X}}$ (this isomorphism holds if and only if the cardinal of *X* is at most 2, as we shall see below). There is one case where [*a* < *b*] and [*b* < *a*] do generate $\\OrdAlgS{\\{a,b\\}}$, namely when addition in $\\Scal$ is idempotent, when 1 + 1 = 1.\n[prop:basis-totals] $\\Totals{X}$ is a basis of $\\OrdAlgS{X}$ for all *X* if and only if addition in $\\Scal$ is idempotent.\nBy Proposition [prop:totals], we know that $\\Totals{X}$ is always a linearly independent family, so all we have to prove is that it generates $\\OrdAlgS{X}$ if and only if 1 + 1 = 1 in $\\Scal$.\nFirstly, assume that $\\Totals{X}$ generates $\\OrdAlgS{X}$ for all *X*. Then, for *X* = {*a*, *b*}, there are two scalars $λ,\\mu∈\\Scal$ such that $[a\\conc b]\\obseqvλ[a<b]+\\mu[a>b]$. Then we have \n$$\\inter{[a\\conc b]}{[a<b]}\n = λ\\inter{[a<b]}{[a<b]} + \\mu\\inter{[a>b]}{[a<b]} = λ$$ \n but by definition we have $\\inter{[a\\conc b]}{[a<b]}=1$, so *λ* = 1. Similarly, we get *μ* = 1, and so $[a\\conc b]=[a<b]+[a>b]$. As a consequence, we have \n$$1 = \\inter{[a\\conc b]}{[a\\conc b]}\n = \\inter{[a<b]}{[a\\conc b]} + \\inter{[a>b]}{[a\\conc b]}\n = 1 + 1.$$ \nReciprocally, assume $\\Scal$ satisfies 1 + 1 = 1. Let *X* be an arbitrary finite set and let $r∈\\Ord{X}$. Let *u* = ∑*i* = 1*k**t**i* be the sum of all total orders that are compatible with *r*. Consider an arbitrary order $s∈\\Ord{X}$. Then we have $\\inter{u}{s}=\\sum\\_{i=1}^k\\inter{t\\_i}{s}$ and each term of this sum is 0 or 1. If *s* is compatible with *r*, then there is a total order *t* that extends both *r* and *s*, so *t* is one of the *t**i*; since *s* and *t* are compatible, the sum contains at least one 1 so $\\inter{u}{s}=1=\\inter{r}{s}$. If *s* is incompatible with *r*, then it is incompatible with any order that contains *r*, and in particular it is incompatible with all the *t**i*, so $\\inter{u}{s}=0=\\inter{r}{s}$. As a consequence we have $r\\obseqv u$, which proves that $\\Totals{X}$ generates $\\OrdAlgS{X}$.\nIn the general case, without any hypothesis on the semiring $\\Scal$, it happens that the family of all orders over *X* is not linearly independent, as soon as *X* has at least three points.\n[prop:split] For all semiring $\\Scal$, in $\\PreOrdAlgS[\\Scal]{\\{x,y,z\\}}$ we have \n$$\\tikzformula{\n \\eventnode{(0,0)}{x}{left}{x}\n \\eventnode{(0,2)}{y}{left}{y}\n \\eventnode{(1,1)}{z}{right}{z}\n \\draw (x) -- (y); }\n + \\tikzformula{\n \\eventnode{(0,0)}{x}{left}{x}\n \\eventnode{(0,2)}{y}{left}{y}\n \\eventnode{(0,1)}{z}{right}{z}\n \\draw (x) -- (y); }\n = \\tikzformula{\n \\eventnode{(0,0)}{x}{left}{x}\n \\eventnode{(0,2)}{y}{left}{y}\n \\eventnode{(1,1)}{z}{right}{z}\n \\draw (z) -- (x) -- (y); }\n + \\tikzformula{\n \\eventnode{(0,0)}{x}{left}{x}\n \\eventnode{(0,2)}{y}{left}{y}\n \\eventnode{(1,1)}{z}{right}{z}\n \\draw (x) -- (y) -- (z); }$$ \nWe use the following notations: *a* : = [*x* < *y*], *b* : = [*x* < *z* < *y*], *c* : = [*x* < *y*, *x* < *z*], *d* : = [*x* < *y*, *z* < *y*], so that the equation we prove is *a* + *b* = *c* + *d*. Let *s* be a partial order over {*x*, *y*, *z*}. First remark that $s\\comp a$ if and only if $s\\comp c$ or $s\\comp d$. Indeed, assume $s\\comp a$, then there is a total order *t* that contains *a* and *s*. If *x* < *t**z* then [*x* < *z*] ⊆ *t* so $a\\sync{}[x<z]\\subseteq t$, hence $s\\comp a\\sync{}[x<z]=c$. Otherwise *z* < *t**x* < *t**y* so *z* < *t**y* then $s\\comp d$. Reciprocally, if $s\\comp c$ or $s\\comp d$ then $s\\comp a$ since *a* is included in *c* and *d*. Secondly, remark that $s\\comp b$ if and only if $s\\comp c$ and $s\\comp d$. Indeed, assume that $s\\comp c$ and $s\\comp d$. Let $s'=s\\sync c=s\\sync a\\sync{}[x<z]$. Suppose $s'\\incomp [z<y]$, then *y* < *s*ʹ*z*. By hypothesis we cannot have $y<\\_{a\\sync s}z$, so (*y*, *z*) occurs in *s*ʹ but not in $(s\\sync a)\\cup[x<z]$, which implies $y<\\_{s\\sync a}x$. This contradicts the hypothesis *x* < *a**y*, hence $s'\\comp[z<y]$, so $s\\comp a\\sync{}[x<z]\\sync{}[z<y]=b$. The reciprocal implication is immediate since *a* ⊆ *b* and *d* ⊆ *b*. As a consequence of the two remarks above, we have $\\inter{a}{s}=1$ if and only if $\\inter{c}{s}=1$ or $\\inter{d}{s}=1$, which is equivalent to $\\inter{(c+d)}{s}∈\\{1,2\\}$. Moreover, $\\inter{(c+d)}{s}=2$ if and only if $\\inter{c}{s}=1$ and $\\inter{d}{s}=1$, which is equivalent to $\\inter{b}{s}=1$. Therefore $\\inter{(a+b)}{s}=\\inter{(c+d)}{s}$.\nThis proposition applies to orders on three points, but the exact same argument applies in any larger context, since the proof never uses the fact that there are no other points than *x*, *y*, *z*. So for any play *r* and points $x,y,z∈\\web{r}$ such that *x* < *r**y*, $x\\conc\\_rz$ and $y\\conc\\_rz$ we have \n$$r + (r\\sync{}[x<z<y]) \\obseqv (r\\sync{}[x<z]) + (r\\sync{}[z<y]).$$ \n This can also be deduced from Proposition [prop:split] using the partial composition operators defined in Section [sec:partial]. When $\\Scal$ is a ring, it allows us to express each of the patterns of the equation in Proposition [prop:split] as a linear combination of the others with coefficients 1 and − 1. This implies that for each of these patterns, the set of all orders over *X* that do not contain the considered pattern generates $\\OrdAlgS{X}$. In each case, the forbidden pattern defines a particular class of orders, respectively weak total orders (as of Proposition [prop:weak] below), orders of height at most 2 and forests with roots up or down.\n[prop:weak] Let (*X*, ≤) be a partially ordered set. The following conditions are equivalent:\n* For all *x*, *y*, *z*∈*X*, if *x* < *y* then *x* < *z* or *z* < *y*.\n* The relation $\\conceq$ is an equivalence.\n* There is a totally ordered set (*Y*, ≤) and a function *f* : *X* → *Y* such that, for all *x*, *y*∈*X*, *x* < *y* if and only if *f*(*x*) < *f*(*y*).\nLet $\\Weak{X}$ be the set of orders that satisfy these conditions, called *weak total orders* over *X*.\nFirstly, assume that for all *x*, *y*, *z*∈*X*, if *x* < *y* then *x* < *z* or *z* < *y*. It is clear that $\\conceq$ is always reflexive and symmetric. Let *x*, *y*, *z*∈*X* such that $x\\conceq z$ and $z\\conceq y$. If *x* < *y*, then by hypothesis we must have *x* < *z* or *z* < *y*, which contradicts the hypothesis on *x*, *y*, *z*. Similarly we cannot have *y* < *x*, so $x\\conceq y$. Therefore $\\conceq$ is transitive and it is an equivalence relation.\nSecondly, assume $\\conceq$ is an equivalence relation. Let *Y* be the set of equivalence classes of $\\conceq$. Define the relation ⊑ on *Y* as *A* ⊑ *B* if *a*≤*b* for some *a*∈*A* and *b*∈*B*. The relation ⊑ is reflexive since for all *A*∈*Y*, for any *a*∈*A* we have *a*≤*a* so *A* ⊑ *A*. Assume *A* ⊑ *B* and *B* ⊑ *A* for some *A*, *B*∈*Y*, then there are *a*, *a*ʹ∈*A* and *b*, *b*ʹ∈*B* such that *a*≤*b* and *b*ʹ≤*a*ʹ; if *a* < *b*ʹ then *a* < *a*ʹ which contradicts $a\\conceq a'$, similarly if *b*ʹ < *a* then *b*ʹ < *b* which contradicts $b'\\conceq b$, so $a\\conceq b'$, which implies that *A* and *B* are the same class, therefore ⊑ is antisymmetric. Assume *A* ⊑ *B* and *B* ⊑ *C* for some *A*, *B*, *C*∈*Y*, then there are *a*∈*A*, *b*, *b*ʹ∈*B* and *c*∈*C* such that *a*≤*b* and *b*ʹ≤*c*; if $a\\conceq c$ then *A* = *C* hence *A* ⊑ *C*, otherwise we must have *a* < *c* or *c* < *a*, but the second case implies *b*ʹ≤*c* < *a*≤*b* which contradicts $b\\conceq b'$, so *a* < *c* and *A* ⊑ *C*, hence ⊑ is transitive. Totality is immediate: if *A* and *B* are two distinct classes, then every pair (*a*, *b*)∈*A*×*B* is comparable. Let *f* be the function that maps each element of *X* to its class. If *x* < *y* then *f*(*x*) ⊏ *f*(*y*) by definition. Reciprocally, if *f*(*x*) ⊏ *f*(*y*), then *x* and *y* must be comparable (since they are in distinct classes), and *y* < *x* would imply *f*(*y*) ⊏ *f*(*x*), so *x* < *y*.\nFinally, assume there is *f* : *X* → *Y* where *Y* is totally ordered such that *x* < *y* if and only if *f*(*x*) < *f*(*y*). Let *x*, *y*, *z* be such that *x* < *y*, then *f*(*x*) < *f*(*y*). Since the order on *Y* is total, we must have either *f*(*x*) < *f*(*z*) or *f*(*z*) < *f*(*y*) (or both), hence *x* < *z* or *z* < *y*.\nIn other words, a weak total order is a total order over sets of mutually incomparable points. Interestingly, this kind of order was considered long ago in scheduling theory as the possibility to label events with time stamps in a possibly non-injective manner. It turns out that weak total orders form a basis.\nLet $r∈\\Ord{X}$. Two elements *a*, *b*∈*X* are equivalent in *r*, written *a* ∼ *r**b*, if for all *c*∈*X*∖{*a*, *b*}, *a* < *r**c* if and only if *b* < *r**c*, and *c* < *r**a* if and only if *c* < *r**b*. For a pair *a* ∼ *r**b* with *a* ≠ *b*, let *r*/(*a* ∼ *b*) be the order *r* ∩ (*X*∖{*b*})2 over *X*∖{*b*}.\nLet *a*, *b*∈*X* with *a*≠*b*. For each $r∈\\Ord{X∖\\{b\\}}$, define the relations\nra~b := r ∪ ∪, ra<b := ra~b ∪ {(a,b)}, ra>b := ra~b ∪ {(b,a)}.\nClearly, *r**a* ∼ *b*, *r**a* < *b* and *r**a* > *b* are partial orders over *X* in which *a* and *b* are equivalent.\n[lemma:coh-up] Let *a*, *b* be two distinct elements of *X*. For all $r∈\\Weak{X}$ and $s∈\\Ord{X∖\\{b\\}}$,\n* if *a* < *r**b* then $r\\comp s\\_{a\\sim b}$ if and only if $r\\comp s\\_{a<b}$, moreover $r\\incomp s\\_{a>b}$,\n* if *a* > *r**b* then $r\\comp s\\_{a\\sim b}$ if and only if $r\\comp s\\_{a>b}$, moreover $r\\incomp s\\_{a<b}$,\n* if $a\\conc\\_rb$ then $r\\comp s\\_{a\\sim b}$ if and only if $r\\comp s\\_{a<b}$ if and only if $r\\comp s\\_{a>b}$.\nIf *a* < *r**b*, we have *r* ∪ *s**a* ∼ *b* = *r* ∪ *s**a* < *b*, since *s**a* ∼ *b* and *s**a* < *b* only differ on (*a*, *b*), so the compatibility of the two pairs is equivalent to this union being acyclic. The same argument applies to the case *a* > *r**b*. For the case $a\\conc\\_rb$, first assume $r\\comp s\\_{a\\sim b}$ and let $t=r\\sync s\\_{a\\sim b}$. By definition of weak orders, we have *a* ∼ *r**b*. If *a* < *t**b* then there exists a sequence *a* = *a*0, …, *a**n* = *b* such that for each *i* < *n*, *a**i* < *r**a**i* + 1 or *a**i* < *s**a* ∼ *b**a**i* + 1, but since *a* and *b* are equivalent in both *r* and *s**a* ∼ *b*, we can replace *b* with *a* in this sequence, which leads to the contradiction *a* < *t**a*. By the same argument we cannot have *b* < *t**a*, so $a\\conc\\_tb$. We thus have $t\\comp[a<b]$ hence $s\\_{a<b}=s\\_{a\\sim b}\\sync{}[a<b]\\comp r$, and similarly $r\\comp s\\_{a>b}$. The reverse implications are immediate since *s**a* ∼ *b* is included in both *s**a* < *b* and *s**a* > *b*.\n[prop:basis-weak] If $\\Scal$ is a ring, then for all finite set *X*, $\\Weak{X}$ is a basis of $\\OrdAlgS{X}$.\nLet *Z*(*X*) be the submodule of all the $u∈\\PreOrdAlgS{X}$ such that for all order *r* over *X*, $\\inter{u}{r}=0$. We actually prove the fact that $\\PreOrdAlgS{X}$ is isomorphic to the direct sum $\\Scal^{\\Weak{X}}⊕ Z(X)$, which is equivalent since by definition $\\OrdAlgS{X}$ is $\\PreOrdAlgS{X}/Z(X)$ when the semiring $\\Scal$ is a ring.\nWe first prove that for all order *r* over *X* there is an $s∈\\Scal^{\\Weak{X}}$ such that *r* − *s*∈*Z*(*X*). Let $N(r)=\\set{(a,b,c)∈X^3}{a<\\_rb,a\\conc\\_r c,b\\conc\\_r c}$, we proceed by induction on $\\sharp N(r)$. If *r* = ∅, then by Proposition [prop:weak] we have $r∈\\Weak{X}$, so we can set *s* = *r*. Otherwise, consider a triple (*a*, *b*, *c*)∈*N*(*r*). Define the orders $r\\_1:=r\\sync{}[a<c]$, $r\\_2:=r\\sync{}[c<b]$ and $r\\_3:=r\\sync{}[a<c<b]$. By Proposition [prop:split], we have *r*1 + *r*2 − *r*3 − *r*∈*Z*(*X*). Besides, for each *i*∈{1, 2, 3}, clearly *N*(*r**i*) ⊂ *N*(*r*) and (*a*, *b*, *c*)∈*N*(*r*)∖*N*(*r**i*), so $\\sharp N(r\\_i)<\\sharp N(r)$. We can then apply the induction hypothesis to get an $s\\_i∈\\Scal^{\\Weak{X}}$ such that *r**i* − *s**i*∈*Z*(*X*). We can then conclude by setting *s* : = *s*1 + *s*2 − *s*3.\nAs a consequence we have $\\PreOrdAlgS{X}=\\Scal^{\\Weak{X}}+Z(X)$, and we now prove that this sum is direct by proving $\\Scal^{\\Weak{X}}\\cap Z(X)=\\{0\\}$. We proceed by recurrence on the size of *X*. If *X* has 0 or 1 element, then the only order over *X* is the trivial order *t*, and $\\inter{t}{t}=1\\neq0$, so *Z*(*X*) = {0} and the result trivially holds. Now let *n*≥2, suppose the result holds for all *X* with at most *n* − 1 points, and let $u∈\\Scal^{\\Weak{X}}\\cap Z(X)$. We now prove that *u* is the zero function.\nLet *r* be a weak total order that is not a total order, let *a*, *b*∈*X* such that $a\\conc\\_rb$. Let *X*ʹ = *X*∖{*b*}. Define $u'∈\\Scal^{\\Weak{X'}}$ by *u*ʹ(*t*) = *u*(*t**a* ∼ *b*) for all $t\\in\\Ord{X'}$, so that *u*(*r*) = *u*ʹ(*r*/(*a* ∼ *b*)). For any orders $s∈\\Weak{X}$ and $t∈\\Ord{X'}$, by Lemma [lemma:coh-up] we have that $\\inter{s}{(t\\_{a<b}+t\\_{a>b}-t\\_{a\\sim b})}$ is 0 if *a* and *b* are comparable in *s*, otherwise it is equal to $\\inter{s}{t\\_{a\\sim b}}$, which is itself equal to $\\inter{s/(a\\sim b)}{t}$ by restriction to *X*ʹ. Let *s*ʹ = *s*/(*a* ∼ *b*), we have \n$$\\inter{u}{(t\\_{a<b}+t\\_{a>b}-t\\_{a\\sim b})}\n = \\sum\\_{s∈\\Weak{X},a\\conc\\_sb}u(s)\\inter{s}{t\\_{a\\sim b}}\n = \\sum\\_{s∈\\Weak{X},a\\conc\\_sb}u'(s')\\inter{s'}{t}$$ \n The mapping *s* ↦ *s*/(*a* ∼ *b*) is a bijection from weak total orders over *X* such that $a\\conc b$ to weak total orders over *X*ʹ, so the latter sum is equal to $\\sum\\_{s'∈\\Weak{X'}}u'(s')\\inter{s'}{t}=\\inter{u'}{t}$. Besides, *u* is in *Z*(*X*) so $\\inter{u}{(t\\_{a<b}+t\\_{a>b}-t\\_{a\\sim b})}=0$, which implies $\\inter{u'}{t}=0$. This holds for all *t*, so *u*ʹ∈*Z*(*X*ʹ). By construction we have $u'∈\\Scal^{\\Weak{X'}}$ so *u*ʹ is in $\\Scal^{\\Weak{X'}}\\cap Z(X')$. By the induction hypothesis this is {0}, so *u*ʹ = 0 and as a consequence we have *u*(*r*) = *u*ʹ(*r*/(*a* ∼ *b*)) = 0.\nBy the argument above, we thus know that *u*(*r*) = 0 as soon as *r* is not a total order. In other words, *u* is a linear combination of total orders. From Proposition [prop:totals] we know that total orders are linearly independent in $\\OrdAlgS{X}$, so we can conclude that *u* = 0.\nAs a consequence, weak total orders on subsets of $\\web{X}$ form a basis of the static order algebra $\\OrdAlg{X}$. We can extend this property to arbitrary order algebras using the representation property.\n[thm:basis] Let *X* be an arena. Then $\\OrdAlg{X}$ has a basis (*b**i*)*i*∈*I* made of plays if\n* $\\Scal$ is idempotent, then the *b**i* are the orbits of totally ordered plays under $\\pgroup[X]$, or\n* $\\Scal$ is a regular ring, then the *b**i* are the orbits of weakly totally ordered plays under $\\pgroup[X]$.\nIn both cases, if $\\Scal$ is rational, then there exists a family of vectors (*b**i*\\*)*i*∈*I* such that for all *i*, *j*∈*I*, $\\outcome{b\\_i\\psync b^\\*\\_j}$ is 1 if *i* = *j* and 0 otherwise.\nPropositions [prop:basis-totals] and [prop:basis-weak] provide bases of the appropriate kinds for strict static order algebras. By Proposition [prop:direct-sum], these yield bases for static order algebras. In each case, call the elements of these bases *base plays*. A permutation of a (weak) total order is always an order of the same kind, so from the fact that base plays generate $\\OrdAlg{\\web{X}}$, we deduce that they also generate $\\OrdAlg{X}$. Now consider two linear combinations *u* = ∑*i*∈*I**λ**i**r**i* and *v* = ∑*i*∈*I**μ**i**r**i*, where the *r**i* are distinct base plays for $\\web{X}$ and representants (as of Definition [def:choice]), and suppose $u\\obseqv v$. By Proposition [prop:represent], we can deduce $\\sum\\_{i∈I}λ\\_i\\saturate r\\_i\n \\obseqv[\\web{X}]\\sum\\_{i∈I}\\mu\\_i\\saturate r\\_i$, and this equivalence is an equality since both sides are linear combinations of base plays. Now consider any *i*∈*I*. In $\\sum\\_{i∈I}λ\\_i\\saturate r\\_i$, the coefficient of *r**i* is $\\multiplicity[X]{r\\_i}$, so the equality above implies $\\multiplicity[X]{r\\_i}λ\\_i=\\multiplicity[X]{r\\_i}\\mu\\_i$, and subsequently *λ**i* = *μ**i* since $\\multiplicity[X]{r\\_i}$ is a non-zero integer and $\\Scal$ is regular. Hence representants of base plays form a basis of $\\OrdAlg{X}$.\nIf $\\Scal$ is an idempotent semiring, then by Proposition [prop:basis-totals] the family (*b**i*)*i*∈*I* is made of total orders, so if we set *b**i*\\* = *b**i* for each *i* we have the expected property.\nNow suppose $\\Scal$ is a rational ring. Let *A* be a representant finite subset of $\\web{X}$. Call *a*1, …, *a**n* the subset of the basis whose plays have support *A*, and let *M* = (*m**i**j*) be the *n*×*n* matrix such that $m\\_{ij}=\\outcome{a\\_i\\psync a\\_j}$. *M* has coefficients in natural numbers, and since the family (*a**i*) is linearly independent by hypothesis, *M* is invertible in $\\Rat$. Since $\\Scal$ is a regular ring, it is an algebra over $\\Rat$, so *M* is also invertible in $\\Scal$. Let *M*− 1 = (*m*ʹ*i**j*) and let *a**i*\\* : = ∑*j* = 1*n**m*ʹ*i**j**a**i*, then by construction $\\outcome{a\\_i\\psync a^\\*\\_j}$ is 1 if *i* = *j* and 0 otherwise.\nObserve that if $\\Scal$ is a rational ring, then in particular it is an algebra over $\\Rat$, then $\\OrdAlg[\\Scal]{X}=\\Scal\\otimes\\OrdAlg[\\Rat]{X}$ as $\\Rat$-algebras, since all plays decompose uniquely as linear combinations of base plays with integer coefficients. The outcome in $\\OrdAlg[\\Scal]{X}$ then appears as the tensor of the identity over $\\Scal$ and the outcome over $\\OrdAlg[\\Rat]{X}$. The algebra $\\OrdAlg[\\Rat]{X}$ further decomposes into the direct sum of the strict order algebras $\\OrdAlgS[\\Rat]{Y}$ for all representant subset *Y* with the permutation group induced over it. This is particularly useful since the $\\OrdAlgS[\\Rat]{Y}$ are finite dimensional vector spaces over $\\Rat$.\nOn the other hand, if $\\Scal$ is neither idempotent nor a regular ring, it is possible that there is no base. For instance, if $\\Scal=\\Nat$, then clearly a play *s* cannot be decomposed as a non-trivial sum of vectors, so any generating family must contain all plays, but then the equation of proposition [prop:split] states that they are not linearly independent.\nLogical structure\n=================\nIn this section, we describe constructions on order algebras. Although order algebras themselves have some interesting structure, the actual objects we are interested in are submodules of such algebras, hereafter called types, which enjoy better properties.\nA *type* over an arena *X* is a submodule of $\\OrdAlg{X}$ generated by a family of plays in $\\Plays{X}$. A type is *strict* if it does not contain the empty play. The notation *A* : *X* is used to represent the fact that *A* is a type over *X*. A *morphism* between types *A* : *X* and *B* : *Y* is a linear map *f* from $\\OrdAlg{X}$ to $\\OrdAlg{Y}$ such that *f*(*A*) ⊂ *B*.\nThe requirement that types are generated by plays is justified by the idea that a type should be a constraint on the behaviours of processes, and that such a constraint should boil down to a constraint on the shape of plays that a process can exhibit. We could also define a type over *X* simply as a subset *S* of $\\Plays{X}$, but the definition as submodules makes it clear that observationally equivalent vectors should belong to the same types, even if one is a combination of plays in *S* while the other is not (this can happen even if *S* is closed under permutations, because of the equation of Proposition [prop:split]).\nThe intended meaning of order algebras is that vectors, that is linear combinations of plays, represent processes. Then types impose constraints on the possible behaviours of processes, based on the possible interactions scenarii they may exhibit. For instance, we can define the type of processes that perform three actions of label *a*, as the submodule generated by the plays that contain three points in the orbit *a*. Similarly, we could define the type of all plays that include as many *a*’s as *b*’s.\nTyped may also used in particular to impose well-formedness conditions. For instance, when modelling a calculus like that includes communication of bound names, one wants to impose that any play that contains an event on a bound name also contains the event that communicates this name.\n[ex:type] Note that the condition of being a submodule of $\\OrdAlg{X}$ imposes non-trivial conditions. For instance, consider the CCS algebra, as of Example [ex:ccs], with $\\Scal$ idempotent. We can define the type of processes in which all actions *a* are causally independent of all actions *b*, as generated by the plays where all occurrences of *a* are incomparable with all occurrences of *b*. This type does not contain the processes *a*.*b* and *b*.*a*, obviously, but it does contain their sum *a*.*b* + *b*.*a*, which is observationally equivalent to the parallel composition $a\\para b$.\n[prop:type-basis] If $\\Scal$ is a rational ring, then for all type *A* : *X* there is a family of plays (*c**i*)*i*∈*I* and a family of vectors (*c**i*\\*)*i*∈*I* in $\\OrdAlg{X}$ such that (*c**i*)*i*∈*I* is a basis of *A* and for all *i*, *j*∈*I*, $\\outcome{c\\_i\\psync c^\\*\\_j}$ is 1 if *i* = *j* and 0 otherwise.\nRecall that if $\\Scal$ is a rational ring, then it is an algebra over $\\Rat$ and $\\OrdAlg[\\Scal]{X}$ can be seen as the $\\Rat$-algebra $\\Scal\\otimes\\OrdAlg[\\Rat]{X}$. Since *A* is generated by plays, we can then decompose it as $\\Scal\\otimes A'$ for a type *A*ʹ in $\\OrdAlg[\\Rat]{X}$, so it is enough to prove the result in the case $\\Scal=\\Rat$. In this case *A* is a subspace of the vector space $\\OrdAlg[\\Rat]{X}$, so it is a standard result that from the generating family we can extract a basis.\nNow assume that (*c**i*)*i*∈*I* is a basis of *A*, and consider a particular base play *c**n*. Set $J:=\\set{i∈I}{\\web{c\\_i}=\\web{c\\_n}}$. Then (*c**i*)*i*∈*J* is a basis of the intersection of *A* and $\\OrdAlgS{\\web{c\\_n},\\rpgroup{\\web{c\\_n}}}$, the strict order algebra over $\\web{c\\_n}$ with the induced permutation group, which is a finite-dimensional $\\Rat$-vector space. Let *f* be a linear form over this algebra such that *f*(*c**n*) = 1 and for all *j*∈*J*∖{*i*}, *f*(*c**j*) = 0. Using the bases (*b**n*) and (*b**n*\\*) from Theorem [thm:basis] we can define $c^\\*\\_n=\\sum\\_{\\web{b\\_k}=\\web{c\\_n}}f(b\\_k)b^\\*\\_k$ and check that for all vector *x* with $\\web{x}=\\web{c\\_n}$ we have $f(x)=\\outcome{c^\\*\\_n\\psync x}$. Then *c**n*\\* satisfies the expected condition.\nProducts and linear maps\n------------------------\nWhen combining order algebras, we need a notion of combination of arenas. Disjoint union is the simplest way, and also the most sensible one:\nLet (*X**i*)*i* ∈ *I* be a family of arenas with pairwise disjoint webs. Define the sum of the family (*X**i*) as \n$$\\sum\\_{i∈I} X\\_i :=\n \\Bigl( \\biguplus\\_{i\\in I} X\\_i, \\prod\\_{i∈I} \\pgroup[X\\_i] \\Bigr)\n \\quad \\text{with} \\quad\n σ⋅x := σ\\_j⋅x\n \\quad \\text{for all }\n σ∈\\prod\\_{i∈I} \\pgroup[X\\_i] \\text{ and }\n x∈\\web{X\\_j}.$$ \n We use equivalently the infix notation *X*1 + *X*2 + ⋯ + *X**n* for finite sums.\n[ex:sum-csp] Following on our first examples, if *X**A* and *X**B* are the arenas used for modelling CSP processes over alphabets *A* and *B* respectively (see Example [ex:csp]), then assuming *A* and *B* are disjoint the webs $\\web{X\\_A}$ and $\\web{X\\_B}$ are disjoint too and *X**A* + *X**B* is actually the arena for processes in the alphabet *A* ⊎ *B*.\nThis sum can be seen as a coproduct in a suitable category of arenas. At the level of order algebras, however, this operation is not a Cartesian product or coproduct, and not even a tensor product in the sense of $\\Scal$-algebras, because the algebra $\\OrdAlg{X+Y}$ contains more plays than those that appear as disjoint unions of a play in $\\web{X}$ and one in $\\web{Y}$. However, $\\OrdAlg{X+Y}$ contains products and tensors as submodules, hence our definition of types. In all statements below, unless explicitly stated, different arenas are always supposed to be disjoint.\nFor all types *A* : *X* and *B* : *Y*, *A* + *B* is a type over *X* + *Y* that is isomorphic to the direct sum and Cartesian product of *A* and *B*.\n*A* is a submodule of $\\OrdAlg{X}$, which is itself obviously a submodule of $\\OrdAlg{X+Y}$. Similarly, *B* is a submodule of $\\OrdAlg{X+Y}$, and since *X* and *Y* are disjoint, so are *A* and *B*, since no permutation in *X* + *Y* can map a point of $\\web{X}$ to a point of $\\web{Y}$. Hence the submodule generated by $A+B=\\Vect{A\\cup B}$ in $\\OrdAlg{X+Y}$ is a direct sum of *A* and *B*.\nLet *X* be an arena, let *Y* be a subset of $\\web{X}$ closed under permutations in $\\pgroup[X]$. Restriction to *Y* is the linear map $\\restric{Y}{}$ over $\\PreOrdAlg{X}$ such that for all $r∈\\Plays{X}$, \n$$\\restric{Y}{r} :=\n \\outcome{r} ⋅ (\\web{r}\\cap Y, \\order{r}\\cap Y^2)$$ \nRestriction of a play *r* to a given subset $Y\\subset\\web{X}$ amounts to ignore the part of *r* that happens outside *Y*, considering that events in $\\web{X}∖ Y$ are private, hence unobservable. The fact that *Y* must be closed under permutations is in accordance with the intuition that two plays are indistinguishable when they are permutations of each other.\nFollowing on example [ex:sum-csp], $\\restric{X\\_A}:\\PreOrdAlg{X\\_A+X\\_B}→\\PreOrdAlg{X\\_A}$ precisely represents CSP’s restriction operator that maps a trace *t* to the restricted trace $t\\upharpoonright A$.\nHence, as we shall see (in detail in Section [sec-traces]), restriction does *not* correspond to the hiding operator $\\new{}$ of the -calculus and related languages. Indeed, the externally observable behaviours of $\\new{u}P$ are those of *P* that do not involve an event on *u*, which to mapping to 0 all plays in *P* that contain an event on *u*, before actually restricting to the arena that does not contain events on *u*.\nNote that in the definition we impose a coefficient $\\outcome{r}$ on the restricted play. Since the outcome $\\outcome{r}$ is 0 or 1 for any play *r*, this amounts to imposing that $\\restric{Y}{r}$ be 0 if *r* is inconsistent. This condition is necessary because we want outcomes to be preserved by restriction: if a play is inconsistent, it means that it contains some deadlock, and hiding the place where this occurs surely should not resolve the deadlock. For instance, an inconsistent play like $(\\tikzinline{\n \\eventnode{(0,0)}{a}{left}{a}\n \\eventnode{(1,0)}{b}{right}{b}\n \\draw (a) to [bend left] (b) to [bend left] (a); })$, when restricted to {*a*}, would yield the consistent play $(\\tikzinline{ \\eventnode{(0,0)}{a}{left}{a} })$.\nIncidentally, this implies that $\\restric{X}{}:\\PreOrdAlg{X}→\\PreOrdAlg{X}$ is not the identity, because it collapses all inconsistent plays to 0. However, up to observational equivalence, it is the identity.\n[prop:restric-compat] Restriction is compatible with observational equivalence.\nFirst observe that, since *Y* is supposed to be closed by permutations, restriction commutes with permutations, hence $\\psplit(\\restric{Y}{u})=\\restric{Y}{\\psplit(u)}$. Then by the representation property (Proposition [prop:represent]), if $u\\obseqv v$ then $\\psplit(u)\\obseqv[\\web{X}]\\psplit(v)$, so it suffices to prove that restriction is compatible with observational equivalence in static arenas.\nLet *Z* be a finite subset of $\\web{X}∖ Y$, define $\\extend{Z}$ as the linear map such that for all $t∈\\Plays{X}$, $\\extend{Z}{t}$ is the play on $\\web{t}\\cup Z$, whose preorder relation is $\\order{t}$ extended as the identity relation on *Z*. Let *r* be a play in $\\Plays{X}$ such that $\\web{r}∖ Y=Z$. Suppose *r* is acyclic, then for all play *s* with $\\web{s}\\cup Z=\\web{r}$ the relation $\\order{r}\\cup\\order{s}$ is acyclic if and only if $(\\order{r}\\cap Y^2)\\cup\\order{s}$ is acyclic, so we have $\\outcome{r\\sync\\extend{Z}{s}}=\\outcome{\\restric{Y}{r}\\sync s}$. If *r* is not acyclic, then the equality holds too since both sides are 0. The equality extends trivially to all plays *s* such that $\\web{s}\\subset Y$.\nConsider a pair $u\\obseqv[\\web{X}]v$ and a play $s∈\\Plays{X}$ with $\\web{s}\\subset Y$. By Proposition [prop:direct-sum] we can decompose *u* as $\\sum\\_{C∈\\PartsFin{\\web{X}}}u\\_C$ with $u\\_C∈\\PreOrdAlgS{C}$ and similarly for *v* so that for each *C* we have $u\\_C\\obseqv[\\web{X}]v\\_C$. For a given *C*, let *Z* = *C*∖*Y*, then by linearity of the equation in the previous paragraph we get $\\outcome{\\restric{Y}{u\\_Z}\\sync s}=\\outcome{u\\_Z\\sync\\extend{Z}{s}}$ for all play *s* with $\\web{s}\\subset Y$, and by the equivalence of *u**Z* and *v**Z* we get $\\outcome{\\restric{Y}{u\\_Z}\\sync s}=\\outcome{\\restric{Y}{v\\_Z}\\sync s}$. This trivially holds too if $\\web{s}\\not\\subset Z$, so we get the equivalence $\\restric{Y}{u\\_Z}\\obseqv\\restric{Y}{v\\_Z}$, and we deduce $\\restric{Y}{u}\\obseqv\\restric{Y}{v}$ by linearity.\nLet *X*, *Y*, *Z* be three arenas with pairwise disjoint supports. Define *partial static synchronisation* along *X* as the bilinear map $\\sync[X]$ from $\\PreOrdAlg{X+Y}×\\PreOrdAlg{X+Z}$ to $\\PreOrdAlg{X+Y+Z}$ such that for all $r∈\\Plays{X+Y}$ and $s∈\\Plays{X+Z}$, \n$$r \\sync[X] s :=\n \\begin{cases}\n \\bigl( \\web{r}\\cup\\web{s}, (\\order{r}\\cup\\order{s})^\\* \\bigr)\n &\\text{if } \\web{r}\\cap\\web{X} = \\web{s}\\cap\\web{X} \\\\\n 0 &\\text{otherwise}\n \\end{cases}$$ \n Deduce partial permuted synchronisation as \n$$r \\psync[X] s := \\multiplicity[X]{\\restric{X}{s}}\n \\sum\\_{s'∈\\porbit[X]{s}} r \\sync[X] s'$$ \nConsider an arena *X* containing at least two interchangeable actions labelled *a*1, *a*2 and arenas *Y* and *Z* containing events *b* and *c* respectively. Then we have the partial static synchronisation \n$$\\tikzformula{\n \\eventnode{(0,0)}{b}{left}{b}\n \\eventnode{(0,1)}{a1}{left}{a\\_1}\n \\eventnode{(1,0)}{a2}{right}{a\\_2}\n \\draw (b) -- (a1); }\n \\sync[X]\n \\tikzformula{\n \\eventnode{(0,0)}{a1}{left}{a1}\n \\eventnode{(0,1)}{c}{left}{c}\n \\eventnode{(1,0)}{a2}{right}{a\\_2}\n \\draw (a1) -- (c); }\n = \\tikzformula{\n \\eventnode{(0,0)}{b}{left}{b}\n \\eventnode{(0,1)}{a1}{left}{a1}\n \\eventnode{(0,2)}{c}{left}{c}\n \\eventnode{(1,0)}{a2}{right}{a\\_2}\n \\draw (b) -- (c); }$$ \n and the partial permuted synchronisation \n$$\\tikzformula{\n \\eventnode{(0,0)}{b}{left}{b}\n \\eventnode{(0,1)}{a1}{left}{a\\_1}\n \\eventnode{(1,0)}{a2}{right}{a\\_2}\n \\draw (b) -- (a1); }\n \\psync[X]\n \\tikzformula{\n \\eventnode{(0,0)}{a1}{left}{a1}\n \\eventnode{(0,1)}{c}{left}{c}\n \\eventnode{(1,0)}{a2}{right}{a\\_2}\n \\draw (a1) -- (c); }\n = \\tikzformula{\n \\eventnode{(0,0)}{b}{left}{b}\n \\eventnode{(0,1)}{a1}{left}{a1}\n \\eventnode{(0,2)}{c}{left}{c}\n \\eventnode{(1,0)}{a2}{right}{a\\_2}\n \\draw (b) -- (c); }\n + \\tikzformula{\n \\eventnode{(0,0)}{b}{left}{b}\n \\eventnode{(0,1)}{a1}{left}{a1}\n \\eventnode{(1,0)}{a2}{right}{a\\_2}\n \\eventnode{(1,1)}{c}{right}{c}\n \\draw (b) -- (a1) (a2) -- (c); }.$$ \n The factor $\\multiplicity[X]{\\restric{X}{s}}$ (which is 1 in this example) plays the same role as in full synchronisation (Definition [def:psync]), remarking that we only apply permutations on the *X* part.\nLet *X*, *Y*, *Z* be three arenas with pairwise disjoint supports. Partial synchronisation along *X* is associative as \n$$(u\\psync[X]v)\\psync[X+Y+Z]w\n = u\\psync[X+Y](v\\psync[X+Z]w)$$ \n for all $u∈\\PreOrdAlg{X+Y}$, $v∈\\PreOrdAlg{X+Z}$ and $w∈\\PreOrdAlg{X+Y+Z}$. It is compatible with observational equivalence and commutative up to equivalence.\nLet *X*, *Y*, *Z* be three disjoint arenas. Consider three plays $r∈\\Plays{X+Y}$, $s∈\\Plays{X+Z}$ and $t∈\\Plays{X+Y+Z}$. The partial synchronisations $(r\\sync[X]s)\\sync[X+Y+Z]t$ and $r\\sync[X+Y](s\\sync[X+Z]t)$ are non-zero if and only if we can define \n$$A=\\web{r}\\cap\\web{X}=\\web{s}\\cap\\web{X}=\\web{t}\\cap\\web{X}, \\quad\n B=\\web{r}\\cap\\web{Y}=\\web{t}\\cap\\web{Y}, \\quad\n C=\\web{s}\\cap\\web{Z}=\\web{t}\\cap\\web{Z},$$ \n and in this case the result is the play on *A* ∪ *B* ∪ *C* whose preorder relation is $(\\order{r}\\cup\\order{s}\\cup\\order{t})^\\*$, so we have \n$$(r\\sync[X]s)\\sync[X+Y+Z]t = r\\sync[X+Y](s\\sync[X+Z]t).$$ \n Assume representants are chosen in each arena in such a way that for $D\\subset\\web{X}$ and $E\\subset\\web{Y}$, $\\repr{D\\cup E}=\\repr{D}\\cup\\repr{E}$, and similarly for *X* + *Z* and *X* + *Y* + *Z*. Choosing representants this way is always possible since permutations of *X*, *Y* and *Z* are independent in the sum arenas. Suppose *r*, *s*, *t* and *A*, *B*, *C* are representants. $\\rpgroup{A}$ is the same in all sums of arenas that involve *X*, and similarly for $\\rpgroup{B}$ and $\\rpgroup{C}$. Moreover we have $\\rpgroup{A\\cup B\\cup C}=\\rpgroup{A}×\\rpgroup{B}×\\rpgroup{C}$, and similarly for other unions, so by similar considerations as for permuted synchronisation, we get\n \n$$\\begin{gathered}\n (r\\psync[X]s)\\psync[X+Y+Z]t\n = \\sum (r\\sync[X]σ\\_As)\\sync[X+Y+Z]σ'\\_Aσ\\_Bσ\\_Ct\n = \\sum r\\sync[X+Y](σ\\_As\\sync[X+Z]σ'\\_Aσ\\_Bσ\\_Ct) \\\\\n = \\sum r\\sync[X+Y]σ\\_Aσ\\_B(s\\sync[X+Z]σ'\\_Aσ\\_Ct)\n = r\\psync[X+Y](s\\psync[X+Z]t)\n \\end{gathered}$$ \nwhere the sums are indexed on $(σ\\_A,σ'\\_A,σ\\_B,σ\\_C)\n ∈\\rpgroup{A}×\\rpgroup{A}×\\rpgroup{B}×\\rpgroup{C}$. Partial synchronisation commutes with permutations so this equality extends to plays that are not representants, and by linearity is extends to arbitrary vectors.\nNow consider $u,u'∈\\PreOrdAlg{X+Y}$, $v∈\\PreOrdAlg{X+Z}$ and $w∈\\PreOrdAlg{X+Y+Z}$, and suppose $u\\obseqv u'$. By the same arguments as in the proof of Proposition [prop:restric-compat], we get the equality $\\outcome{u\\psync[X+Y](v\\psync[X+Z]w)}\n =\\outcome{u\\psync\\restric{X+Y}(v\\psync[X+Z]w)}$, then $u\\obseqv u'$ implies that these are equal to $\\outcome{u'\\psync[X+Y](v\\psync[X+Z]w)}$, and applying associativity on this we can deduce $u\\psync[X]v\\obseqv u'\\psync[X]v$. Commutativity of partial synchronisation is obvious, and it yields the compatibility with observational equivalence on the right.\nBy similar arguments, we prove other “localized” associativities, the general case being \n$$( u \\psync[A+B] v ) \\psync[A+C+D] w\n = u \\psync[A+B+C] ( v \\psync[A+D] w )$$ \n for $u∈\\OrdAlg{A+B+C+E}$, $v∈\\OrdAlg{A+B+D+F}$ and $w∈\\OrdAlg{A+C+D+G}$, where *A*, *B*, *C*, *D*, *E*, *F*, *G* are seven (!) pairwise disjoint arenas. Although this formulation is frighteningly heavy, the point is rather simple: when partially synchronising two vectors *u* and *v*, synchronise them along the arenas they have in common, and the result will be on the union of the arenas of *u* and *v*.\nThe simplest case of partial synchronisation is when “synchronising” two vectors $u∈\\OrdAlg{X}$ and $v∈\\OrdAlg{Y}$ along the empty arena, yielding $u\\psync[\\emptyset]v∈\\OrdAlg{X+Y}$. In this case, *u* and *v* are essentially kept independent, which in particular implies \n$$\\outcome{u\\psync[\\emptyset]v} = \\outcome{u}\\outcome{v}.$$ \n This is deduced by linearity from the case of plays, remarking that for $r∈\\Plays{X}$ and $s∈\\Plays{Y}$, $r\\psync[\\emptyset]s$ is the disjoint union of *r* and *s*, which is consistent if and only if *r* and *s* are consistent.\nLet *X* and *Y* be two arenas. Define the bilinear map ⊗ from $\\OrdAlg{X}×\\OrdAlg{Y}$ to $\\OrdAlg{X+Y}$ as $u\\otimes v:=u\\psync[\\emptyset]v$. For two types *A* : *X* and *B* : *Y*, define *A* ⊗ *B* as the submodule of $\\OrdAlg{X+Y}$ generated by the image of *A*×*B* by ⊗ .\nSimply put, *A* ⊗ *B* is the $\\Scal$-module consisting of processes that can be written as juxtapositions of a process in *A* and a process in *B* with no scheduling constraint between them, or as a sums of such things, *up to observational equivalence*. As illustrated in Example [ex:type], this does not imply that any vector ∑*i*∈*I**λ**i**r**i*∈*A* ⊗ *B* is syntactically a sum of $u\\_i\\psync[∅]v\\_i$ with *u**i*∈*A* and *v**i*∈*B*.\nIf $\\Scal$ is a rational ring, then for all types *A* : *X* and *B* : *Y*, *A* ⊗ *B* is the tensor product of *A* and *B* in the sense of $\\Scal$-algebras.\nBy Proposition [prop:type-basis] the types *A* and *B* have bases (*b**i*)*i*∈*I* and (*c**j*)*j*∈*J*, and there are families of vectors (*b**i*\\*)*i*∈*I* and (*c**j*\\*)*j*∈*J* such that each *b**n*\\* identifies *b**n* among the elements of (*b**i*)*i*∈*I*, and similarly for *c**n*\\*. We prove that the vectors $b\\_i\\psync[\\emptyset]c\\_j$ are linearly independent. Consider a linear combination $u=\\sum\\_{(i,j)∈I× J}λ\\_{ij}(b\\_i\\psync c\\_j)$ in $\\OrdAlg{X+Y}$. For each (*m*, *n*)∈*I*×*J* we have\n \n$$\\begin{gathered}\n \\outcome{ u \\psync[X+Y] (b^\\*\\_m \\psync[\\emptyset] c^\\*\\_n) }\n = \\sum\\_{(i,j)∈I× J} λ\\_{mn}\n \\outcome{ (b\\_{i} \\psync[\\emptyset] c\\_{j})\n \\psync[X+Y] (b^\\*\\_m \\psync[\\emptyset] c^\\*\\_n) } \\\\\n = \\sum\\_{(i,j)∈I× J} λ\\_{mn}\n \\outcome{ (b\\_{i} \\psync[X] b^\\*\\_m)\n \\psync[\\emptyset] (c\\_{j} \\psync[Y] c^\\*\\_n) }\n = \\sum\\_{(i,j)∈I× J} λ\\_{mn}\n \\outcome{ b\\_{i} \\psync[X] b^\\*\\_m }\n \\outcome{ c\\_{j} \\psync[Y] c^\\*\\_n }\n \\end{gathered}$$ \nusing the associativity properties stated above. By definition of *b**m*\\* and *c**n*\\*, the only non-zero term in the final sum is for (*i*, *j*) = (*m*, *n*), and this term is *λ**m**n*. Applying this on every (*m*, *n*) implies the unicity of the decomposition of *u* on the *b**m* ⊗ *c**n*. So the *b**m* ⊗ *c**n* form a linearly independent family, which proves that *A* ⊗ *B* is isomorphic to the tensor product of *A* and *B*, as $\\Scal$-modules. The associativity property ensures that they are also isomorphic as $\\Scal$-algebras.\nLet *X*, *Y* and *Z* be three arenas. Let $u∈\\OrdAlg{X+Y}$ and $v∈\\OrdAlg{Y+Z}$. Composition of *u* and *v* through *Y* is the vector $u\\compo[Y]v:=\\restric{X+Z}{(u\\psync[Y]v)}∈\\OrdAlg{X+Z}$.\nLet *A* : *X* and *B* : *Y* be two types. The type $A\\multimap B:X+Y$ is the submodule of $\\OrdAlg{X+Y}$ generated by all plays *r* such that for all *u*∈*A*, $r\\compo[X]u∈B$.\nBy the remarks above, we get associativity of composition. In the special case where *X* is the empty arena, $\\OrdAlg{X+Y}$ is equal to $\\OrdAlg{Y}$ and $u\\compo[Y]v$ is a vector in *Z*, so *v* induces a linear map from $\\OrdAlg{Y}$ to $\\OrdAlg{Z}$. However, this mapping from vectors of $\\OrdAlg{X+Y}$ to linear maps from $\\OrdAlg{X}$ to $\\OrdAlg{Y}$ is neither injective nor surjective.\nIt is easy to check the standard adjunction $A\\multimap(B\\multimap C)=(A\\otimes B)\\multimap C$ for all types *A*, *B*, *C* of pairwise disjoint supports. Moreover, if we call $\\one$ the non-trivial type over the empty arena, which is isomorphic to $\\Scal$, we have for all type *A* that $A\\otimes\\one=\\one\\otimes A=\\one\\multimap A=A$.\nBialgebraic structure\n---------------------\n[def:indexing] Let *X* and *Y* be two arenas. Define the *indexing* of *Y* by *X* as the arena \n$$X\\indexing Y := \\bigl( \\web{X}×\\web{Y},\n \\pgroup[X]×(\\pgroup[Y])^{\\web{X}} \\bigr)$$ \n where permutations act as \n(*σ*, *φ*)(*x*, *y*) : = (*σ**x*, *φ*(*x*) *y*) \nWe interpret indexing as follows: $\\web{X\\indexing Y}$ consists of copies of *Y* indexed by points of *X*. A permutation in $X\\indexing Y$ consists in permuting each copy independently, using the function $φ:\\web{X}\\to\\pgroup[Y]$ that provides a permutation for each copy, and then permuting the copies themselves using a permutation in *X*.\nNote that we easily get the equality $(X+Y)\\indexing Z=(X\\indexing Z)+(Y\\indexing Z)$, however $X\\indexing(Y+Z)$ is not equal to $(X\\indexing Y)+(X\\indexing Z)$, since permutations of copies in the former operate the same way on the copies of *X* and those of *Y*, while in the latter they may not. There is also an isomorphism between $(X\\indexing Y)\\indexing Z$ and $X\\indexing(Y\\indexing Z)$, and these appear as $(\\web{X}×\\web{Y}×\\web{Z},\n\\pgroup[X]×(\\pgroup[Y])^{\\web{X}}×(\\pgroup[Z])^{\\web{X}×\\web{Y}})$.\nThe structure of the indexing arena is used only for identifying and permuting copies, in particular we will not consider plays on this arena. The primary purpose of indexing is to build an arena in which the symmetric algebra over a given type will fit. It also generalises the direct sum when the indexing arena is static.\nLet $\\PNat$ be the arena with $\\web{\\PNat}=\\Nat$, the set of natural numbers, and $\\pgroup[\\PNat]=\\Perm{\\Nat}$, the group of all permutations of $\\Nat$. For all arena *X*, define $\\sharp X:=\\PNat\\indexing X$.\nSo the arena $\\PNat\\indexing X$ contains a countable number of interchangeable copies of *X*. Another useful construct is the following: identifying each integer *n* with the set {0, …, *n* − 1}, which is in turn identified with the static arena with this set as the web, the arena $n\\indexing X$ is isomorphic to the sum *X* + ⋯ + *X* with *n* independent copies of *X*. If *n* = 0, this yields the empty arena ∅. Then $n\\indexing\\sharp X=n\\indexing\\PNat\\indexing X$ contains a countable set of copies of *X*, partitioned into *n* countable classes of interchangeable copies.\n[def:gamma-delta] Let *n* be a strictly positive integer, let *φ* be a bijection from $n×\\Nat$ to $\\Nat$. For all arena *X*, define the function $γ^n\\_φ:\\Plays{n\\indexing\\sharp X}\\to\\Plays{\\sharp X}$ as \n$$\\web{γ^n\\_φs} := \\set[2]{ (φ(i),x) }{ (i,x) ∈\\web{s} }\n \\quad\\text{and}\\quad\n (φ(i),x) ≤\\_{γ^n\\_φ{s}} (φ(j),y)\n \\text{ iff } (i,x) ≤\\_s (j,y).$$ \n Define the linear map $δ^n:\\PreOrdAlg{\\sharp X}\\to\\PreOrdAlg{n\\indexing\\sharp X}$ as \n$$δ^ns := \\sum\\_{c\\,:\\,\\pi\\_1(\\web{s})\\to n} c\\bullet s\n \\quad\\text{with}\\quad\n \\begin{array}{l}\n \\web{c\\bullet s} := \\set[2]{ ((c(i),i),x) }{ (i,x)∈\\web{s}, i∈A } \\\\\n ((c(i),i),x) ≤\\_{c\\bullet s} ((c(j),j),y) \\text{ iff } (i,x) ≤\\_s (j,y)\n \\end{array}$$ \n where *π*1 is the first projection, so $\\pi\\_1(\\web{s})=\\set{i}{(i,x)∈\\web{s}}$.\nThe function *γ**φ**n* is a simple renaming of the copies of *X* using the function *φ*, which extends the bijection $φ:n×\\Nat\\to\\Nat$ to a bijection between $\\Plays{n\\indexing\\sharp X}$ and $\\Plays{\\sharp X}$. As explained below, this bijection is compatible with observational equivalence, but its quotient is not injective. Instead, it fuses the *n* independent copies of $\\sharp X$ into one, which makes events from different copies interchangeable.\nThe linear map *δ**n* acts as a non-deterministic inverse operation. Given a play *s* in $\\Plays{\\sharp X}$, it enumerates all possible ways of partitioning the events of *s* into *n* identified subsets. The function *c* represents such a choice, and *c* • *r* applies this choice to the play *r*.\nAs we shall see, the operators *γ**φ**n* and *δ**n* are very similar to a multiplication and comultiplication in a bialgebra. They are analogous to concatenation and deconcatenation, which give a bialgebraic structure to tensor algebras .\n[prop:gamma-delta] Let *X* be an arena and let *n* be a strictly positive integer. The maps *γ**φ**n* and *δ* are compatible with observational equivalence and the quotient map of *γ**φ**n* is independent of *φ*. For all vectors $u∈\\OrdAlg{n\\indexing\\sharp X}$ and $v∈\\OrdAlg{\\sharp X}$ we have \n$$γ^n(u) \\psync[\\sharp X] v \\obseqv[\\sharp X]\n u \\psync[n\\indexing\\sharp X] δ^n(v).$$ \nObserve that for any permutation $σ∈\\pgroup[n\\indexing\\Nat\\indexing\\sharp X]$ ( a family of independent permutations on each copy of *X* in $n\\indexing\\sharp X$) there is a permutation $σ'∈\\pgroup[\\Nat\\indexing\\sharp X]$ such that $γ^n\\_φ\\circσ=σ'\\circγ^n\\_φ$, and the other way around for *δ**n*. As a consequence, by Proposition [prop:represent], we can deduce the expected result from the case where *X* is static. Then all considered permutations are in $\\Perm{n×\\Nat}$ and $\\Perm{\\Nat}$.\nThe map *γ**φ**n* decomposes as the injection of $\\PreOrdAlg{(n\\indexing\\PNat)\\indexing X}$ into $\\PreOrdAlg{\\Perm{n×\\Nat}\\indexing X}$ and the renaming of $\\Perm{n×\\Nat}\\indexing X$ into $\\PNat\\indexing X$ through *φ*. The former consists in growing the permutation group on a fixed web and the latter is an isomorphism, so both are compatible with observational equivalence. For *δ**n*, given a permutation $σ∈\\Perm{\\Nat}$ and a play *s*, for all choice function *c* for *σ**s* we have $c\\bulletσ s=σ'(c'\\bullet s)$ with $c'=c\\circσ$ and *σ*ʹ(*i*, *j*) = (*i*, *σ*(*j*)), which establishes a bijection between the choices of *s* and those of *σ**s*. From this we can conclude that *δ**n* is compatible with observational equivalence.\nLet $r∈\\Plays{n\\indexing\\sharp X}$, let $s∈\\Plays{\\sharp X}$ and let *φ* be a bijection from $n×\\Nat$ to $\\Nat$. Assume *s* is a representant. First suppose $\\repr{\\web{γ^n\\_φ(r)}}\\neq\\web{s}$, then $γ^n(r)\\psync s$ is zero. Suppose that there is a choice *c* such that $\\repr{\\web{c\\bullet s}}=\\web{r}$, then we get a permutation $σ∈\\pgroup[n\\indexing\\sharp X]$ that induces a bijection from $\\web{c\\bullet s}$ to $\\web{r}$. By definition *σ* is a bijection between the pairs (*c*(*i*), *i*) and $\\pi\\_1(\\web{r})$, which can be extended into a bijection *ψ* from $n×\\Nat$ to $\\Nat$, such that $\\repr{\\web{γ^n\\_φ(r)}}=\\web{s}$. This contradicts the hypothesis $\\repr{\\web{γ^n\\_φ(r)}}\\neq\\web{s}$ since *γ**φ**n*(*r*) and *γ**ψ**n*(*r*) are necessarily permutations of each other, from the remarks above. Hence for all *c* we have $\\repr{\\web{c\\bullet s}}\\neq\\web{r}$, so $r\\psyncδ^n(s)=0$, and the equality holds.\nNow suppose $\\repr{\\web{γ^n\\_φ(r)}}=\\web{s}$. Applying a suitable permutation to *r* and choosing *φ* appropriately (we know from the above remarks that these operations are allowed) we can assume that *γ**φ**n*(*r*) is a representant, so $\\web{γ^n\\_φ(r)}=\\web{s}$, and $γ^n\\_φ(r)\\psync s=\\sum\\_{σ∈G}φ r\\syncσ s\n \\obseqv\\sum\\_{σ∈G}r\\syncφ^{-1}σ s$, where *G* is the group of permutations of $\\web{s}$ induced by $\\pgroup[\\sharp X]$, that is the symmetric group of $\\pi\\_1(\\web{s})$. For each *σ*∈*G*, the function *π*1*φ*− 1*σ* is a choice function *c**σ* over $\\pi\\_1(\\web{s})$, and *σ*\\*(*i*, *j*) : = (*i*, *σ*− 1*φ*(*i*, *j*)) is a permutation in $\\pgroup[n\\indexing\\Nat]$ such that *σ*\\**φ*− 1*σ*(*i*) = (*c**τ*(*i*), *i*), hence *σ*\\**φ*− 1*σ*(*s*) = *c**σ* • *s*. By partitioning the sum for *τ*∈*G* according to choice functions, we get $γ^n\\_φ(r)\\psync s \\obseqv\n \\sum\\_c\\sum\\_{σ∈G,c\\_σ=c}r\\syncσ^{\\*{-1}}(c\\_σ\\bullet s)$. By construction, for a fixed *c*, we have $\\set{σ'^{\\*{-1}}}{σ'∈G,c\\_{σ'}=c}\n =\\pgroup[n\\indexing\\Nat]σ^{\\*{-1}}$, so we get $γ^n\\_φ(r)\\psync s\\obseqv[\\sharp X]\n \\sum\\_cr\\psync[n\\indexing\\sharp X](c\\bullet s)\n =r\\psync[n\\indexing\\sharp X]δ^n(s)$, from which we conclude by linearity.\nAs a consequence, $\\OrdAlg{\\sharp X}$ has the structure of a commutative algebra with *γ*2 as the multiplication and the empty play as the unit. The *δ*2 does not make it a bialgebra in general, because for an arbitrary $u∈\\OrdAlg{\\sharp X}$, $δ^2(u)∈\\OrdAlg{\\sharp X+\\sharp X}$ has no reason to be in the tensor product $\\OrdAlg{\\sharp X}\\"},"data_id":{"kind":"string","value":"arxiv_0184772"}}},{"rowIdx":46198,"cells":{"content":{"kind":"string","value":" A. 2010, ApJ, 712, 294 Evans, N. J., II 2008, Pathways Through an Eclectic Universe, 390, 52 Fraternali, F. & Binney, J. J. 2008, MNRAS, 386, 935 Frisch U. 1995, Turbulence – The Legacy of A.N. 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D., & Leroy, A. 2008,, 136, 2563 Wolf-Chase G.A., Barsony M., O’Linger J. 2000, AJ, 120, 1467 Wolfire M.G., Hollenbach D., McKee C.F., Tielens A.G.G.M., & Bakes E.L.O. 1995, ApJ, 443, 152 Wong T. & Blitz L. 2002, ApJ, 569, 157\nlll Parameter & Unit & Explanation \n*G* = 5 × 10− 15 & pc3yr− 1M⊙− 1 & gravitation constant \n*κ* & yr− 1 & epicyclic frequency \n*Q* & & Toomre parameter \n*R* & pc &galactocentric radius \n*H* & pc & thickness of the gas disk \n*H*\\* & pc & thickness of the stellar disk \n$R\\_{\\rm break}$ & pc & break radius between star formation regimes \n$l\\_{\\rm cl}$ & pc & cloud size \n$v\\_{\\rm rot}$ & pcyr− 1 & rotation velocity \n$\\Omega=v\\_{\\rm rot}/R$ & yr− 1 & angular velocity \n$\\Phi\\_{\\rm V}$ & & volume filling factor \n$\\Phi\\_{\\rm A}=\\Phi\\_{\\rm V}\\,H/l\\_{\\rm cl}$ & & area filling factor \n*ρ* & M⊙pc− 3 & disk midplane gas density \n$\\rho\\_{\\rm cl}=\\rho/\\Phi\\_{\\rm V}$ & M⊙pc− 3 & cloud density \n$\\dot{\\rho}\\_{\\*}$ & M⊙pc− 3yr− 1 & star formation rate \nΣ & M⊙pc− 2 & gas surface density \nΣ\\* & M⊙pc− 2 & stellar surface density \n$\\dot{\\Sigma}\\_{\\*}$ & M⊙pc− 2yr− 1 & star formation rate \n*ξ* = 4.6 × 10− 8 & pc2yr− 2 & constant relating SN energy input to SF \n$\\dot{M}$ & M⊙yr− 1 & disk mass accretion rate \n$v\\_{\\rm turb}$ & pcyr− 1 & gas turbulent velocity dispersion \n$v\\_{\\rm rad}$ & pcyr− 1 & gas radial velocity \n$v\\_{\\rm disp}^{\\*}$ & pcyr− 1 & stellar vertical velocity dispersion \n*ν* & pc2yr− 1 & viscosity \n$f\\_{\\rm mol}=\\Sigma\\_{\\rm H\\_{2}}/(\\Sigma\\_{\\rm HI}+\\Sigma\\_{\\rm H\\_{2}})$ & & molecular fraction \n*α* & yrM⊙pc− 3 & constant of molecule formation timescale \n$l\\_{\\rm driv}$ & pc & turbulent driving length scale \n$l\\_{\\rm diss}$ & pc & turbulent dissipation length scale \n*δ* & & scaling between driving and dissipation length scale \n$SFE=\\dot{\\Sigma}\\_{\\*}/\\Sigma$ & yr− 1 & star formation efficiency \n$t\\_{\\rm ff}^{l}$ & yr & cloud free fall timescale at size *l* \n$t\\_{\\rm turb}^{l}$ & yr & cloud turbulent timescale at size *l* \n$t\\_{\\rm mol}^{l}$ & yr & cloud molecule formation timescale at size *l* \n*η* & & star formation efficiency per free fall time \n*γ* & & $t\\_{\\rm sf}^{l}= \\gamma t\\_{\\rm mol}^{l}$ at the break radius \nlccccccccc Galaxy & Dist. & i & P.A. & Morph. & $M\\_{\\rm B}$ & $R\\_{\\rm 25}$ & log *M*\\* & SFR & *l*\\* \n& (Mpc) & (∘) & (∘) & & (mag) & (kpc) & (M⊙) & (M⊙yr− 1) & (kpc) \nIC 2574 & 4.0 & 53 & 56 & Irr & -18.0 & 7.5 & 8.7 & 0.07 & 2.1 \nNGC 4214 & 2.9 & 44 & 65 & Irr & -17.4 & 2.9 & 8.8 & 0.11 & 0.7 \nNGC 2976 & 3.6 & 65 & 335 & Sc & -17.8 & 3.8 & 9.1 & 0.09 & 0.9 \nNGC 7793 & 3.9 & 50 & 290 & Scd & -18.7 & 6.0 & 9.5 & 0.24 & 1.3 \nNGC 2403 & 3.2 & 63 & 124 & SBc & -19.4 & 7.3 & 9.7 & 0.38 & 1.6 \nNGC 0925 & 9.2 & 66 & 287 & SBcd & -20.0 & 14.2 & 9.9 & 0.56 & 4.1 \nNGC 0628 & 7.3 & 7 & 20 & Sc & -20.0 & 10.4 & 10.1 & 0.81 & 2.3 \nNGC 3198 & 13.8 & 72 & 215 & SBc & -20.7 & 13.0 & 10.1 & 0.93 & 3.2 \nNGC 3184 & 11.1 & 16 & 179 & SBc & -19.9 & 11.9 & 10.3 & 0.90 & 2.4 \nNGC 4736 & 4.7 & 41 & 296 & Sab & -20.0 & 5.3 & 10.3 & 0.48 & 1.1 \nNGC 3351 & 10.1 & 41 & 192 & SBb & -19.7 & 10.6 & 10.4 & 0.94 & 2.2 \nNGC 6946 & 5.9 & 33 & 243 & SBc & -20.9 & 9.8 & 10.5 & 3.24 & 2.5 \nNGC 3627 & 9.3 & 62 & 173 & SBb & -20.8 & 13.9 & 10.6 & 2.22 & 2.8 \nNGC 5194 & 8.0 & 20 & 172 & SBc & -21.1 & 9.0 & 10.6 & 3.13 & 2.8 \nNGC 3521 & 10.7 & 73 & 340 & SBbc & -20.9 & 12.9 & 10.7 & 2.10 & 2.9 \nNGC 2841 & 14.1 & 74 & 153 & Sb & -21.2 & 14.2 & 10.8 & 0.74 & 4.0 \nNGC 5055 & 10.1 & 59 & 102 & Sbc & -20.6 & 17.4 & 10.8 & 2.12 & 3.2 \nNGC 7331 & 14.7 & 76 & 168 & SAb & -21.7 & 19.6 & 10.9 & 2.99 & 3.3 \nlccccccccc Galaxy & $\\dot{M}$ & *δ* & *γ* & $R\\_{\\rm break}$ & $c\\_{\\rm H\\_2}$ & $c\\_{\\rm v\\_{\\rm disp}}$ & $c\\_{\\rm SFE}$ & *g* & $< t\\_{\\rm ff} >$ \n& (M⊙yr− 1) & & & (kpc) & & & & & (Myr) \nKM05 & & & & & & & & \nIC2574 & 0.22 & 2.0 & 0.07 & 1.1 & 0.63 & 0.79 & 0.71 & 0.98 & 29.6 \nNGC4214 & 0.08 & 1.3 & 0.11 & 1.3 & 0.63 & 0.79 & 0.71 & 0.15 & 9.5 \nNGC2976 & 0.31 & 3.2 & - & - & 0.63 & 1.00 & 1.00 & 2.59 & 4.1 \nNGC7793 & 0.06 & 1.4 & 0.11 & 3.9 & 0.63 & 0.79 & 0.71 & 0.69 & 6.0 \nNGC2403 & 0.22 & 1.8 & 0.07 & 4.9 & 1.00 & 1.00 & 0.71 & 0.31 & 8.8 \nNGC0925 & 0.11 & 2.0 & 0.07 & 9.1 & 0.63 & 0.79 & 0.71 & 0.97 & 12.5 \nNGC0628 & 0.08 & 1.6 & 0.13 & 7.6 & 0.63 & 1.00 & 1.00 & 0.95 & 4.7 \nNGC3198 & 0.11 & 1.8 & 0.07 & 11.0 & 0.63 & 0.79 & 1.00 & 0.94 & 7.2 \nNGC3184 & 0.11 & 2.0 & 0.10 & 10.0 & 0.63 & 1.00 & 1.41 & 1.24 & 3.8 \nNGC4736 & 0.89 & 2.0 & - & - & 0.63 & 1.26 & 1.41 & 2.34 & 7.7 \nNGC3351 & 0.06 & 1.6 & 0.10 & 8.6 & 0.63 & 1.00 & 0.71 & 0.79 & 5.8 \nNGC6946 & 0.32 & 2.5 & - & - & 0.63 & 1.00 & 1.00 & 1.47 & 3.0 \nNGC3627 & 0.22 & 2.0 & - & - & 0.63 & 0.79 & 1.00 & 0.47 & 4.4 \nNGC5194 & 1.26 & 1.0 & - & - & 0.63 & 0.79 & 0.71 & 3.29 & 22.1 \nNGC3521 & 0.22 & 1.8 & - & - & 0.63 & 0.79 & 1.41 & 5.82 & 4.9 \nNGC2841 & 0.08 & 2.0 & 0.17 & 11.3 & 0.63 & 0.79 & 0.71 & 0.42 & 6.0 \nNGC5055 & 0.32 & 2.0 & 0.07 & 12.5 & 0.63 & 0.79 & 1.41 & 7.83 & 5.9 \nNGC7331 & 0.22 & 2.0 & 0.09 & 13.9 & 0.63 & 0.79 & 1.41 & 1.37 & 4.4 \nVB03 & & & & & & & & \nIC 2574 & 0.16 & 8.9 & 1.00 & 0.00 & 0.63 & 0.79 & 0.71 & 0.21 & 5.2 \nNGC 4214 & 0.06 & 1.0 & 1.00 & 0.00 & 0.63 & 0.79 & 0.71 & 0.21 & 11.2 \nNGC 2976 & 0.32 & 3.5 & - & - & 0.63 & 1.00 & 0.71 & 2.44 & 4.1 \nNGC 7793 & 0.16 & 1.0 & 1.00 & 0.00 & 0.63 & 1.00 & 0.71 & 0.33 & 12.5 \nNGC 2403 & 0.22 & 1.3 & 1.00 & 0.00 & 0.63 & 1.00 & 0.71 & 0.11 & 14.2 \nNGC 0925 & 0.16 & 1.0 & 0.33 & 0.00 & 0.63 & 0.79 & 0.71 & 0.41 & 32.0 \nNGC 0628 & 0.08 & 1.8 & 1.00 & 0.00 & 0.63 & 1.00 & 1.00 & 1.01 & 4.6 \nNGC 3198 & 0.11 & 2.5 & 0.33 & 4.3 & 1.00 & 0.79 & 1.00 & 0.93 & 5.1 \nNGC 3184 & 0.16 & 4.0 & 0.33 & 6.7 & 1.58 & 1.00 & 1.41 & 1.31 & 2.3 \nNGC 4736 & 0.63 & 7.9 & - & - & 1.58 & 1.26 & 1.41 & 2.59 & 1.7 \nNGC 3351 & 0.06 & 1.8 & 0.33 & 2.7 & 0.63 & 1.00 & 0.71 & 0.84 & 5.1 \nNGC 6946 & 0.45 & 7.9 & - & - & 1.58 & 1.00 & 1.00 & 1.13 & 1.2 \nNGC 3627 & 0.32 & 8.9 & - & - & 1.58 & 0.79 & 0.71 & 0.40 & 1.3 \nNGC 5194 & 0.06 & 8.9 & - & - & 1.58 & 0.79 & 0.71 & 4.30 & 0.8 \nNGC 3521 & 0.32 & 8.9 & 0.25 & 11.7 & 1.58 & 0.79 & 1.41 & 4.22 & 1.1 \nNGC 2841 & 0.11 & 3.5 & 0.33 & 5.1 & 1.00 & 0.79 & 0.71 & 0.41 & 4.3 \nNGC 5055 & 0.89 & 8.9 & 1.00 & 5.1 & 1.58 & 1.00 & 1.41 & 4.06 & 2.0 \nNGC 7331 & 0.22 & 4.5 & 0.50 & 6.8 & 1.58 & 0.79 & 1.41 & 0.67 & 2.0 \n---\n1. Only three galaxies, IC 2574, NGC 4214, and NGC 2976, have rotation velocities smaller than 100 kms− 1. We calculated the models for these galaxies using up to 3 times smaller effective yields. The results where not significantly different compared to the models based on Eq. [eq:alphacb] with the exception that the molecular fraction decreased by the same factor as the effective yield.[↩](#fnref1)\n2. This decline is somewhat slowed down by gas replenishment from dying stars.[↩](#fnref2)\n\nSustaining star formation rates in spiral galaxiesSupernova-driven turbulent accretion disk models applied to THINGS galaxies\n=============================================================================================================================\nGas disks of spiral galaxies can be described as clumpy accretion disks without a coupling of viscosity to the actual thermal state of the gas. The model description of a turbulent disk consisting of emerging and spreading clumps contains free parameters, which can be constrained by observations of molecular gas, atomic gas and the star formation rate for individual galaxies. Radial profiles of 18 nearby spiral galaxies from THINGS, HERACLES, SINGS, and GALEX data are used to compare the observed star formation efficiency, molecular fraction, and velocity dispersion to the model. The observed radially decreasing velocity dispersion can be reproduced by the model. In the framework of this model the decrease in the inner disk is due to the stellar mass distribution which dominates the gravitational potential. Introducing a radial break in the star formation efficiency into the model improves the fits significantly. This change in star formation regime is realized by replacing the free fall time in the prescription of the star formation rate with the molecule formation timescale. Depending on the star formation prescription, the break radius is located near the transition region between the molecular-gas-dominated and atomic-gas-dominated parts of the galactic disk or closer to the optical radius. It is found that only less massive galaxies ($\\log M\n({\\rm M}\\_{\\odot}) \\la 10)$) can balance gas loss via star formation by radial gas accretion within the disk. These galaxies can thus access their gas reservoirs with large angular momentum. On the other hand, the star formation of massive galaxies is determined by the external gas mass accretion rate from a putative spherical halo of ionized gas or from satellite accretion. In the absence of this external accretion, star formation slowly exhausts the gas within the optical disk within the star formation timescale.\nIntroduction\n============\nGas accretion plays an important role for the evolution of galactic disks. The star formation rate in the solar neighborhood has been approximately constant over the last ∼ 10 Gyr, while the local gas depletion time is ≈ 2 Gyr. This suggests that the gas consumed by star formation must be replenished via external or internal accretion. Continuous addition of metal-poor gas would also explain the discrepancy between the observed stellar metallicity distribution in the solar neighborhood and that predicted by closed-box models of chemical evolution.\nThe need for accretion may also be seen from observations of other galaxies. Large spiral galaxies like the Milky Way typically have star formation rates of a few M⊙/yr and gas reservoirs of order 109 M⊙ in the star-forming part of the galaxy. Over these regions, gas depletion times are 1–2 Gyr, so that for these galaxies to survive in their present configuration, roughly the same amount of mass consumed by star formation must arrive at the galaxy center via some type of accretion. This accretion can be external, infalling gas from the galactic halo, or internal, radial gas transport via spiral arms and gas viscosity. In the absence of external accretion the spiral galaxy might deplete its gas and become a lenticular galaxy.\nTurbulent galactic disks have been studied numerically and analytically in recent years. used 3D, global hydrodynamic simulations of the central part of a late type galaxy to study the density distribution of the inhomogeneous ISM. They found that the density distribution is well fitted by a single lognormal function over a wide density range. first noticed that the Schmidt-Kennicutt law of the star formation rate can be reproduced if star formation occurs in dense gas above a critical density. The simulations of did not include feedback from supernova (SN) explosions.\nIn a series of articles,,, and studied the 2D evolution of an inhomogeneous ISM in a galactic disk without SN feedback.\n emphasize the importance of gravity due to gas and stars for driving interstellar turbulence. When strong feedback from SN is included, the fraction of dense gas and thus star formation is reduced compared to the models of and turbulence is driven by expanding shells in overdense regions. On the other hand,, who studied gaseous galactic disks by means of 3D, high-resolution hydrodynamical simulations with and without SN feedback, found that SN feedback is an important driver of turbulence in galaxies with star formation rates ≥ 10− 3 M⊙yr− 1kpc− 2. This star formation rate is found to galactic radii ∼ *R*25 in most of the nearby spiral galaxies investigated by.\n note that the gas disk thickness is important for setting the index of the Schmidt-Kennicutt law. studied the relationships among pressure, vertical distribution of gas, and the fraction of dense gas in 2D hydrodynamic simulations of a starforming galactic disk with SN feedback. They found that vertical hydrostatic equilibrium gives a good estimate for the mean midplane pressure of the ISM. In their models they recover the observed ratio $R\\_{\\rm mol}=M\\_{\\rm H\\_2}/M\\_{\\rm HI}$ if the gas surface density is proportional to the epicyclic frequency. state that the empirical result that $R\\_{\\rm mol}$ is proportional to the mean midplane pressure of the ISM implies that the epicyclic frequency, the gas surface density, and the star formation rate per unit volume are interdependent, most probably due to a galaxy evolution with the Toomre parameter *Q* close to unity. If gravitational instability of a clumpy and turbulent gas disk is taken into account, turbulence drives the disc to a regime of transition between instability at small scales and stability in the classical sense (Toomre *Q* ∼ 1; ).\nMost recently, have developed an analytical model for the evolution of a thin disk of gas and stars with an arbitrary rotation curve that is kept in a state of marginal gravitational instability (*Q* ∼ 1) and energy equilibrium due to the balance between energy released by accretion and energy lost due to decay of turbulence. Equilibrium disks of this kind have been investigated by. showed that disks initially out of equilibrium evolve into it on timescales comparable to the orbital period if the external gas accretion rate is high.\nIn this article we use the analytical models of to estimate the radial mass accretion rate of a sample of nearby spiral galaxies. The VB03 model treats galaxies as clumpy accretion disks using a simplified description of turbulence driven by supernova (SN) explosions. Because we focus on modeling the galactic disk within the optical radius, we neglect gravitational instabilities as energy source for turbulence. A gas disk without star formation might generate turbulence via gravitational instabilities, but once stars form, we assume that SN dominate energy injection. The formalism of our equilibrium model is close to that of. The model naturally links the physical properties of the galactic gas disk (surface density, molecular fraction, star formation rate) to the midplane pressure which mainly depends on the stellar surface density, epicyclic frequency, the Toomre parameter, and the gas mass accretion rate. Furthermore, we assume the classical scale-independent Toomre stability criterion.\nWe compare these models to multiwavelength observations of 18 spiral galaxies from the THINGS sample compiled by. From the comparison, we estimate the radial mass accretion rate of each galaxy.\nThe VB03 model considers the gas disk of a galaxy as a single turbulent medium. The dissipation timescale of turbulent kinetic energy on galactic scales is about the crossing time, so that for a typical driving scale length of ∼ 100 pc of the turbulent flow the dissipation time $\\tau\\_{\\rm d}\n\\sim 10$ Myr. Therefore in order to maintain turbulence, an efficient, continuous, driving mechanism is needed. Possible candidates for such a driving mechanism are:\n* gravitational instabilities like spiral arms or rotational shear,\n* magneto-rotational instabilities,\n* protostellar outflows,\n* stellar winds,\n* UV radiation,\n* supernova (SN) explosions,\n* external accretion.\nOf these only SN explosions can balance the energy loss due to turbulent energy dissipation if the driving length scale is ∼ 100 pc and the star formation rate is ≥ 10− 3 M⊙yr− 1kpc− 2. In cases where the driving length scale is of the order of the disk thickness ( ∼ 500 pc), the energy input due to gravitational instabilities can maintain turbulence in the interstellar medium (ISM). The VB03 model and this paper consider only turbulence driven by SN.\nIn the VB03 model, SN-driven turbulence sets the disk structure and the disk structure determines the star formation rate, which in turn determines the SN rate. As a result, the conversion of gas into stars is of prime importance to the model. Despite a large variety of observations in different wavelengths, star formation in galaxies is not well understood. ISM turbulence in turn probably controls the star formation process to some degree, but large scale gravitational instabilities, thermal instabilities, and H2 formation may all be important.\nIn the last decades it was thought that star formation is controlled by the interplay between self-gravitation and magnetic fields. However, the interstellar medium (ISM) is of turbulent nature. Its structure is usually described as hierarchical over length scales of several magnitudes up to ∼ 100 pc. The neutral phase of the ISM is not uniform but has a fractal structure.\n developed an analytical model for clumpy accretion disks and included a simplified description of turbulence in the disk. In contrast to classical accretion disk theory, VB03 do not use the “thermostat” mechanism, which implies a direct coupling between the heat produced by viscous friction and the viscosity itself, which is assumed to be proportional to the thermal sound speed. Thus, the viscosity, which is responsible for the gas heating, depends itself on the gas temperature. This leads to an equilibrium corresponding to a thermostat mechanism. Instead, VB03 use energy flux conservation, where the energy flux provided by SNe is transferred through a turbulent cascade to smaller scales where it is dissipated. The SN energy flux is expected to be proportional to the local star formation rate. In particular, the star formation recipe takes the turbulent nature of the ISM into account. VB03 showed that it is possible to reproduce the the local and global gas properties of the Galaxy with this model. It was, however, not possible at that time to compare observed radial profile to the VB03 model.\nHere, we extend this comparison to a sample of 18 nearby spiral galaxies. presented radial profiles of the star formation rate and gas surface density (CO and Hi) of these galaxies to study their star formation efficiencies. They showed that the SFE of H2 alone is nearly constant (H2 depletion time of 1.9 Gyr) at their 800 pc resolution. Where the interstellar medium (ISM) is mostly Hi, however, the SFE decreases with increasing radius, a decline reasonably described by an exponential with scale length 0.2R25–0.25R25.\nIn this article we extend the formalism of VB03 (Sec. [sec:model]) by adding a break in the star formation rate (Sec. [sec:break]). Beyond this break the relevant timescale for star formation is no longer the free fall time of the most massive self-gravitating cloud at a given galactocentric radius, but the molecular formation timescale. In Sec. [sec:results] and [sec:discussion] the radial profiles presented in are compared and discussed to the extended VB03 model. We discuss the implications of our findings in Sec. [sec:how] and give our conclusions in Sec. [sec:conclusions].\nThe analytical model [sec:model]\n================================\nSince the model is described in detail in VB03, we summarize here only the basic idea and the resulting expressions for the relevant disk properties.\nThe model considers the warm, cold, and molecular phases of the ISM as a single, turbulent gas. We assume this gas to be in vertical hydrostatic equilibrium, with the midplane pressure balancing the weight of the gas and stellar disk. The gas is taken to be clumpy, so that the local density is enhanced relative to the average density of the disk. Using this local density, we calculate two timescales relevant to star formation: the free fall of time of an individual clump and the characteristic timescale for H2 to form on grains. The longer of these (modulo an empirical scaling factor) is taken as the governing timescale for star formation. The star formation rate is used to calculate the rate of energy injection by supernovae. This rate is related to the turbulent velocity dispersion (an observable) and the driving scale of turbulence. These quantities in turn provide estimates of the clumpiness of gas in the disk (i.e., the contrast between local and average density) and the rate at which viscosity moves matter inward.\nThe model relies on several empirical calibrations: e.g., the relationship between star formation rate and energy injected into the ISM by supernovae, the H2 formation timescale (and its dependence on metallicity), and the turbulent dimension of the ISM (used to relate the driving length scale to the characteristic cloud size modulo a free parameter). As far as possible, these are drawn from observations of the Milky Way.\nWe are left with three free parameters. First, there is an unknown scaling factor relating the driving length of turbulence to the size of gravitationally bound clumps, which we call *δ*. Second, the point at which the star formation timescale transitions from the free fall time to the H2 formation time is *a priori* unknown. Third, the mass accretion rate, $\\dot{M}$, which is related to the driving length and turbulent velocity, is a free parameter. In Section [sec:results] via comparison to the present sample of galaxies, we constrain *δ*. In practice, we also treat the Toomre *Q* parameter of the gas as a “semi-free” parameter, allowing it to change somewhat from the observed value.\nWith these assumptions in hand, we may fit the model to a galaxy by comparing the observed kinematics, Hi, CO, and star formation rate profiles to those predicted by the model. The resulting fit yields the disk mass accretion rate and several other parameters that may be checked against expectations: the free fall time (or density) for the largest self-gravitating structures, the driving length of turbulence, and the approximate velocity of radial inflow.\nIn the remainder of this section, we discuss our assumptions in slightly more detail, justify them via comparison to observation and theory, and note the physics that we neglect.\n*ISM:* Following, e.g.,, we view the ISM as a single turbulent gas. In this picture, the warm, cold, and molecular phases of the ISM are a single entity. Locally, the exact phase of the gas may depend on the local pressure, metallicity, stellar radiation field, stellar winds, and shocks. Here we view these factors as secondary, making a few simplifying assumptions. The equilibrium between the different phases of the ISM and the equilibrium between turbulence and star formation depends on three local timescales: the turbulent crossing time $t\\_{\\rm turb}^{l}$, the molecule formation timescale $t\\_{\\rm mol}^{l}$, and the local free fall timescale $t\\_{\\rm ff}^{l}$ of a cloud.\n*Supernova-driven Turbulence:* First, we assume that the gas is turbulent, so that the turbulent velocity is the relevant one throughout the disk (making the exact temperature of the gas is largely irrelevant). We assume that this turbulence is driven by SNe and that they input their energy in turbulent eddies that have a characteristic length scale, $l\\_{\\rm driv}$, and a characteristic velocity, $v\\_{\\rm turb}$. This driving length scale may be the characteristic length scale of a SN bubble, but it does not have to be so. It may be set by the interaction of multiple SN bubbles or of a SN with the surrounding ISM. We note that based on simulations, the assumption of a single driving scale may be a simplification. The VB03 model does not address the spatial inhomogeneity of the turbulent driving nor the mechanics of turbulent driving and dissipation. It is assumed that the energy input rate into the ISM due to SNe, $\\dot{E}\\_{\\rm SN}$, is cascaded to smaller scales without losses by turbulence. At scales smaller than the size of the largest selfgravitating clouds the energy is dissipated via cloud contraction and star formation. We refer to for a review of these topics. We limit our analytical model to the first energy sink which is the scale where the clouds become selfgravitating.\nWe can connect the energy input into the ISM by SNe directly to the star formation rate. With the assumption of a constant initial mass function (IMF) independent of environment one can write\n \n$$\\label{eq:energyflux}\n \\frac{\\dot{E}\\_{\\rm SN}}{\\Delta A}=\\xi\\,\\dot{\\Sigma}\\_{\\*} \n = \\xi\\,\\dot{\\rho}\\_{\\*} l\\_{\\rm driv}=\\Sigma \\nu \\frac{v\\_{\\rm turb}^{2}}{l\\_{\\rm driv}^{2}}\\,$$ \nwhere Δ*A* is the unit surface element of the disk. The factor of proportionality *ξ* relates the local SN energy input to the local star formation rate and is assumed to be independent of local conditions. *ξ* is normalized using Galactic observations by integrating over the Galactic disk and results in *ξ* = 4.6 × 10− 8 (pc/yr)2 (see VB03). The adopted energy that is injected into the ISM is $E^{\\rm kin}\\_{\\rm SN}=10^{50}$ ergs based on numerical studies by. The final two parts of Equation [eq:energyflux] assume that stars form over a characteristic scale equal to the driving length and equate energy output from SNe with the energy transported by turbulence (see VB03).\n*Star Formation in Molecular Clouds:* Second, we assume that stars form out of gravitationally bound clouds. Where the timescale to form H2 is short, we take the local gravitational free fall time, given by\n \n$$t^{\\rm l}\\_{\\rm ff}=\\sqrt{\\frac{3\\pi}{32G\\rho\\_{\\rm cl}}}\\,\n \\label{eq:localff}$$ \nto be the relevant timescale for star formation. Here *G* is the gravitational constant and $\\rho\\_{\\rm cl}$ the density of a single cloud.\nCloud collapse, and thus star formation can only proceed if enough molecules form during the cloud collapse to allow the gas to continue cooling. Therefore, where the timescale for H2 formation is long (compared to the free fall time), we view this as the relevant timescale for star formation. We take the characteristic time to form H2 out of H to be approximately\n \n$$\\label{eq:molform}\nt\\_{\\rm mol}^{l}=\\alpha / \\rho\\_{\\rm cl}\\,$$ \nwhere *α* is a coefficient that depends on metallicity and temperature and $\\rho\\_{\\rm cl}$ is the density of a single cloud. Unlike some other recent numerical and theoretical treatments of H2 abundance by and, we take no account of the rate of destruction of H2 via the UV radiation field. Instead, we assume H2 to be destroyed via the collapse and subsequent star formation of a self-gravitating cloud.\nThe coefficient of the molecular formation timescale *α* is assumed to be metallicity dependent. Because we admit external gas accretion, the metallicity of the star-forming ISM mainly depends on the ratio of accretion to star formation rate *a*. Small *a* < 1 lead to a metallicity derived from a closed box model, whereas in the case of *a* > 1 the metallicity equals the true stellar yield $y\\_{\\rm true}$. For gas fractions higher than 0.04 the difference between the two solutions is less than a factor of two. Moreover, showed that the effective yield $y\\_{\\rm eff}=Z\\_{\\rm gas}/\\ln(1/f\\_{\\rm gas})$, where $Z\\_{\\rm gas}$ is the gas metallicity and $f\\_{\\rm gas}$ the gas fraction, for disk galaxies with a rotation velocity higher than 100 kms− 1 is approximately constant, i.e., for these galaxies a closed box model can be applied. We thus feel confident to estimate the metallicity based on a closed box model using the gas fraction [1](#fn1):\n \n$$\\label{eq:alphacb}\n\\alpha=7.2 \\times 10^{7} \\times \\big( \\log(\\frac{\\Sigma\\_{\\*}+\\Sigma}{\\Sigma})\n\\big)^{-1}\\ {\\rm yr\\,M\\_{\\odot}pc^{-3}},$$ \nwhere Σ\\* is the stellar surface density and log refers to a natural logarithm. The calibration of the effective yield is based on the comparison of the observed O/H profiles with those of the closed box model (Sect. [sec:results]). Adopting a stellar and gas surface density of Σ\\* = 40 M⊙pc− 2 and $\\Sigma\\_{\\rm\n gas}=10\\,$ M⊙pc− 2 at the solar radius of the Galaxy yields *α*⊙ = 4.5 × 107 yrM⊙pc− 3, which is twice the value used by.\nThis approach assumes that molecular clouds are relatively short lived, appearing and disappearing over roughly a free fall time (equivalently, by our construction, a turbulent crossing time); otherwise they might reach chemical equilibrium even when the H2 formation time is long compared to the free fall time. Accordingly, we estimate the molecular fraction in the disk from the ratio of a cloud lifetime (i.e., the crossing or free-fall time) to the H2 formation time scale: $f\\_{\\rm mol}=\\frac{\\Sigma\\_{\\rm H\\_{2}}}{\\Sigma\\_{\\rm HI}+\\Sigma\\_{\\rm H\\_{2}}}=t\\_{\\rm turb}^{l}/t\\_{\\rm mol}^{l}$.\nNote that the molecular formation timescale, in particular, is fairly approximate — as noted above we have neglected radiation field and temperature dependences. Therefore we allow an extra factor *γ* when comparing the two to derive the relevant timescale, with the transition between the two regimes at $t\\_{\\rm ff}^l = \\gamma\nt\\_{\\rm mol}^l$. We derive a value of *γ* = 0.4 ± 0.3 for the VB03 star formation prescription and *γ* = 0.12 ± 0.06 for the star formation prescription following. comparing the model to the whole set of observations in Section [sec:results].\n*Vertical Disk Structure:* In the model, the disk scale height is determined unambiguously by the assumption of hydrostatic equilibrium and the turbulent pressure :\n \n$$p\\_{\\rm turb}=\\rho v\\_{\\rm turb}^{2} = \\frac{\\pi}{2} G \\Sigma ( \\Sigma + \\Sigma\\_{\\*} \\frac{v\\_{\\rm turb}}{v\\_{\\rm disp}^{\\*}})~,\n\\label{eq:pressure}$$ \nwhere *ρ* is the average density, $v\\_{\\rm turb}$ the gas turbulent velocity in the disk, $v\\_{\\rm disp}^{\\*}$ the stellar vertical velocity dispersion, and Σ the surface density of gas and stars. The stellar velocity dispersion is calculated by $v\\_{\\rm disp}^{\\*}=\\sqrt{2 \\pi G \\Sigma\\_{\\*} H\\_{\\*}}$, where the stellar vertical height is taken to be *H*\\* = *l*\\*/7.3 with *l*\\* being the stellar radial scale length. We neglect thermal, cosmic ray, and magnetic pressure. During the fitting procedure we realized that the local pressure equilibrium is of prime importance for the goodness of the fits.\n*Treatment as an Accretion Disk:* The turbulent motion of clouds is expected to redistribute angular momentum in the gas disk like an effective viscosity would do. This allows accretion of gas towards the center and makes it possible to treat the disk as an accretion disk. This gaseous turbulent accretion disk rotates in a given gravitational potential Φ with an angular velocity $\\Omega=\\sqrt{R^{-1}\\frac{{\\rm\n d}\\Phi}{{\\rm d}R}}$, where *R* is the disk radius. The disk has an effective turbulent viscosity that is responsible for mass accretion and outward angular momentum transport. In this case, the turbulent velocity is driven by SN explosions, which stir the disk and lead to viscous transport of angular momentum. In addition, star formation removes gas from the viscous evolution. Following, the evolution of the gas surface density is given by\n \n$$\\frac{\\partial \\Sigma}{\\partial t}=-\\frac{1}{R}\\frac{\\partial}{\\partial R}\\left(\n\\frac{(\\partial/\\partial R)[\\nu \\Sigma R^3 ({\\rm d}\\Omega/{\\rm }dR)]}{({\\rm d}/{\\rm d}R)(R^2 \\Omega)}\\right)\n-\\dot{\\Sigma}\\_{\\*}+\\dot{\\Sigma}\\_{\\rm ext}\\,\n\\label{eq:linpringle}$$ \nwhere *ν* is the gas disk viscosity, Ω the angular velocity, and $\\dot{\\Sigma}\\_{\\rm ext}$ is the external mass accretion rate. In contrast to we assume a continuous and non-zero external gas mass accretion rate. By approximating ∂/∂*R* ∼ 1/*R*, the global viscous evolution becomes\n \n$$\\frac{\\partial \\Sigma}{\\partial t} \\sim \\frac{\\Sigma \\nu}{R^2}-\\dot{\\Sigma}\\_{\\*}+\\dot{\\Sigma}\\_{\\rm ext}\\.\n\\label{eq:parsigma}$$ \nIf the external and disk mass accretion rate keeps the combined Toomre parameter of the gas and stars smoothed over a few rotation periods to $Q\\_{\\rm tot} \\sim 1$, the gas surface density will only vary slowly with changes in the dark halo mass distribution (via Ω) and the stellar disk structure, the gas loss due to star formation is balanced by external accretion as suggested by and, and the gas disk can be regarded as being stationary (∂Σ/∂*t* = 0). Indeed, local spiral galaxies show a $Q\\_{\\rm tot}$ not too far away from unity (1.3-2.5; ). For such a stationary gas disk, where star formation is balanced by external accretion, the local mass and momentum conservations yield:\n \n$$\\nu \\Sigma=\\frac{\\dot{M}}{2\\pi}\\,\n\\label{eq:transport}$$ \nwhere $\\dot{M}$ is the mass accretion rate within the disk. In the absence of external mass accretion, the gas disk can be assumed to be stationary as long as the star formation timescale *t*\\* exceeds the viscous timescale *t**ν* = *R*2/*ν*. For $\\dot{\\Sigma}\\_{\\rm ext} < \\dot{\\Sigma}\\_{\\*}$ and *t*\\* < *t**ν* Eq. [eq:transport] is not valid. In this case the gas disk is rapidly turned into stars within the gas consumption time (2 Gyr, ). Since most spiral galaxies still have a significant amount of gas, we think that spiral galaxies are generally not in this state. Solving the time dependent Eq. [eq:linpringle] is beyond the scope of this work and we apply Eq. [eq:transport]. Radial integration of Eq. [eq:parsigma] then gives\n \n$$\\frac{\\partial M}{\\partial t} \\sim \\dot{M}-\\dot{M}\\_{\\*}+\\dot{M}\\_{\\rm ext}\\,$$ \nwhere *M* is the disk gas mass.\nThe viscosity is related to the driving length scale and characteristic velocity of the SN-driven turbulence by $\\nu=v\\_{\\rm turb}l\\_{\\rm\n driv}$ (VB03). Because the lifetime of a collapsing and starforming cloud ($t\\_{\\rm ff}^{l} < t\\_{\\rm turb}^{l}$) is smaller than the turnover time of the large-scale eddy ($l\\_{\\rm driv}/v\\_{\\rm turb}$), the turbulent and clumpy ISM can be treated as one entity for the viscosity description.\n*Clumpiness:* A critical factor in the model is the relationship between the density of individual clouds, $\\rho\\_{\\rm cl}$, and the average density of the disk, *ρ*. It is the density of individual clouds that is relevant to the timescale for star formation. In this model, the two are related by the volume filling factor, $\\Phi\\_{\\rm V}$ so that $\\rho\\_{\\rm cl}=\\Phi\\_{\\rm\n V}^{-1}\\rho$.\nHere $\\rho\\_{\\rm cl}$ refers to the density of the largest self-gravitating structures in the disk, so that for these structures the turbulent crossing time and gravitational free fall time are equal. The scale of such a cloud, $l\\_{\\rm cl}$ is smaller than the driving length scale $l\\_{\\rm driv}$ by a factor *δ*, which we do not know *a priori*.\nShear due to differential galactic rotation could stabilize clouds, modifying the timescale for collapse. However, this effect is mainly important when the ratio of the cloud to disk surface density is lower than the ratio of cloud to disk velocity dispersion, which is not the case over most of the disk in a typical spiral. Typical GMC surface densities are ∼ 200 M⊙pc− 2, whereas disk surface densities only exceed 100 M⊙pc− 2 in the very center of spiral galaxies. The ratio of velocity dispersion being about 0.5, clouds at galactic radii larger than a few kpc are not stabilized by shear from galactic rotation.\nWe can calculate the turbulent timescale for the cloud, $t\\_{\\rm turb}^l$, for a fractal ISM:\n \n$$t\\_{\\rm turb}^{l}=\\delta^{-\\frac{2}{3}-\\frac{3-D}{3}}\n\\,l\\_{\\rm driv}/v\\_{\\rm turb}\\,$$ \nwhere *D* is the fractal dimension of the ISM. We assume *D* = 2 for a compressible, self-gravitating fluid, which is close to the findings of. Once *δ* and thus $t\\_{\\rm turb}^l$ are specified, we can solve for the density of the corresponding scale by setting $t\\_{\\rm ff}^l = t\\_{\\rm turb}^l$. The volume filling factor is then defined by comparing $\\rho\\_{\\rm cl}$ and *ρ*.\nBecause the relationship between the driving length scale, $l\\_{\\rm driv}$ and $l\\_{\\rm cl}$ is not known beforehand, we treat this as a free parameter. Once the volume filling factor is known (from *δ* or $l\\_{\\rm cl}$), we can calculate the local star formation rate, $\\dot{\\rho\\_\\*}$, via\n \n$$\\label{eq:starform}\n\\dot{\\rho}\\_{\\*} = \\eta \\frac{\\rho}{t\\_{\\rm sf}^{\\rm l}}\\,$$ \nwhere $t\\_{\\rm sf}^l$ is the local timescale for star formation, either the free fall or molecular formation timescale, depending on local conditions. We assume *η* = 7 × 10− 3 in consistency with. Since in our model the lifetime of a cloud is the free-fall as suggested by or the molecule formation timescale, this implies that during the cloud lifetime about 1% of the cloud mass turns into stars.\nIn VB03 a different star formation prescription is used:\n \n$$\\label{eq:starformm}\n\\dot{\\rho}\\_{\\*} = \\Phi\\_{\\rm V} \\frac{\\rho}{t\\_{\\rm sf}^{\\rm l}}\\.$$ \nThe vertically integrated star formation rate in the inner disk where $t\\_{\\rm sf}^{\\rm l}=t\\_{\\rm ff}^{\\rm l}=t\\_{\\rm turb}^{\\rm l}=\\delta^{-1} t\\_{\\rm turb}$ is\n \n$$\\label{eq:starformm1}\n\\dot{\\Sigma}\\_{\\*} = \\Phi\\_{\\rm V} \\frac{\\rho}{t\\_{\\rm ff}^{\\rm l}} l\\_{\\rm driv} = \n\\delta \\Phi\\_{\\rm V} \\rho v\\_{\\rm turb}\\,$$ \ni.e. it is the mass flux density of the turbulent ISM into the regions of star formation. This alternative description yields fitting results that are as good as those based on Eq. [eq:starform] (in practice $\\eta \\sim \\Phi\\_{\\rm V} \\sim 10^{-2}$).\nA Two Part Model [sec:break]\n----------------------------\nVB03 assumed that $t\\_{\\rm sf}^l$ in Equation [eq:starform] is always the local free fall time. As described above, here we consider two regimes, adopting instead the H2 formation time when this becomes long compared to $t\\_{\\rm ff}^l$. Therefore, the present model has two parts. We label the first, where $t\\_{\\rm sf}^l = t\\_{\\rm ff}^l$ the *inner disk*. We label the second, where $t\\_{\\rm sf}^l = t\\_{\\rm mol}^l$ the *outer disk*. The change in regime should occur at the radius where $t\\_{\\rm ff}^l = \\gamma t\\_{\\rm mol}^l$. Here *γ* is a (free) scaling parameter that reflects a level of uncertainty in the application of the molecular formation timescale.\nAn alternative interpretation of the change in star formation regime is that (i) the star formation efficiency per free fall time or (ii) the volume or area filling factor of star forming regions change from the inner to the outer disk. The latter possibility is motivated by the work of who show that gaseous spiral arms formed in the inner disk region can propagate into the outer gas disk creating overdensities where stars can form. Since these gaseous arms occupy a smaller fraction of the disk area with increasing galactocentric radius, the azimuthally averaged surface and area filling factors decrease. Having this picture in mind, one can write for the outer disk\n \n$$\\dot{\\rho}\\_{\\*}=\\eta \\frac{\\rho}{t\\_{\\rm mol}^{\\rm l}}=\\eta f\\_{\\rm mol} \\frac{\\rho}{t\\_{\\rm ff}^{\\rm l}}\n=\\tilde{\\eta} \\frac{\\rho}{t\\_{\\rm ff}^{\\rm l}}$$ \nand\n \n$$\\dot{\\rho}\\_{\\*}=\\Phi\\_{\\rm V} \\frac{\\rho}{t\\_{\\rm mol}^{\\rm l}}=\\Phi\\_{\\rm V} f\\_{\\rm mol} \\frac{\\rho}{t\\_{\\rm ff}^{\\rm l}}\n=\\tilde{\\Phi}\\_{\\rm V} \\frac{\\rho}{t\\_{\\rm ff}^{\\rm l}}\\.$$ \nIn this formulation the relevant local star formation timescale is still the local free fall time of the clouds.\nModel calculations [sec:calc]\n-----------------------------\nThe VB03 model yields the following system of equations to describe a turbulent clumpy galactic accretion disk:\n \n$$\\nu = v\\_{\\rm turb} l\\_{\\rm driv}\\,$$ \n \n$$\\nu \\Sigma = \\frac{\\dot{M}}{2\\pi}\\,$$ \n \nΣ = *ρ* *H*, \n \n$$p\\_{\\rm turb}=\\rho v\\_{\\rm turb}^{2} = \\frac{\\pi}{2} G \\Sigma ( \\Sigma + \\Sigma\\_{\\*} \\frac{v\\_{\\rm turb}}{v\\_{\\rm disp}^{\\*}})~,$$ \n \n$$Q = \\frac{v\\_{\\rm turb} \\Omega}{\\pi G \\Sigma}\\,$$ \n \n$$\\Sigma \\nu \\frac{v\\_{\\rm turb}^{2}}{l\\_{\\rm driv}^{2}} = \\xi\\,\\dot{\\Sigma}\\_{\\*}\\,$$ \n \n$$\\dot{\\Sigma}\\_{\\*} = \\Phi\\_{\\rm V} \\frac{\\rho}{t\\_{\\rm SF}^{\\rm l}} l\\_{\\rm driv}\\ {\\rm (VB03)\\ or}\\ =\\eta \\frac{\\rho}{{t\\_{\\rm SF}^{\\rm l}}} l\\_{\\rm driv}\\ {\\rm (KM05)}\\,$$ \n \n$$t\\_{\\rm SF}^{\\rm l}=\\sqrt{\\frac{3\\pi}{32G\\rho\\_{\\rm cl}}}\\ {\\rm or}\\ =\\frac{\\alpha}{\\rho\\_{\\rm cl}}\\.$$ \nThe meaning of the variables is given in Table [tab:parameters]. In the case of a selfgravitating gas disk (Σ ≫ Σ\\*) or a dominating stellar disk (Σ\\* ≫ Σ) the set of equations can be solved analytically (we use $\\rho v\\_{\\rm turb}^{2} = \\pi G \\Sigma^{2}$ and $\\rho v\\_{\\rm turb}^{2} = \\pi G \\Sigma \\Sigma\\_{\\*}$). In Appendix [sec:modelequations] we give the equations relating several observables ($v\\_{\\rm turb}$, Σ, $SFE = \\dot{\\Sigma\\_{\\*}}/\\Sigma$, and $f\\_{\\rm mol} $ and $f\\_{\\rm mol}=\\Sigma\\_{{\\rm H}\\_2}/(\\Sigma\\_{{\\rm H}\\_2}+\\Sigma\\_{{\\rm HI}})$) and other quantities of interest ($l\\_{\\rm driv}$, *ν*, Φ*V*) to our free parameters ($\\dot{M}$, *δ*), assumed constants (*ξ*, *η*), the approximated metallicity (*α*), the “fundamental” observables (stellar surface density Σ\\*, radius *R* and rotation Ω), and the “quasi-free” parameter Toomre *Q* of the gas (which we treat as free, but compare to observations).\nFor the global comparison between the observed and the model radial profiles we solve the set of equations given above numerically and we use\n \n$$\\frac{1}{t\\_{\\rm SF}^{\\rm l}}=\\frac{1}{\\sqrt{\\frac{3\\pi}{32G\\rho\\_{\\rm cl}}+\\alpha^{2}/\\rho\\_{\\rm cl}^{2}}}\\.$$ \nComparison with Observations\n============================\nWe compare the model described above to radial profiles of gas and star formation for 18 nearby spiral galaxies.\nData [sec:data]\n---------------\nThe galaxies we compare to are all part of the THINGS survey. To carry out the comparison, we use radial profiles of the Hi velocity dispersion measured via the second moment presented by ; rotation curves derived by ; radial profiles of atomic (Hi) and molecular gas (traced via CO) surface density based on maps by,, and ; and profiles of star formation rate surface density derived by from a combination of 24*μ*m and FUV intensity. We use stellar surface density estimates derived from SINGS 3.6*μ*m imaging by assuming a constant mass-to-light ratio.\nThe profiles of atomic and molecular gas are combined into profiles of the molecular fraction, $f\\_{\\rm mol}=\\Sigma\\_{\\rm H\\_2}/(\\Sigma\\_{\\rm\nH\\_2}+\\Sigma\\_{\\rm HI})$, and the profiles of star formation rate, Hi, and molecular gas are combined into profiles of star formation efficiency, $SFE = \\dot{\\Sigma}\\_{\\rm \\*} / \\Sigma$. Most of the profiles are drawn from and details on methodology and data can be found there.\nThe following uncertainties affect the data in addition to the uncertainty in the mean value over the azimuthal ring estimated from the RMS scatter in that ring:\n*SFR and SFE*: there is a ∼ 50% uncertainty based on inter-comparison of different star formation rate tracers. Half of this ( ∼ 0.1 dex) is internal to the galaxy, the other half ( ∼ 0.1 dex) is galaxy-to-galaxy scatter (due to, e.g., the star formation history and dust properties).\n*Hi*: THINGS is expected to recover the true Hi flux with 10% accuracy.\n*CO*: The global CO-H2 conversion factor is uncertain by a factor of 2, which dwarfs any uncertainty in the calibration. An additional systematic uncertainty is the dependence of the conversion factor on local conditions. The sample selection somewhat minimizes this concern. The CO data have been masked before constructing profiles, which can lead to artificially steep profiles at the edge of the CO emitting region.\n*Velocity dispersion*: the scatter in the dispersion within a given ring is about 30%. Five of the 18 galaxies are too inclined to derive reliable velocity dispersions. Several regions have very high velocity dispersions, > 20 km s− 1, which we expect are due at least partially to non-circular motions like bars, outflows, or streaming along spiral arms. We do not use these data in the comparison.\nProcedure\n---------\nThe comparison is done in the following way:\n1. The Toomre parameter of the gas is calculated using the observed Hi dispersion velocity $v\\_{\\rm disp}$, rotation velocity $v\\_{\\rm rot}$, and total gas surface density $\\Sigma\\_{\\rm gas}$:\n \n$$Q=\\frac{v\\_{\\rm disp} v\\_{\\rm rot}}{\\pi G \\Sigma\\_{\\rm gas} R}\\.$$ \nThis approximate form of *Q* is used to avoid taking the (noisy) derivative of the observed rotation curve. In two cases, we adjust *Q* somewhat from the measured value: (i) in the inner part (*R* ≤ 4 kpc) of the disk of NGC 3351 where we set *Q* = 8 and (ii) in the whole disk of NGC 2841 where we also assume *Q* = 8. For both galaxies this corresponds to an approximately constant velocity dispersion of ∼ 12-15 kms− 1. In cases without a measured velocity dispersion, we adopt a flat *Q* (NGC 2841, NGC 3351), an exponentially declining *Q* (NGC 3627), or a constant velocity dispersion of 15 kms− 1 (NGC 3521, NGC 7331).\n2. We solve the set of equations given in Sect. [sec:calc] using a grid for values of *δ*, $\\dot{M}$, and *γ*. To account for the overall calibration uncertainties (Sect. [sec:data]), we calculate the models on the *δ*-$\\dot{M}$-*γ*-grid multiplying all points of $\\Sigma\\_{\\rm H\\_2}$ by (0.63, 1., 1.58), $v\\_{\\rm turb}$ by (0.79, 1., 1.26), and *S**F**E* by (0.71, 1., 1.41), allowing for all permutations. For each set of parameters the goodness *g* of the fit is determined for each observable profile:\n \n$$g = \\frac{1}{N} \\sum\\_{i=1}^{N} \\frac{(V^{\\rm obs}\\_{i}-V^{\\rm model}\\_{i})^2}{\\sigma^2}\\,$$ \nwhere *N* is the number of points of a radial profile, $V\\_{\\rm obs/model}$ are the observed and the model points of the profiles, and *σ* is the uncertainty of a point given by. At the end we calculate a total goodness of the fit by taking the mean of the goodnesses of the individual profiles. This does not correspond to a traditional *χ*2, because the uncertainties of the points of a given profile are correlated.\n3. The final *δ* and $\\dot{M}$ are determined by the mean of all *δ* and $\\dot{M}$ yielding with min(*g*) ≤ *g* ≤ 1.1 min(*g*). We verified that these values are not significantly different from those derived from the minimum of the total goodness. In practice, the velocity dispersion provides the strongest constraint on $\\dot{M}$, because it has a small error (it is shown in linear scale in Figs. [fig:radialprofilesKM05] and [fig:radialprofilesVB03]), followed by the molecular fraction and the SFE. Increasing $\\dot{M}$ leads to an increasing velocity dispersion (Eq. [eq:vturbsgz3] and [eq:vturbsgz]) and decreasing $f\\_{\\rm mol}$ (Eq. [eq:fmol3] and [eq:fmol]) and SFE (Eq. [eq:sfe3] and [eq:sfe]). Typically, a variation in $\\dot{M}$ by a factor of 3 leads to changes in $v\\_{\\rm turb}$, $f\\_{\\rm mol}$, and SFE that are larger than the error bars (cf. Sect. [sec:how]).\nResults [sec:results]\n=====================\nWe have applied the VB03 model using the KM05 (Eq. [eq:starform]) and VB03 (Eq. [eq:starformm]) star formation descriptions to 18 spiral galaxies of the sample of (Table [tab:galaxies]). The Hi rotation curves are from. We realized that in the framework of the VB03 star formation prescription the free fall times of the largest selfgravitating clouds are significantly larger than the collapse times derived by for NGC 0925, NGC 2403, and NGC 7793. Since the model fits are sensitive to the local vertical pressure equilibrium and the assumed correlation between the vertical and radial stellar scale height has a large scatter, we prefer a gravitational potential dominated by the stellar disk for these galaxies yielding local free fall times in agreement with.\nThe resulting radial profiles are presented in Fig. [fig:radialprofilesKM05] for the KM05 star formation prescription and in Fig. [fig:radialprofilesVB03] for the VB03 star formation prescription. The order of the plots (from top left to bottom right) is: Hi rotation curve, observed and fitted stellar surface density profile Σ\\* (the assumed Σ\\* for the fit is shown as a solid line), model (dashed) and observed (solid) total gas density profile Σ, Toomre *Q* parameter of stars+gas (dash-dotted) and of the gas derived from observations (dotted) and assumed for the fit (solid), driving scale length $l\\_{\\rm driv}$, Hi velocity dispersion (solid) and model turbulent velocity dispersion (dashed), observed (solid) and modeled (dashed) star formation efficiency, observed (solid) and modeled (dashed) molecular fraction, star formation (solid) and viscous (dashed) timescales, free fall timescale of the most massive selfgravitating gas clouds, radial velocity of the gas within the disk, and the observed (solid) and model (dashed) metallicity. The observed metallicity profiles are derived from $12+\\log({\\rm O/H})$ assuming a solar oxygen abundance of $12+\\log({\\rm O/H})=8.9$. For the calculation of the combined Toomre parameter (stars+gas) $Q\\_{\\rm tot}$ we limited the scale of instabilities to 3 kpc.\nIn agreement with we find that $Q\\_{\\rm tot}$ is close to one for most of the galaxies. Only NGC 3521 and NGC 7331 show $Q\\_{\\rm tot} \\sim 2$ and for NGC 2841 $Q\\_{\\rm tot} \\sim 3$. The Toomre parameter of the gas ranges between 2 and 4 for most of the galaxies and these galaxies can thus be assumed in a quasi equilibrium state where the mass accretion (radially in the disk and external) balances star formation to keep $Q\\_{\\rm tot} \\sim 1$. Moreover, for all galaxies with masses smaller than 1010 M⊙, the viscous timescale is smaller than or comparable to the star formation timescale. For these galaxies our assumption of a stationary accretion disk (Eq. [eq:transport]) is justified without the need of a high external accretion rate. For more massive galaxies Eq. [eq:transport] is a convenient parametrization which leads to a meaningful mean viscosity as discussed in Sect. [sec:model]. The parameters derived from the fitting procedure are the disk mass accretion rate $\\dot{M}$, the scaling between the driving and dissipation length scale *δ*, and the break radius of the star formation efficiency $R\\_{\\rm break}$ which are given in Table [tab:galaxies1] for the KM05 and VB03 star formation prescriptions.\nWe find acceptable fits for all galaxies (KM05 and VB03), except for NGC 5194 (M 51). NGC 2976, NGC 4736, NGC 5194, NGC 3521, and NGC 5055 show *g* > 1.5. For NGC 4736 we suspect a different mass accretion rate between the inner (*R* < 2 kpc) and outer disk. This would violate our assumption of stationarity, i.e. a radially constant mass accretion rate. For NGC 5055 (*g* ∼ 4), the deviations between the model and the observations occur in the central part of the galaxies where the velocity dispersion is overestimated. For NGC 2976 and NGC 3521 the disagreement between the model and observations occurs in the outer disk region. The KM05 star formation prescription provides better fits (smaller *g*) for 4 galaxies (NGC 4214, NGC 4736, NGC 3351, and NGC 5194). The profiles of the other 14 galaxies are better fitted by the VB03 star formation prescription.\nWhereas the KM05 star formation prescription yields a constant *δ* = 1.9 ± 0.5, *δ* increases with galaxy mass in the VB03 star formation prescription. Introducing a *γ* > 0 leads to a significant improvement of the fit (about a two times smaller goodness). We find *γ* = 0.10 ± 0.03 for the KM05 star formation prescription. VB03 shows a bimodal distribution of *γ* with 5 galaxies having *γ* = 0.33 and 6 galaxies having *γ* = 1.0. Overall *γ* = 0.65 ± 0.34 for the VB03 star formation prescription. This translates into $R\\_{\\rm break}/l\\_{\\*}=3.0 \\pm 1.1$ and $R\\_{\\rm break}/R\\_{25}=0.7 \\pm 0.2$ for the KM05 star formation prescription and for massive galaxies (log(*M*\\*) > 10) $R\\_{\\rm break}/l\\_{\\*}=2.0 \\pm 1.0$ and $R\\_{\\rm break}/R\\_{25}=0.4 \\pm 0.2$ for the VB03 star formation prescription. The break radius of the VB03 star formation prescription is thus close to the transition between a mostly-Hi and a mostly-H2 ISM, whereas it is closer to the optical radius in the KM05 star formation prescription.\n*Total gas surface density:* For most of the sample galaxies the total gas surface density profile is well reproduced. In the framework of the KM05 star formation prescription the total gas surface density is underestimated in the outer disk of IC 2574, NGC 7793, and NGC 0925; it is overestimated in the outer disk of NGC 4736. In the framework of the VB03 star formation prescription the total gas surface density is somewhat underestimated in IC 2574, NGC 0925, and greatly underestimated in NGC 5194; it is overestimated in the outer disk of NGC 4736. All these differences are within ∼ 30%, except for NGC 5194 for the VB03 star formation prescription.\n*Velocity dispersion:* Velocity dispersion profiles are available for 13 out of our 18 sample galaxies. We do not try to fit dispersion velocities higher than 20 kms− 1. These tend to appear in regions where geometry confuses the measurement (e.g., the central parts of inclined disks) or regions of very high SFR, where we worry that outflows or other non-disk structures render the measured velocity dispersion inappropriate for comparison to the model. For examples of our concerns, see the position-velocity diagrams of or compare the results of gaussian fitting to moment methods treating the same galaxy. Both reveal the presence of asymmetric, multi-component Hi profiles in the central parts of some galaxies. We can reproduce the observed profiles within the error bars for all galaxies (KM05 and VB03 star formation prescriptions), except NGC 5194 with the VB03 star formation prescription.\n*Star formation efficiency:* The star formation efficiency in the inner disk of all galaxies is well reproduced by both models. The star formation efficiency of the outer disks of NGC 2976, NGC 0925, NGC 3198, NGC 5055, and NGC 7331 is better reproduced by the VB03 than by the KM05 star formation prescription. On the other hand, that of NGC 4214, NGC 6946, NGC 5194, and NGC 2841 is better reproduced by the KM05 star formation prescription. NGC 2976 and NGC 4736 show very low surface density gas in this region leading to *Q* > 10. Since the star formation efficiency is proportional to *Q*− 1 in the outer disk, the mismatch between model and data might be due to our incomplete knowledge of *Q* in these regions.\n*Molecular fraction:* For 15 out of the 18 sample galaxies CO observations are available. The VB03 star formation prescription reproduces the molecular fractions of almost all galaxies (except NGC 4736 and NGC 5194) roughly within the error bars. There are deviations between the KM05 star formation prescription and observed molecular fractions for NGC 5194, NGC3521, NGC 5055. In NGC 3184, NGC 4736, NGC 6946, and NGC 5194 the decline of the observed molecular fraction at the outer edge of the CO distribution is much steeper than predicted by the model. This is mostly due to sensitivity, because the CO data cubes were clipped at given S/N levels.\nDiscussion [sec:discussion]\n===========================\nSince the VB03 model is axisymmetric, it does not include non-axisymmetric structures like spiral arms and bars. Whereas spiral arms do not change the given picture considerably, the existence of bars do have a strong impact on radial gas flows. Despite the importance of a bar in the evolution of galactic disks, the fact that the VB03 model does not include their effect does not affect our conclusions.\nIn the following we discuss aspects related to different radial profiles:\n*Star formation law:* Fig. [fig:hibigiel] shows the star formation rate per unit area as a function of the total gas surface density (Hi+H2) of our model. The distribution agrees well with observed distributions from THINGS and the literature. The absolute values and the shape, i.e. the ’knee’, are reproduced by both the KM05 and VB03 models.\n*Velocity dispersion:* As shown in, the velocity dispersion of spiral galaxies decreases with increasing galactocentric radius. In the following, We note that the VB03 model for a stellar disk dominated gravitational potential yields a radially declining velocity dispersion, whereas a selfgravitating gas disk leads to a constant velocity dispersion in the inner disk ($R < R\\_{\\rm break}$) and a declining velocity dispersion in the outer disk ($R > R\\_{\\rm break}$). Thus, it is in principle possible to determine these different regimes from the radial behavior of the observed velocity dispersion.\n*Molecular fraction:* With the model radial profiles we can investigate the dependence of the star formation rate per unit surface area on the molecular and total surface density (Fig. [fig:molbigiel]). The correlation between the star formation rate and the molecular gas surface density is linear with a molecular depletion timescale of ∼ 1.5 Gyr for the KM05 star formation prescription and ∼ 2 Gyr for the VB03 star formation prescription. The correlation between the total gas surface density and the star formation rate is steep where Hi is dominating (Σ < 20 M⊙yr− 1) and flattens for higher gas surface densities. The VB03 model reproduces results by and who found (i) an average molecular gas depletion timescale of 2 Gyr and (ii) a critical gas surface density of 14 M⊙pc− 2 for the change between gas predominantly in molecular form and gas predominantly in atomic form.\n*Driving length scale:* The derived driving scale lengths are monotonically increasing with increasing radius. Typical values are 100-300 pc in the inner half of the optical disk. At the optical radius the driving length scale is about 400-800 pc. The radial increase and the large values at the optical radius are consistent with sizes of Hi shells observed in the Galaxy.\n*Radial gas motions:* Radial gas motions with velocity $v\\_{\\rm rad}$ can be estimated with the mass accretion rate $\\dot{M}$ and the gas surface density $\\Sigma\\_{\\rm gas}$ :\n \n$$\\dot{M}=2 \\pi R \\Sigma\\_{\\rm gas} (-v\\_{\\rm rad})$$ \nThe radial profiles of the radial velocity are presented in Figs. [fig:radialprofilesKM05] and [fig:radialprofilesVB03]. Typically, we find radial velocities smaller than 2 kms− 1. This is consistent with the findings of who found non-circular motions smaller than 5 kms− 1 in these galaxies. Moreover, we observe a general increase of radial velocities with decreasing galactocentric radius. A strongly increasing *Q* in the outer disks of NGC 2976, NGC 4736, and NGC 3627 due to a strongly decreasing gas surface density leads to steeply increasing radial velocities (up to 10 kms− 1) in these galaxies. This is again consistent with the profiles of non-circular motions derived by.\n*Cloud free fall time estimates:* The free-fall time only depends on the cloud density $\\rho\\_{\\rm cl}$ (Eq. [eq:localff]). Typical densities of giant molecular clouds are between 100-1000 cm− 3 leading to free fall times of 1.6-5.1 Myr. estimated a characteristic timescale for star formation in the spiral arms of disk galaxies, going from atomic hydrogen (Hi) to dust-enshrouded massive stars. Their free fall time estimates vary between 1 and 6 Gyr. The VB03 model yields the physical properties of the most massive selfgravitating clouds at a given galactocentric radius via the local free-fall ($t\\_{\\rm ff}^{l}$), turbulent ($t\\_{\\rm turb}^{l}$), and molecule formation ($t\\_{\\rm mol}^{l}$) timescales. We recall that for selfgravitating clouds $t\\_{\\rm ff}^{l}=t\\_{\\rm turb}^{l}$. The radial profiles of the local free fall time for our sample galaxies are shown in Figs. [fig:radialprofilesKM05] and [fig:radialprofilesVB03]. They are calculated using Eq. [eq:localff] and [eq:phiv]. For the comparison with the free fall timescales derived from observations we calculated the mean and the standard deviation of the radial profiles within *l*\\* ≤ *R* ≤ 2 *l*\\*. Our free fall timescales (Table [tab:galaxies1]) are in good agreement with those derived from observations except for those of NGC 2403, NGC 0925, (both star formation prescriptions) and NGC 7793 (VB03 star formation prescription). By modifying Eq. [eq:alphacb] to better match the observed metallicity profile:\n \n$$\\alpha=1.6 \\times 7.2 \\times 10^{7} \\times \\big( \\log(\\frac{\\Sigma\\_{\\*}+\\Sigma}{\\Sigma}) \\big)^{-1}\\ \n{\\rm yr\\,M\\_{\\odot}pc^{-3}}\\,$$ \nwe obtain the following parameters: NGC 7793: *δ* = 1, $\\dot{M}$=0.06 M⊙yr− 1; NGC 2403: *δ* = 4, $\\dot{M}$=0.22 M⊙yr− 1; NGC 0925: *δ* = 9, $\\dot{M}$=0.31 M⊙yr− 1. With these parameters the free fall timescales of these three galaxies are in good agreement with expectations. We note that in this case *γ* = 0.33 for NGC 7793 and NGC 2403.\nHow to sustain the star formation rate [sec:how]\n================================================\nCan these galaxies sustain their star formation rates by radial transport of gas within the galactic disk? To answer this question one has to compare the local star formation rate and the local viscous timescale *t**ν* = *R*2/*ν* using Eq. [eq:nu] and [eq:nu1]. The local timescale comparison is presented in Figs. [fig:radialprofilesKM05] and [fig:radialprofilesVB03]. The global comparison of the mean fraction < *t**ν*/*t*\\* > calculated over *l*\\* ≤ *R* ≤ *R*25 is shown in Fig. [fig:sfrmdot]. The KM05 and VB03 star formation prescriptions yield consistent results for this fraction within the galactic disks.\nThe local viscous and star formation timescales are $t\\_{\\nu}=R^2/\\nu=2 \\pi \\Sigma R^2 / \\dot{M}$ and $t\\_\\*=\\Sigma / \\dot{\\Sigma}\\_{\\*}$. The mean fraction is then approximately $<t\\_{\\nu}/t\\_{\\*}> \\simeq <2 \\pi \\Sigma\\_\\* R^2 / \\dot{M}> \\simeq \\dot{M}\\_{\\*}/\\dot{M}$. The important point for the global comparison is that the derived mass accretion rates for almost all sample galaxies lie in the range between ∼ 0.1 M⊙yr− 1 and 0.6 M⊙yr− 1, whereas the star formation rates vary between 0.1 M⊙yr− 1 and 3 M⊙yr− 1. In the following we show why the mass accretion rate shows this behavior. Using $\\nu=v\\_{\\rm turb}l\\_{\\rm driv}$ in Eq. [eq:energyflux] leads to\n \n$$\\label{eq:eq1}\nl\\_{\\rm driv}=\\frac{\\Sigma v\\_{\\rm turb}^{3}}{\\xi \\dot{\\Sigma\\_{\\*}}}\\.$$ \nThe expression for the mass accretion rate then becomes\n \n$$\\label{eq:eq2}\n\\dot{M}=2 \\pi \\nu \\Sigma = 2 \\pi v\\_{\\rm turb} l\\_{\\rm driv} \\Sigma = \\frac{2 \\pi \\Sigma^{2} v\\_{\\rm turb}^{4}}{\\xi \\dot{\\Sigma}\\_{\\*}}\\.$$ \nAssuming typical values at *R*25/2, $v\\_{\\rm turb}=10$ kms− 1, $l\\_{\\rm driv}=100$ pc, Σ = 10 M⊙pc− 2, yields a mass accretion rate of $\\dot{M}=0.07$ M⊙yr− 1. For an error estimate we use the observational uncertainties given in Sec. [sec:data]. \nBecause of the large uncertainties associated with the molecular gas, we estimate the error of the mass accretion rate at galactic radii larger than *R*25/2 where the gas is predominantly in atomic form. We assume Σ = 10 M⊙pc− 2, ΔΣ = 1 M⊙pc− 2, $\\dot{\\Sigma}\\_{\\*}=5 \\times 10^{-10}$ M⊙pc− 2yr− 1, $\\Delta \\dot{\\Sigma}\\_{\\*}=2.5 \\times 10^{-10}$ M⊙pc− 2yr− 1, $v\\_{\\rm turb}=10$ kms− 1, and $\\Delta v\\_{\\rm turb}=3$ kms− 1. This leads to $\\log(\\dot{M} ({\\rm M}\\_{\\odot}{\\rm yr}^{-1}))=-0.5 \\pm 0.5$, i.e. we obtain an uncertainty of about a factor of 3.\nThe inspection of Figs. [fig:radialprofilesKM05], [fig:radialprofilesVB03], and [fig:sfrmdot] gives the consistent answer that only the less massive galaxies ($\\log M\\_{\\*} ({\\rm M}\\_{\\odot}) \\la 10$) can sustain the gas loss due to star formation by radial gas transport within the galactic disks. These galaxies can even have access to their gas reservoir beyond the optical radius. On the other hand, the radial gas transport in the massive spiral galaxies might not be sufficient to balance the gas loss due to star formation. This implies that, whereas a massive galaxy needs spherical infall from a putative gas halo or has to wait for infall with an angular momentum close to that of its disk to replenish its gas content, a less massive galaxy can live with large angular momentum accretion, because its mass accretion rate even at *R* > *R*25 is large enough for using this gas to sustain its star formation rate. The star formation rate of the massive galaxies is thus set by the amount of external accretion. In the absence of such an external gas accretion galaxies will slowly consume their gas, the gas surface density will decrease, the Toomre parameter *Q* of the gas will increase, and the star formation rate will decline[2](#fn2). Examples in our sample might be NGC 3351 and NGC 2841. Given that most of the massive galaxies of our sample do not show this behavior suggests that these galaxies experience mass accretion with rates comparable to their star formation rates (1-3 M⊙yr− 1) from a putative spherical halo of ionized gas or from satellite accretion leading to a temporarily enhanced mass accretion rate within the disk.\nConclusions [sec:conclusions]\n=============================\nThe theory of clumpy gas disks (VB03) provides analytic expressions for large-scale and small-scale properties of galactic gas disks. Large-scale properties considered are the gas surface density, density, disk height, turbulent driving length scale, velocity dispersion, gas viscosity, volume filling factor, and molecular fraction. Small-scale properties are the mass, size, density, turbulent, free-fall, molecular formation timescales of the most massive selfgravitating gas clouds. These quantities depend on the stellar surface density, the angular velocity Ω, the disk radius *R*, and 4 free parameters, which are the Toomre parameter *Q* of the gas, the mass accretion rate $\\dot{M}$, the ratio *δ* between the driving length scale of turbulence and the cloud size, and the radius, at which the local star formation timescale is no longer the cloud free-fall timescale, but the molecule formation timescale. We determine these free parameters using three independent measurements of the radial profiles of the (i) neutral gas (Hi), molecular gas (CO), and star formation rate (FUV + 24 *μ*m). A sample of 18 mostly spiral galaxies from is used in the analysis. Based on the simultaneous VB03 model fitting of the radial profiles of the total gas surface density, velocity dispersion, star formation efficiency, and molecular fraction, we conclude that\n1. the model star formation efficiency is very sensitive to the description of local pressure equilibrium in the disk midplane (Eq. [eq:pressure]);\n2. the fits of all radial profiles are acceptable for all galaxies except NGC 5194 (M 51). The model-derived free-fall timescales of selfgravitating clouds are in good agreement with expectations from observations;\n3. the observed radially decreasing gas velocity dispersion can be reproduced by the model. In the framework of the VB03 model the decrease in the inner disk is due to the stellar mass distribution which dominates the gravitational potential. A selfgravitating gas disk yields a constant velocity dispersion in the inner disk, whereas it leads to a radially decreasing velocity dispersion in the outer disk. It might be thus possible to identify the different regimes from the radial behavior of the gas velocity dispersion;\n4. Introducing a change in star formation regime into the model improves the fits significantly. This change is realized by replacing the free fall time in the prescription of the star formation rate with the molecule formation timescale;\n5. depending on the star formation prescription, the best-fit break between regimes in the model is located near the transition region between the molecular-gas-dominated and atomic-gas-dominated parts of the galactic disk or closer to the optical radius;\n6. the viscous timescale is smaller than or comparable to the star formation timescale for galaxies less massive than 1010 M⊙, whereas it is much higher for the massive galaxies;\n7. as a consequence less massive galaxies can balance the gas loss due to star formation by radial gas inflow within the galactic disk. In this way these galaxies can even use the gas reservoir outside the optical radius. This is impossible for massive galaxies. The star formation rate of massive galaxies is determined by the external gas mass accretion rate. Massive galaxies depend thus on external infall with an angular momentum close to that of the disk, whereas less massive galaxies can use large angular momentum gas located beyond the optical radius for star formation.\nModel equations [sec:modelequations]\n====================================\nStar formation recipe according to (Eq. [eq:starform]) [sec:app1]\n------------------------------------------------------------------\nThe following equations are appropriate for the *inner disk* regime where the local free fall time is the limiting timescale for star formation and Σ\\* ≫ Σ. For the actual numbers we assume $\\dot{M}=0.2$ M⊙yr− 1, *Q* = 2, Σ\\* = 100 M⊙yr− 1, *δ* = 2, *η* = 7 × 10− 3, $\\alpha=7.2 \\times 10^{7} \\times \\big( \\log(\\frac{\\Sigma\\_{\\*}+\\Sigma}{\\Sigma})\n\\big)^{-1}\\ {\\rm yr\\,M\\_{\\odot}pc^{-3}}=2 \\times 10^{7}$ yrM⊙pc− 3, *γ* = 0.2, and Ω = 1.8 × 10− 8 yr− 1.\n \n$$v\\_{\\rm turb}=1.08\\,G^{\\frac{1}{3}}\\dot{M}^{\\frac{1}{6}}Q^{\\frac{1}{6}}\\delta^{\\frac{1}{6}}\n\\xi^{\\frac{1}{6}}\\eta^{\\frac{1}{6}}\\Sigma\\_{\\*}^{\\frac{1}{6}} \\Omega^{-\\frac{1}{6}}=19~{\\rm km\\,s}^{-1}\\,\n\\label{eq:vturbsgz3}$$ \n \n$$\\phi\\_{\\rm V}=0.50\\,G^{\\frac{2}{3}}\\dot{M}^{\\frac{5}{6}}\\delta^{-\\frac{19}{6}}Q^{-\\frac{1}{6}}\n\\xi^{-\\frac{7}{6}}\\eta^{-\\frac{7}{6}}\\Sigma\\_{\\*}^{-\\frac{1}{6}}\\Omega^{\\frac{1}{6}}=0.01\\,\n\\label{eq:phiv3}$$ \n \n$$\\Sigma=0.50\\,G^{-\\frac{2}{3}}\\dot{M}^{\\frac{1}{6}}Q^{-\\frac{5}{6}}\\delta^{\\frac{1}{6}}\n\\xi^{\\frac{1}{6}}\\eta^{\\frac{1}{6}}\\Sigma\\_{\\*}^{\\frac{1}{6}}\\Omega^{\\frac{5}{6}}=16~{\\rm M\\_{\\odot}pc^{-2}}\\,\n\\label{eq:sigmasgz3}$$ \n \n$$\\label{eq:starform3}\n\\dot{\\Sigma}\\_{\\*}=\\eta \\delta \\Sigma\\_{\\*} Q^{-1} \\Omega=1.3 \\times 10^{-8}~{\\rm M\\_{\\odot}yr^{-1}pc^{-2}}\\,$$ \n \n$$l\\_{\\rm driv}=0.43\\,G^{\\frac{1}{3}}\\dot{M}^{\\frac{2}{3}}Q^{\\frac{2}{3}}\\delta^{-\\frac{1}{3}}\n\\xi^{-\\frac{1}{3}}\\eta^{-\\frac{1}{3}}\\Sigma\\_{\\*}^{-\\frac{1}{3}}\\Omega^{-\\frac{2}{3}}=140~{\\rm pc}\\,\n\\label{eq:ldriv3}$$ \n \n$$\\nu=0.46\\,G^{\\frac{2}{3}}\\dot{M}^{\\frac{5}{6}}Q^{\\frac{5}{6}}\\delta^{-\\frac{1}{6}}\\xi^{-\\frac{1}{6}}\n\\eta^{-\\frac{1}{6}}\\Sigma\\_{\\*}^{-\\frac{1}{6}}\\Omega^{-\\frac{5}{6}}=2.8 \\times 10^{-3}~{\\rm pc^{2}yr^{-1}}\\,\n\\label{eq:nu3}$$ \n \n$$\\label{eq:sfe3}\nSFE=1.99\\,G^{\\frac{2}{3}}\\dot{M}^{-\\frac{1}{6}}Q^{-\\frac{1}{6}}\\delta^{\\frac{5}{6}}\\xi^{-\\frac{1}{6}}\n\\eta^{\\frac{5}{6}}\\Sigma\\_{\\*}^{\\frac{5}{6}}\\Omega^{\\frac{1}{6}}=7.7 \\times 10^{-10}~{\\rm yr^{-1}}\\,$$ \n \n$$\\label{eq:fmol3}\nf\\_{\\rm mol}=0.74\\,G^{-1}\\dot{M}^{-\\frac{1}{2}}Q^{-\\frac{1}{2}}\\delta^{\\frac{3}{2}}\n\\xi^{\\frac{1}{2}}\\eta^{\\frac{1}{2}}\\Sigma\\_{\\*}^{\\frac{1}{2}}\\Omega^{\\frac{1}{2}} \\alpha^{-1}=0.80\\.$$ \nThe following equations are appropriate for the *inner disk* regime where the local free fall time is the limiting timescale for star formation and Σ ≫ Σ\\*:\n \n$$v\\_{\\rm turb}=0.87\\,G^{\\frac{1}{5}}\\dot{M}^{\\frac{1}{5}}\\delta^{\\frac{1}{5}}\n\\xi^{\\frac{1}{5}}\\eta^{\\frac{1}{5}}=12~{\\rm km\\,s}^{-1}\\,\n\\label{eq:vturbsgz}$$ \n \n$$\\phi\\_{\\rm V}=0.62\\,G^{\\frac{4}{5}}\\dot{M}^{\\frac{4}{5}}\\delta^{-\\frac{16}{5}}\n\\xi^{-\\frac{6}{5}}\\eta^{-\\frac{6}{5}}=0.02\\,\n\\label{eq:phiv}$$ \n \n$$\\Sigma=0.28\\,G^{-\\frac{4}{5}}\\dot{M}^{\\frac{1}{5}}Q^{-1}\\delta^{\\frac{1}{5}}\n\\xi^{\\frac{1}{5}}\\eta^{\\frac{1}{5}}\\Omega=7~{\\rm M\\_{\\odot}pc^{-2}}\\,\n\\label{eq:sigmasgz1}$$ \n \n$$\\label{eq:starform1}\n\\dot{\\Sigma}\\_{\\*}=0.28\\,G^{-\\frac{4}{5}}\\dot{M}^{\\frac{1}{5}}Q^{-2}\\delta^{\\frac{6}{5}}\\xi^{\\frac{1}{5}}\\eta^{\\frac{6}{5}}\\Omega^{2}\n=9.2 \\times 10^{-10}~{\\rm M\\_{\\odot}yr^{-1}pc^{-2}}\\,$$ \n \n$$l\\_{\\rm driv}=0.66\\,G^{\\frac{3}{5}}\\dot{M}^{\\frac{3}{5}}Q\\delta^{-\\frac{2}{5}}\n\\xi^{-\\frac{2}{5}}\\eta^{-\\frac{2}{5}}\\Omega^{-1}=350~{\\rm pc}\\,\n\\label{eq:ldriv}$$ \n \n$$\\nu=0.57\\,G^{\\frac{4}{5}}\\dot{M}^{\\frac{4}{5}}Q\\delta^{-\\frac{1}{5}}\\xi^{-\\frac{1}{5}}\n\\eta^{-\\frac{1}{5}}\\Omega^{-1}=6.0 \\times 10^{-4}~{\\rm pc^{2}yr^{-1}}\\,\n\\label{eq:nu}$$ \n \n$$\\label{eq:sfe}\nSFE=\\eta \\delta Q^{-1} \\Omega=1.3 \\times 10^{-10}~{\\rm yr^{-1}}\\,$$ \n \n$$\\label{eq:fmol}\nf\\_{\\rm mol}=0.39\\,G^{-\\frac{7}{5}}\\dot{M}^{-\\frac{2}{5}}Q^{-1}\\delta^{\\frac{8}{5}}\n\\xi^{\\frac{3}{5}}\\eta^{\\frac{3}{5}}\\Omega \\alpha^{-1}=0.21\\.$$ \nNote that the turbulent velocity and the volume filling factor are constant and thus independent of the galactocentric radius.\nThe following equations are appropriate for a selfgravitating *outer disk* where the molecular formation timescale is the relevant timescale for star formation and Σ ≫ Σ\\*:\n \n$$v\\_{\\rm turb}=0.54\\,G^{-\\frac{1}{2}}Q^{-\\frac{1}{2}} \\delta \\xi^{\\frac{1}{2}}\\eta^{\\frac{1}{2}}\n(\\gamma \\alpha)^{-\\frac{1}{2}} \\Omega^{\\frac{1}{2}}=12~{\\rm km\\,s}^{-1}\\,\n\\label{eq:vturbsgz1}$$ \n \n$$\\phi\\_{\\rm V}=10.6\\,G^{5}\\dot{M}^{2}Q^{3}\\delta^{-8} \\xi^{-3} \\eta^{-3} \\alpha^{3} \\Omega^{-3}=0.01\\,\n\\label{eq:phiv1}$$ \n \n$$\\Sigma=0.17\\,G^{-\\frac{3}{2}} Q^{-\\frac{3}{2}} \\delta \\xi^{\\frac{1}{2}} \\eta^{\\frac{1}{2}} \n\\alpha^{-\\frac{1}{2}} \\Omega^{\\frac{3}{2}}=7~{\\rm M\\_{\\odot}pc^{-2}}\\,\n\\label{eq:sigmasgz2}$$ \n \n$$\\label{eq:starform2}\n\\dot{\\Sigma}\\_{\\*}=0.02\\,G^{-5}\\dot{M}^{-1}Q^{-5} \\delta^{6} \\xi^{2} \\eta^{3} \\alpha^{-3} \\Omega^{5}=\n1.4 \\times 10^{-9}~{\\rm M\\_{\\odot}yr^{-1}pc^{-2}}\\,$$ \n \n$$l\\_{\\rm driv}=1.7\\,G^{2}\\dot{M} Q^{2} \\delta^{-2} \\xi^{-1} \\eta^{-1} \\alpha \\Omega^{-2}=330~{\\rm pc}\\,\n\\label{eq:ldriv1}$$ \n \n$$\\nu=0.92\\,G^{\\frac{3}{2}}\\dot{M} Q^{\\frac{3}{2}} \\delta^{-\\frac{3}{2}} \\xi^{-\\frac{1}{2}} \\eta^{-\\frac{1}{2}}\n\\alpha^{\\frac{1}{2}}\\Omega^{-\\frac{3}{2}}=3.0 \\times 10^{-3}~{\\rm pc^{2}yr^{-1}}\\,\n\\label{eq:nu1}$$ \n \n$$\\label{eq:sfe1}\nSFE=0.09\\,G^{-\\frac{7}{2}}\\dot{M}^{-1}Q^{-\\frac{7}{2}} \\delta^{5}\n\\xi^{\\frac{3}{2}} \\eta^{\\frac{5}{2}} (\\gamma \\alpha)^{-\\frac{5}{2}}\\Omega^{\\frac{7}{2}}=1.4 \\times 10^{-10}~{\\rm yr^{-1}}\\,$$ \n \n$$\\label{eq:fmol1}\nf\\_{\\rm mol}=0.09\\,G^{-\\frac{7}{2}} \\dot{M}^{-1} Q^{-\\frac{5}{2}} \\delta^{4}\n\\xi^{\\frac{3}{2}} \\eta^{\\frac{3}{2}} \\gamma^{-\\frac{3}{2}} \\alpha^{-\\frac{5}{2}}\\Omega^{\\frac{5}{2}}=0.23\\.$$ \nStar formation according to VB03 (Eq. [eq:starformm])\n-----------------------------------------------------\nThe following equations are appropriate for the *inner disk* regime where the local free fall time is the limiting timescale for star formation and Σ\\* ≫ Σ. For the actual numbers we assume $\\dot{M}=0.2$ M⊙yr− 1, *Q* = 2, Σ\\* = 100 M⊙yr− 1, *δ* = 2, $\\alpha=7.2 \\times 10^{7} \\times \\big( \\log(\\frac{\\Sigma\\_{\\*}+\\Sigma}{\\Sigma})\n\\big)^{-1}\\ {\\rm yr\\,M\\_{\\odot}pc^{-3}}=2 \\times 10^{7}$ yrM⊙pc− 3, *γ* = 0.2, and Ω = 1.8 × 10− 8 yr− 1.\n \n$$v\\_{\\rm turb}=1.02\\,G^{\\frac{5}{13}}\\dot{M}^{\\frac{3}{13}}Q^{\\frac{2}{13}}\\delta^{-\\frac{1}{13}}\n\\xi^{\\frac{1}{13}}\\Sigma\\_{\\*}^{\\frac{2}{13}} \\Omega^{-\\frac{2}{13}}=19~{\\rm km\\,s}^{-1}\\,\n\\label{eq:vturbsgz4}$$ \n \n$$\\phi\\_{\\rm V}=0.73\\,G^{\\frac{4}{13}}\\dot{M}^{\\frac{5}{13}}\\delta^{-\\frac{19}{13}}Q^{-\\frac{1}{13}}\n\\xi^{-\\frac{7}{13}}\\Sigma\\_{\\*}^{-\\frac{1}{13}}\\Omega^{\\frac{1}{13}}=9 \\times 10^{-3}\\,\n\\label{eq:phiv4}$$ \n \n$$\\Sigma=0.33\\,G^{-\\frac{8}{13}}\\dot{M}^{\\frac{3}{13}}Q^{-\\frac{11}{13}}\\delta^{-\\frac{1}{13}}\n\\xi^{\\frac{1}{13}}\\Sigma\\_{\\*}^{\\frac{2}{13}}\\Omega^{\\frac{11}{13}}=12~{\\rm M\\_{\\odot}pc^{-2}}\\,\n\\label{eq:sigmasgz4}$$ \n \n$$\\label{eq:starform4}\n\\dot{\\Sigma}\\_{\\*}= 0.73\\,G^{\\frac{4}{13}}\\dot{M}^{\\frac{5}{13}}Q^{-\\frac{14}{13}}\\delta^{-\\frac{6}{13}}\n\\xi^{-\\frac{7}{13}}\\Sigma\\_{\\*}^{\\frac{12}{13}}\\Omega^{\\frac{14}{13}}=\n1.5 \\times 10^{-8}~{\\rm M\\_{\\odot}yr^{-1}pc^{-2}}\\,$$ \n \n$$l\\_{\\rm driv}=0.48\\,G^{\\frac{3}{13}}\\dot{M}^{\\frac{7}{13}}Q^{\\frac{9}{13}}\\delta^{\\frac{2}{13}}\n\\xi^{-\\frac{2}{13}}\\Sigma\\_{\\*}^{-\\frac{4}{13}}\\Omega^{-\\frac{9}{13}}=140~{\\rm pc}\\,\n\\label{eq:ldriv4}$$ \n \n$$\\nu=0.49\\,G^{\\frac{8}{13}}\\dot{M}^{\\frac{10}{13}}Q^{\\frac{11}{13}}\\delta^{\\frac{1}{13}}\\xi^{-\\frac{1}{13}}\n\\Sigma\\_{\\*}^{-\\frac{2}{13}}\\Omega^{-\\frac{11}{13}}=2.8 \\times 10^{-3}~{\\rm pc^{2}yr^{-1}}\\,\n\\label{eq:nu4}$$ \n \n$$\\label{eq:sfe4}\nSFE=2.23\\,G^{\\frac{12}{13}}\\dot{M}^{\\frac{2}{13}}Q^{-\\frac{3}{13}}\\delta^{-\\frac{5}{13}}\\xi^{-\\frac{8}{13}}\n\\Sigma\\_{\\*}^{\\frac{10}{13}}\\Omega^{\\frac{3}{13}}=1.3 \\times 10^{-9}~{\\rm yr^{-1}}\\,$$ \n \n$$\\label{eq:fmol4}\nf\\_{\\rm mol}=0.63\\,G^{-\\frac{11}{13}}\\dot{M}^{-\\frac{4}{13}}Q^{-\\frac{7}{13}}\\delta^{\\frac{10}{13}}\n\\xi^{\\frac{3}{13}}\\Sigma\\_{\\*}^{\\frac{6}{13}}\\Omega^{\\frac{7}{13}} \\alpha^{-1}=0.88\\.$$ \nThe following equations are appropriate for the *inner disk* regime where the local free fall time is the limiting timescale for star formation and Σ ≫ Σ\\*:\n \n$$v\\_{\\rm turb}=0.82\\,G^{\\frac{3}{11}}\\dot{M}^{\\frac{3}{11}}\\delta^{-\\frac{1}{11}}\n\\xi^{\\frac{1}{11}}=13~{\\rm km\\,s}^{-1}\\,\n\\label{eq:vturbsgz5}$$ \n \n$$\\phi\\_{\\rm V}=0.81\\,G^{\\frac{4}{11}}\\dot{M}^{\\frac{4}{11}}\\delta^{-\\frac{16}{11}}\n\\xi^{-\\frac{6}{11}}=0.01\\,\n\\label{eq:phiv5}$$ \n \n$$\\Sigma=0.26\\,G^{-\\frac{8}{11}}\\dot{M}^{\\frac{3}{11}}Q^{-1}\\delta^{-\\frac{1}{11}}\n\\xi^{\\frac{1}{11}}\\Omega=8~{\\rm M\\_{\\odot}pc^{-2}}\\,\n\\label{eq:sigmasgz5}$$ \n \n$$\\label{eq:starform5}\n\\dot{\\Sigma}\\_{\\*}=0.21\\,G^{-\\frac{4}{11}}\\dot{M}^{\\frac{7}{11}}Q^{-2}\\delta^{-\\frac{6}{11}}\\xi^{-\\frac{5}{11}}\\Omega^{2}=\n1.4 \\times 10^{-9}~{\\rm M\\_{\\odot}yr^{-1}pc^{-2}}\\,$$ \n \n$$l\\_{\\rm driv}=0.71\\,G^{\\frac{5}{11}}\\dot{M}^{\\frac{5}{11}}Q\\delta^{\\frac{2}{11}}\n\\xi^{-\\frac{2}{11}}\\Omega^{-1}=290~{\\rm pc}\\,\n\\label{eq:ldriv5}$$ \n \n$$\\nu=0.61\\,G^{\\frac{8}{11}}\\dot{M}^{\\frac{8}{11}}Q\\delta^{\\frac{1}{11}}\\xi^{-\\frac{1}{11}}\\Omega^{-1}=\n4.1 \\times 10^{-3}~{\\rm pc^{2}yr^{-1}}\\,\n\\label{eq:nu5}$$ \n \n$$\\label{eq:sfe5}\nSFE=0.81\\,G^{\\frac{4}{11}}\\dot{M}^{\\frac{4}{11}}Q^{-1}\\delta^{-\\frac{5}{11}}\\xi^{-\\frac{6}{11}}\\Omega=\n1.9 \\times 10^{-10}~{\\rm yr^{-1}}\\,$$ \n \n$$\\label{eq:fmol5}\nf\\_{\\rm mol}=0.34\\,G^{-\\frac{13}{11}}\\dot{M}^{-\\frac{2}{11}}Q^{-1}\\delta^{\\frac{8}{11}}\n\\xi^{\\frac{3}{11}}\\Omega \\alpha^{-1}=0.27\\.$$ \nNote that the turbulent velocity and the volume filling factor are constant and thus independent of the galactocentric radius.\nThe following equations are appropriate for a selfgravitating *outer disk* where the molecular formation timescale is the relevant timescale for star formation and Σ ≫ Σ\\*:\n \n$$v\\_{\\rm turb}=0.71\\,G^{\\frac{1}{8}}\\dot{M}^{\\frac{1}{4}}Q^{-\\frac{1}{8}}\\xi^{\\frac{1}{8}}\n(\\gamma \\alpha)^{-\\frac{1}{8}} \\Omega^{\\frac{1}{8}}=13~{\\rm km\\,s}^{-1}\\,\n\\label{eq:vturbsgz6}$$ \n \n$$\\phi\\_{\\rm V}=1.95\\,G^{\\frac{5}{4}}\\dot{M}^{\\frac{1}{2}}Q^{\\frac{3}{4}}\\delta^{-2}\n\\xi^{-\\frac{3}{4}} (\\gamma \\alpha)^{\\frac{3}{4}} \\Omega^{-\\frac{3}{4}}=9 \\times 10^{-3}\\,\n\\label{eq:phiv6}$$ \n \n$$\\Sigma=0.23\\,G^{-\\frac{7}{8}}\\dot{M}^{\\frac{1}{4}}Q^{-\\frac{9}{8}}\\xi^{\\frac{1}{8}} \n(\\gamma \\alpha)^{-\\frac{1}{8}} \\Omega^{\\frac{9}{8}}=8~{\\rm M\\_{\\odot}pc^{-2}}\\,\n\\label{eq:sigmasgz6}$$ \n \n$$\\label{eq:starform6}\n\\dot{\\Sigma}\\_{\\*}=0.09\\,G^{-\\frac{5}{4}}\\dot{M}^{\\frac{1}{2}}Q^{-\\frac{11}{4}}\\xi^{-\\frac{1}{4}} \n(\\gamma \\alpha)^{-\\frac{3}{4}} \\Omega^{\\frac{11}{4}}=1.7 \\times 10^{-9}~{\\rm M\\_{\\odot}yr^{-1}pc^{-2}}\\,$$ \n \n$$l\\_{\\rm driv}=0.93\\,G^{\\frac{3}{4}}\\dot{M}^{\\frac{1}{2}}Q^{\\frac{5}{4}}\\xi^{-\\frac{1}{4}}\n(\\gamma \\alpha)^{\\frac{1}{4}}\\Omega^{-\\frac{5}{4}}=270~{\\rm pc}\\,\n\\label{eq:ldriv6}$$ \n \n$$\\nu=0.69\\,G^{\\frac{7}{8}}\\dot{M}^{\\frac{3}{4}}Q^{\\frac{9}{8}} \\xi^{-\\frac{1}{8}}\n(\\gamma \\alpha)^{\\frac{1}{8}}\\Omega^{-\\frac{9}{8}}=\n3.9 \\times 10^{-3}~{\\rm pc^{2}yr^{-1}}\\,\n\\label{eq:nu6}$$ \n \n$$\\label{eq:sfe6}\nSFE=0.39\\,G^{-\\frac{3}{8}}\\dot{M}^{\\frac{1}{4}}Q^{-\\frac{13}{8}}\n\\xi^{-\\frac{3}{8}}(\\gamma \\alpha)^{-\\frac{5}{8}}\\Omega^{\\frac{13}{8}}=\n2.1 \\times 10^{-10}~{\\rm yr^{-1}}\\,$$ \n \n$$\\label{eq:fmol6}\nf\\_{\\rm mol}=0.22\\,G^{-\\frac{13}{8}} \\dot{M}^{-\\frac{1}{4}} Q^{-\\frac{11}{8}} \\delta\n\\xi^{\\frac{3}{8}} \\gamma^{-\\frac{3}{8}} \\alpha^{-\\frac{11}{8}}\\Omega^{\\frac{11}{8}}=0.29\\.$$ \nCritical angular velocity\n-------------------------\nHere we give the expressions for the critical angular velocity $\\Omega\\_{\\rm crit}$ where $t^{\\rm l}\\_{\\rm ff}= \\gamma t^{\\rm l}\\_{\\rm mol}$ (Σ ≫ Σ\\*). 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D., & Leroy, A. 2008,, 136, 2563 Wolf-Chase G.A., Barsony M., O’Linger J. 2000, AJ, 120, 1467 Wolfire M.G., Hollenbach D., McKee C.F., Tielens A.G.G.M., & Bakes E.L.O. 1995, ApJ, 443, 152 Wong T. & Blitz L. 2002, ApJ, 569, 157\nlll Parameter & Unit & Explanation \n*G* = 5 × 10− 15 & pc3yr− 1M⊙− 1 & gravitation constant \n*κ* & yr− 1 & epicyclic frequency \n*Q* & & Toomre parameter \n*R* & pc &galactocentric radius \n*H* & pc & thickness of the gas disk \n*H*\\* & pc & thickness of the stellar disk \n$R\\_{\\rm break}$ & pc & break radius between star formation regimes \n$l\\_{\\rm cl}$ & pc & cloud size \n$v\\_{\\rm rot}$ & pcyr− 1 & rotation velocity \n$\\Omega=v\\_{\\rm rot}/R$ & yr− 1 & angular velocity \n$\\Phi\\_{\\rm V}$ & & volume filling factor \n$\\Phi\\_{\\rm A}=\\Phi\\_{\\rm V}\\,H/l\\_{\\rm cl}$ & & area filling factor \n*ρ* & M⊙pc− 3 & disk midplane gas density \n$\\rho\\_{\\rm cl}=\\rho/\\Phi\\_{\\rm V}$ & M⊙pc− 3 & cloud density \n$\\dot{\\rho}\\_{\\*}$ & M⊙pc− 3yr− 1 & star formation rate \nΣ & M⊙pc− 2 & gas surface density \nΣ\\* & M⊙pc− 2 & stellar surface density \n$\\dot{\\Sigma}\\_{\\*}$ & M⊙pc− 2yr− 1 & star formation rate \n*ξ* = 4.6 × 10− 8 & pc2yr− 2 & constant relating SN energy input to SF \n$\\dot{M}$ & M⊙yr− 1 & disk mass accretion rate \n$v\\_{\\rm turb}$ & pcyr− 1 & gas turbulent velocity dispersion \n$v\\_{\\rm rad}$ & pcyr− 1 & gas radial velocity \n$v\\_{\\rm disp}^{\\*}$ & pcyr− 1 & stellar vertical velocity dispersion \n*ν* & pc2yr− 1 & viscosity \n$f\\_{\\rm mol}=\\Sigma\\_{\\rm H\\_{2}}/(\\Sigma\\_{\\rm HI}+\\Sigma\\_{\\rm H\\_{2}})$ & & molecular fraction \n*α* & yrM⊙pc− 3 & constant of molecule formation timescale \n$l\\_{\\rm driv}$ & pc & turbulent driving length scale \n$l\\_{\\rm diss}$ & pc & turbulent dissipation length scale \n*δ* & & scaling between driving and dissipation length scale \n$SFE=\\dot{\\Sigma}\\_{\\*}/\\Sigma$ & yr− 1 & star formation efficiency \n$t\\_{\\rm ff}^{l}$ & yr & cloud free fall timescale at size *l* \n$t\\_{\\rm turb}^{l}$ & yr & cloud turbulent timescale at size *l* \n$t\\_{\\rm mol}^{l}$ & yr & cloud molecule formation timescale at size *l* \n*η* & & star formation efficiency per free fall time \n*γ* & & $t\\_{\\rm sf}^{l}= \\gamma t\\_{\\rm mol}^{l}$ at the break radius \nlccccccccc Galaxy & Dist. & i & P.A. & Morph. & $M\\_{\\rm B}$ & $R\\_{\\rm 25}$ & log *M*\\* & SFR & *l*\\* \n& (Mpc) & (∘) & (∘) & & (mag) & (kpc) & (M⊙) & (M⊙yr− 1) & (kpc) \nIC 2574 & 4.0 & 53 & 56 & Irr & -18.0 & 7.5 & 8.7 & 0.07 & 2.1 \nNGC 4214 & 2.9 & 44 & 65 & Irr & -17.4 & 2.9 & 8.8 & 0.11 & 0.7 \nNGC 2976 & 3.6 & 65 & 335 & Sc & -17.8 & 3.8 & 9.1 & 0.09 & 0.9 \nNGC 7793 & 3.9 & 50 & 290 & Scd & -18.7 & 6.0 & 9.5 & 0.24 & 1.3 \nNGC 2403 & 3.2 & 63 & 124 & SBc & -19.4 & 7.3 & 9.7 & 0.38 & 1.6 \nNGC 0925 & 9.2 & 66 & 287 & SBcd & -20.0 & 14.2 & 9.9 & 0.56 & 4.1 \nNGC 0628 & 7.3 & 7 & 20 & Sc & -20.0 & 10.4 & 10.1 & 0.81 & 2.3 \nNGC 3198 & 13.8 & 72 & 215 & SBc & -20.7 & 13.0 & 10.1 & 0.93 & 3.2 \nNGC 3184 & 11.1 & 16 & 179 & SBc & -19.9 & 11.9 & 10.3 & 0.90 & 2.4 \nNGC 4736 & 4.7 & 41 & 296 & Sab & -20.0 & 5.3 & 10.3 & 0.48 & 1.1 \nNGC 3351 & 10.1 & 41 & 192 & SBb & -19.7 & 10.6 & 10.4 & 0.94 & 2.2 \nNGC 6946 & 5.9 & 33 & 243 & SBc & -20.9 & 9.8 & 10.5 & 3.24 & 2.5 \nNGC 3627 & 9.3 & 62 & 173 & SBb & -20.8 & 13.9 & 10.6 & 2.22 & 2.8 \nNGC 5194 & 8.0 & 20 & 172 & SBc & -21.1 & 9.0 & 10.6 & 3.13 & 2.8 \nNGC 3521 & 10.7 & 73 & 340 & SBbc & -20.9 & 12.9 & 10.7 & 2.10 & 2.9 \nNGC 2841 & 14.1 & 74 & 153 & Sb & -21.2 & 14.2 & 10.8 & 0.74 & 4.0 \nNGC 5055 & 10.1 & 59 & 102 & Sbc & -20.6 & 17.4 & 10.8 & 2.12 & 3.2 \nNGC 7331 & 14.7 & 76 & 168 & SAb & -21.7 & 19.6 & 10.9 & 2.99 & 3.3 \nlccccccccc Galaxy & $\\dot{M}$ & *δ* & *γ* & $R\\_{\\rm break}$ & $c\\_{\\rm H\\_2}$ & $c\\_{\\rm v\\_{\\rm disp}}$ & $c\\_{\\rm SFE}$ & *g* & $< t\\_{\\rm ff} >$ \n& (M⊙yr− 1) & & & (kpc) & & & & & (Myr) \nKM05 & & & & & & & & \nIC2574 & 0.22 & 2.0 & 0.07 & 1.1 & 0.63 & 0.79 & 0.71 & 0.98 & 29.6 \nNGC4214 & 0.08 & 1.3 & 0.11 & 1.3 & 0.63 & 0.79 & 0.71 & 0.15 & 9.5 \nNGC2976 & 0.31 & 3.2 & - & - & 0.63 & 1.00 & 1.00 & 2.59 & 4.1 \nNGC7793 & 0.06 & 1.4 & 0.11 & 3.9 & 0.63 & 0.79 & 0.71 & 0.69 & 6.0 \nNGC2403 & 0.22 & 1.8 & 0.07 & 4.9 & 1.00 & 1.00 & 0.71 & 0.31 & 8.8 \nNGC0925 & 0.11 & 2.0 & 0.07 & 9.1 & 0.63 & 0.79 & 0.71 & 0.97 & 12.5 \nNGC0628 & 0.08 & 1.6 & 0.13 & 7.6 & 0.63 & 1.00 & 1.00 & 0.95 & 4.7 \nNGC3198 & 0.11 & 1.8 & 0.07 & 11.0 & 0.63 & 0.79 & 1.00 & 0.94 & 7.2 \nNGC3184 & 0.11 & 2.0 & 0.10 & 10.0 & 0.63 & 1.00 & 1.41 & 1.24 & 3.8 \nNGC4736 & 0.89 & 2.0 & - & - & 0.63 & 1.26 & 1.41 & 2.34 & 7.7 \nNGC3351 & 0.06 & 1.6 & 0.10 & 8.6 & 0.63 & 1.00 & 0.71 & 0.79 & 5.8 \nNGC6946 & 0.32 & 2.5 & - & - & 0.63 & 1.00 & 1.00 & 1.47 & 3.0 \nNGC3627 & 0.22 & 2.0 & - & - & 0.63 & 0.79 & 1.00 & 0.47 & 4.4 \nNGC5194 & 1.26 & 1.0 & - & - & 0.63 & 0.79 & 0.71 & 3.29 & 22.1 \nNGC3521 & 0.22 & 1.8 & - & - & 0.63 & 0.79 & 1.41 & 5.82 & 4.9 \nNGC2841 & 0.08 & 2.0 & 0.17 & 11.3 & 0.63 & 0.79 & 0.71 & 0.42 & 6.0 \nNGC5055 & 0.32 & 2.0 & 0.07 & 12.5 & 0.63 & 0.79 & 1.41 & 7.83 & 5.9 \nNGC7331 & 0.22 & 2.0 & 0.09 & 13.9 & 0.63 & 0.79 & 1.41 & 1.37 & 4.4 \nVB03 & & & & & & & & \nIC 2574 & 0.16 & 8.9 & 1.00 & 0.00 & 0.63 & 0.79 & 0.71 & 0.21 & 5.2 \nNGC 4214 & 0.06 & 1.0 & 1.00 & 0.00 & 0.63 & 0.79 & 0.71 & 0.21 & 11.2 \nNGC 2976 & 0.32 & 3.5 & - & - & 0.63 & 1.00 & 0.71 & 2.44 & 4.1 \nNGC 7793 & 0.16 & 1.0 & 1.00 & 0.00 & 0.63 & 1.00 & 0.71 & 0.33 & 12.5 \nNGC 2403 & 0.22 & 1.3 & 1.00 & 0.00 & 0.63 & 1.00 & 0.71 & 0.11 & 14.2 \nNGC 0925 & 0.16 & 1.0 & 0.33 & 0.00 & 0.63 & 0.79 & 0.71 & 0.41 & 32.0 \nNGC 0628 & 0.08 & 1.8 & 1.00 & 0.00 & 0.63 & 1.00 & 1.00 & 1.01 & 4.6 \nNGC 3198 & 0.11 & 2.5 & 0.33 & 4.3 & 1.00 & 0.79 & 1.00 & 0.93 & 5.1 \nNGC 3184 & 0.16 & 4.0 & 0.33 & 6.7 & 1.58 & 1.00 & 1.41 & 1.31 & 2.3 \nNGC 4736 & 0.63 & 7.9 & - & - & 1.58 & 1.26 & 1.41 & 2.59 & 1.7 \nNGC 3351 & 0.06 & 1.8 & 0.33 & 2.7 & 0.63 & 1.00 & 0.71 & 0.84 & 5.1 \nNGC 6946 & 0.45 & 7.9 & - & - & 1.58 & 1.00 & 1.00 & 1.13 & 1.2 \nNGC 3627 & 0.32 & 8.9 & - & - & 1.58 & 0.79 & 0.71 & 0.40 & 1.3 \nNGC 5194 & 0.06 & 8.9 & - & - & 1.58 & 0.79 & 0.71 & 4.30 & 0.8 \nNGC 3521 & 0.32 & 8.9 & 0.25 & 11.7 & 1.58 & 0.79 & 1.41 & 4.22 & 1.1 \nNGC 2841 & 0.11 & 3.5 & 0.33 & 5.1 & 1.00 & 0.79 & 0.71 & 0.41 & 4.3 \nNGC 5055 & 0.89 & 8.9 & 1.00 & 5.1 & 1.58 & 1.00 & 1.41 & 4.06 & 2.0 \nNGC 7331 & 0.22 & 4.5 & 0.50 & 6.8 & 1.58 & 0.79 & 1.41 & 0.67 & 2.0 \n---\n1. Only three galaxies, IC 2574, NGC 4214, and NGC 2976, have rotation velocities smaller than 100 kms− 1. We calculated the models for these galaxies using up to 3 times smaller effective yields. The results where not significantly different compared to the models based on Eq. [eq:alphacb] with the exception that the molecular fraction decreased by the same factor as the effective yield.[↩](#fnref1)\n2. This decline is somewhat slowed down by gas replenishment from dying stars.[↩](#fnref2)\n"},"data_id":{"kind":"string","value":"arxiv_0184773"}}},{"rowIdx":46199,"cells":{"content":{"kind":"string","value":"*6 \n*x*1 & **1.000** & 0.954 & 0.885 & 0.692 & 0.706 & 0.663 \n*x*2 & 0.954 &**1.000** & 0.933 & 0.528 & 0.729 & 0.663 \n*x*3 & 0.885 & 0.933& **1.000** & 0.466 & 0.618 & 0.578 \n*x*4 & 0.692 & 0.528 & 0.466 & **1.000** & 0.477 & 0.794 \n*x*5 & 0.706 & 0.729 & 0.618 & 0.477 & **1.000** & 0.591 \n*x*6 & 0.663 & 0.663 & 0.578 & 0.794 & 0.591 & **1.000** \n[tab:corr]\nAlso, the dataset is composed of some outlying observations (Figure [fig5]): Ferrari enzo (*x*1, *x*2, *x*5), Bentley continental (*x*2), Aston Martin (*x*2), Land Rover discovery (*x*5), Mercedes class S (*x*5), Smart (*x*5, *x*6).\n| | | |\n| --- | --- | --- |\n| capacity *x*1 | power *x*2 | speed *x*3 |\n| image | image | image |\n| weight *x*4 | width *x*5 | length *x*6 |\n| image | image | image |\n[fig5]\nThe overall information (variability) is partitioned over six components (Table [tab6]).\n[h!]\n| c || c | c | c | c | **eigenvalues in %** & Gini *ν* = 2 & Gini *ν* = 4 & Gini *ν* = 6 & Variance \n**Axis 1** & **80.35797** & **83.17172** & **84.84995** &**73.52112** \n**Axis 2** & **12.0761** & **10.58655** & **9.715974** & **14.22349** \n**Axis 3** & 4.132136 & 2.987015 & 3.130199 & **7.26106** \n**Axis 4** & 3.059399 & 2.612411 & 1.519626 & 3.93117 \n**Axis 5** & 0.3332362 & 0.3125735 & 0.2696611 & 0.85727 \n**Axis 6** & 0.04115858 & − 0.3297257 & -0.5145944 & 0.20585 \n**Sum** & **100 %** & **100 %** & **100 %** & **100 %** \n[tab6]\nTwo axes may be chosen to analyze the data. As shown in the previous Section about the simulations, when the data are highly correlated such that two axes are sufficient to project the data, the Gini PCA and the standard PCA yield the same share of information on each axis. However, we can expect some differences for absolute contributions *A**C**T* and relative contributions *R**C**T*.\nThe projection of the data is depicted in Figure [fig6], for each method.\n.6![Gini (\\nu = 2)](Proj_ind_nu2.png \"fig:\")\n.6![Gini (\\nu = 4)](Proj_ind_nu4.png \"fig:\")\n.6![Gini (\\nu = 6)](Proj_ind_nu6.png \"fig:\")\n.6![Variance](Proj_ind_var.png \"fig:\")\n[fig6]\nAs depicted in Figure [fig6], the projection is very similar for each technique. The cars with extraordinary (or very low) abilities are in the same relative position in the four projections: Land Rover Discovery Td5 at the top, Ferrari Enzo at the bottom right, Smart Fortwo coupé at the bottom left. However, when we improve the coefficient of variability *ν* to look for what happens at the tails of the distributions (of the two axes), we see that more cars are distinguishable: Land Rover Defender, Audi TT, BMW Z4, Renault Clio 3.0 V6, Bentley Contiental GT. Consequently, contrary to the case *ν* = 2 or the variance, the projections with *ν* = 4, 6 allow one to find other important observations, which are not outlying observations that contribute to the overall amount of variability. For this purpose, let us first analyze the correlations between the variables and the new axes in order to interpret the results, see Tables [tab7] to [tab10].\nSome slight differences appear between the Gini PCA and the classical one based on the variance. The theoretical Section [s4] indicates that the Gini methodology for *ν* = 2 is equivalent to the variance when the variables are Gaussian. On cars data, we observe this similarity. In each PCA, all variables are correlated with Axis 1 and weight with Axis 2. However, when *ν* increases, the Gini methodology allows outlying observations to be diluted so that some variables may appear to be significant, whereas they are not in the variance case.\n[h!]\n| c | c || l | l | l | l | l | c | **Gini** (*ν* = 2) & capacity & power & speed & weight & width & length \n& correlation & -0.974 & -0.945 & -0.872 & 0.760 & -0.933 & -0.823 \n Axe 1 & & & & & & & \n& ***U*-stat** & **-56.416** & **-25.005** & **-10.055** & **-4.093** & **-24.837** & **-12.626** \n& correlation & -0.032 & -0.241 & -0.405 & 0.510 & 0.183 & -0.379 \n Axe 2 & & & & & & & \n& ***U*-stat** & **-0.112** & **-0.920** & **-1.576** & **2.897** & **0.526** & **1.666** \n[tab7]\n[h!]\n| c | c || l | l | l | l | l | c | **Gini** (*ν* = 4) & capacity & power & speed & weight & width & length \n& correlation & 0.982 & 0.948 & 0.797 & 0.858 & 0.952 & 0.888 \n Axe 1 & & & & & & & \n& ***U*-stat** & **8.990** & **8.758** & **4.805** & **9.657** & **8.517** & **8.182** \n& correlation & -0.021 & 0.207 & 0.516 & -0.279 & -0.147 & -0.246 \n Axe 2 & & & & & & & \n& ***U*-stat** & **-0.095** & **0.817** & **2.299** & **-1.773** & **-0.705** & **-1.200** \n[tab8]\n[h!]\n| c | c || l | l | l | l | l | c | **Gini** (*ν* = 6) & capacity & power & speed & weight & width & length \n& valeurs & -0.781 & -0.759 & -0.598 & -0.730 & -0.755 & -0.701 \n Axe 1 & & & & & & & \n& ***U*-stat** & **-4.036** & **-3.903** & **-3.137** & **-3.125** & **-3.882** & **-3.644** \n& valeurs & 0.019 & -0.170 & -0.570 & 0.153 & 0.125 & 0.218 \n Axe 2 & & & & & & & \n& ***U*-stat** & **0.089** & **-0.734** & **-1.914** & **0.734** & **0.569** & **0.906** \n[tab9]\n| c | c || l | l | l | l | l | c | **Variance** & capacity & power & speed & weight & width & length \n& valeurs & 0.962 & 0.923 & 0.886 & 0.756 & 0.801 & 0.795 \n Axe 1 & & & & & & & \n& ***U*-stat** & **11.802** & **11.322** & **10.866** & **9.282** & **9.825** & **9.752** \n& valeurs & -0.126 & -0.352 & -0.338 & 0.575 & -0.111 & 0.504 \n Axe 2 & & & & & & & \n& ***U*-stat** & **-0.307** & **-0.855** & **-0.821** & **1.396** & **-0.269** & **1.223** \n[tab10]\nTables [tab8] and [tab9] (*ν* = 4, 6) show that Axis 2 is correlated to speed (not weight as in the variance PCA). In this respect the absolute contributions must describe the cars associated with speed on Axis 2. Indeed, the Land Rover discovery, a heavy weight car, is no more available on Axis 2 for the Gini PCA for *ν* = 2, 4, 6 (Figures [fig:8], [fig:9], [fig:10]). Note that the red line in the Figures represents the mean share of the information on each axis, *i.e.* 100%/24 cars = 4.16% of information per car.\n.6![Axis 1 (variance) ](CTA1-var.png \"fig:\")\n.6![Axis 2 (variance) ](CTA2-var.png \"fig:\")\n[fig:7]\n.6![Axis 1 (\\nu=2) ](CTA1_nu2.png \"fig:\")\n.6![Axis 2 (\\nu=2) ](CTA2_nu2.png \"fig:\")\n[fig:8]\n.6![Axis 1 (\\nu=4) ](CTA1_nu4.png \"fig:\")\n.6![Axis 2 (\\nu=4) ](CTA2_nu4.png \"fig:\")\n[fig:9]\n.6![Axis 1 (\\nu=6) ](CTA1_nu6.png \"fig:\")\n.6![Axis 2 (\\nu=6) ](CTA2_nu6.png \"fig:\")\n[fig:10]\nFinally, some cars are not correlated with axis 2 in the standard PCA, see Figures [fig:8]–[fig:10], while this is the case in the Gini PCA. 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Schechtman (2011), A Gini-based methodology for identifying and analyzing time series with non-normal innovations. *SSNR Electronic Journal* July: 1–26.\nTibshirani, R. (1996) Regression shrinkage and selection via the LASSO. *Journal of the Royal statistical society, series B*, **58**:1, 267–288.\n Yaari, M.E. (1987), The Dual Theory of Choice Under Risk, *Econometrica*, **55**: 99–115.\n Yaari, M.E. (1988), A Controversial Proposal Concerning Inequality Measurement, *Journal of Economic Theory*, **44**: 381–397.\nYitzhaki, S. (1991), Calculating Jackknife Variance Estimators for Parameters of the Gini Method. *Journal of Business and Economic Statistics*, **9**: 235–239.\nYitzhaki, S. (2003), Gini’s Mean difference: a superior measure of variability for non-normal distributions, *Metron*, **LXI(2)**: 285-316.\nYitzhaki, S. & Olkin, I. (1991). Concentration indices and concentration curves. *Institute of Mathematical Statistics Lecture Notes*, **19**: 380–392.\nYitzhaki, S. & E. Schechtman (2013), *The Gini Methodology. A Primer on a Statistical Methodology*, Springer.\nZou, H., Hastie, T. & R. Tibshirani (2006), Sparse Principal Component Analysis, *Journal of Computational and Graphical Statistics*, **15**:2, 265-286.\nAppendix\n========\n[h]\n|c ||c c c c c c| cars& capacity *x*1 &power *x*2 &speed *x*3 &weight *x*4 &width *x*5 & length *x*6 \nCitroën C2 1.1 Base &1124 &61 &158 &932 &1659 &3666 \nSmart Fortwo Coupé& 698 &52 &135 &730 &1515 &2500 \nMini 1.6 170 &1598 &170 &218 &1215 &1690 &3625 \nNissan Micra 1.2 65 &1240 &65 &154 &965 &1660 &3715 \nRenault Clio 3.0 V6 &2946 &255 &245 &1400 &1810 &3812 \nAudi A3 1.9 TDI &1896 &105 &187 &1295 &1765 &4203 \nPeugeot 307 1.4 HDI 70 &1398 &70 &160 &1179 &1746 &4202 \nPeugeot 407 3.0 V6 BVA &2946 &211 &229 &1640 &1811 &4676 \nMercedes Classe C 270 CDI &2685 &170 &230 &1600 &1728 &4528 \nBMW 530d& 2993 &218 &245 &1595 &1846 &4841 \nJaguar S-Type 2.7 V6 Bi-Turbo& 2720 &207 &230 &1722 &1818 &4905 \nBMW 745i &4398 &333 &250 &1870 &1902 &5029 \nMercedes Classe S 400 CDI &3966 &260 &250 &1915 &2092 &5038 \nCitroën C3 Pluriel 1.6i &1587 &110 &185 &1177 &1700 &3934 \nBMW Z4 2.5i &2494 &192 &235 &1260 &1781 &4091 \nAudi TT 1.8T 180 &1781 &180 &228 &1280 &1764 &4041 \nAston Martin Vanquish &5935 &460 &306 &1835 &1923 &4665 \nBentley Continental GT &5998 &560 &318 &2385 &1918 &4804 \nFerrari Enzo &5998 &660 &350 &1365 &2650 &4700 \nRenault Scenic 1.9 dCi 120 &1870 &120 &188 &1430 &1805 &4259 \nVolkswagen Touran 1.9 TDI 105 &1896 &105 &180 &1498 &1794 &4391 \nLand Rover Defender Td5 &2495 &122 &135 &1695 &1790 &3883 \nLand Rover Discovery Td5 &2495 &138 &157 &2175 &2190 &4705 \nNissan X-Trail 2.2 dCi &2184 &136 &180 &1520 &1765 &4455 \n---\n1. Université de Nîmes – e-mail: stephane.mussard@unimes.fr, Research fellow Grédi University of Sherbrooke, and Liser Luxembourg.[↩](#fnref1)\n2. See for an overview of the “Gini methodology”.[↩](#fnref2)\n3. See for the link between variability (risk) measures and the Gini correlation index.[↩](#fnref3)\n4. In most cases, PCA is performed on scaled (and centered) variables, otherwise variables with large scales might alter interpretations. Thus, it will make sense, later on, to assume that components of ${\\boldsymbol{X}}$ have identical distributions. At least the first two moments will be equal.[foot3][↩](#fnref4)\n5. but other norms have also been considered in statistical literature, such as the Froebenius norm in the Eckart-Young theorem, or the maximum of singular values – also called 2-(induced)-norm.[↩](#fnref5)\n6. In risk analysis *ν* ∈ (0, 1) denotes risk lover decision makers (max-max approach), whereas *ν* > 1 stands for risk averse decision makers, and *ν* → ∞ extreme risk aversion (max-min approach).[↩](#fnref6)\n7. By definition *G**M**D**ν*(**x**⋅ ℓ, **x**⋅ ℓ) ≥ 0 for all ℓ = 1, …, *K*. As we impose that **x**⋅ ℓ ≠ *c***1***N*, the condition becomes *G**M**D**ν*(**x**⋅ ℓ, **x**⋅ ℓ) > 0.[↩](#fnref7)\n8. We have: \n$$\\bar{\\mathbf{f}}\\_{\\cdot k} = 1/N\\sum\\_{i=1}^N f\\_{ik} = 1/N \\left[ \\sum\\_{i=1}^N {\\mathbf{z}}^{\\top}\\_{i\\cdot} \\mathbf{b}\\_{\\cdot k} \\right] = 1/N\\left[ \\sum\\_{i=1}^N {\\mathbf{z}}^{\\top}\\_{i\\cdot} \\right]\\mathbf{b}\\_{\\cdot k} = \\mathbf{0}.$$ \n[↩](#fnref8)\n9. Strictly speaking and suggests probability distortion functions Ψ : [0, 1] → [0, 1], which does not necessarily coincide to our case.[↩](#fnref9)\n10. As indicated by, the efficient Jackknife method may be used to find the variance of any *U*-statistics.[↩](#fnref10)\n11. An R markdown for Gini PCA is available: \nData from Michel Tenenhaus’s website (see also the Appendix): \n[↩](#fnref11)\n\nPrincipal Component Analysis:A Generalized Gini Approach\n========================================================\nA principal component analysis based on the generalized Gini correlation index is proposed (Gini PCA). The Gini PCA generalizes the standard PCA based on the variance. It is shown, in the Gaussian case, that the standard PCA is equivalent to the Gini PCA. It is also proven that the dimensionality reduction based on the generalized Gini correlation matrix, that relies on city-block distances, is robust to outliers. Monte Carlo simulations and an application on cars data (with outliers) show the robustness of the Gini PCA and provide different interpretations of the results compared with the variance PCA.\n**keywords** : Generalized Gini ; PCA ; Robustness.\nIntroduction\n============\nThis late decade, a line of research has been developed and focused on the Gini methodology, see for a general review of different Gini approaches applied in Statistics and in Econometrics.[2](#fn2) Among the Gini tools, the Gini regression has received a large audience since the Gini regression initiated by. Gini regressions have been generalized by in different areas and particularly in time series analysis. and investigated ARMA processes with an identification and an estimation procedure based on Gini autocovariance functions. This robust Gini approach has been shown to be relevant to heavy tailed distributions such as Pareto processes. Also, proposed a unit root test based on Gini regressions to deal with outlying observations in the data.\nIn parallel to the above literature, a second line of research on multidimensional Gini indices arose. This literature paved the way on the valuation of inequality about multiple commodities or dimensions such as education, health, income, etc., that is, to find a real-valued function that quantifies the inequality between the households of a population over each dimension, see among others,,,. More recently, shows that it is possible to construct multidimensional Gini indices by exploring the projection of the data in reduced subspaces based on the Euclidean norm. Accordingly, some notions of linear algebra have been increasingly included in the axiomatization of multidimensional Gini indices.\nIn this paper, in the same vein as in the second line of research mentioned above, we start from the recognition that linear algebra may be closely related to the maximum level of inequality that arises in a given dimension. In data analysis, the variance maximization is mainly used to further analyze projected data in reduced subspaces. The variance criterion implies many problems since it captures a very precise notion of dispersion, which does not always match some basic properties satisfied by variability measures such as the Gini index. Such a property may be, for example, an invariance condition postulating that a dispersion measure remains constant when the data are transformed by monotonic maps.[3](#fn3) Another property typically related to the Gini index is its robustness to outlying observations, see e.g. in the case of linear regressions. Accordingly, it seems natural to analyze multidimensional dispersion with the Gini index, instead of the variance, in order to provide a Principal Components Analysis (PCA) in a Gini sense (Gini PCA).\nIn the field of PCA, and are among the first authors dealing with a ℓ1-norm PCA framework. Their idea was to robustify the standard PCA by means of the Gini Mean Difference metric introduced by, which is a city-block distance measure of variability. The authors employ the Gini Mean Difference as an estimator of the standard deviation of each variable before running the singular value decomposition leading to a robust PCA. In the same vein, make use of a rotational ℓ1 norm PCA to robustify the variance-covariance matrix in such a way that the PCA is rotational invariant. Recent PCAs derive latent variables thanks to regressions based on *elastic net* (a ℓ1 regularization) that improves the quality of the regression curve estimation, see.\nIn this paper, it is shown that the variance may be seen as an inappropriate criterion for dimensionality reduction in the case of data contamination or outlying observations. A generalized Gini PCA is investigated by means of Gini correlations matrices. These matrices contain generalized Gini correlation coefficients (see ) based on the Gini covariance operator introduced by and. The generalized Gini correlation coefficients are: (i) bounded, (ii) invariant to monotonic transformations, (iii) and symmetric whenever the variables are exchangeable. It is shown that the standard PCA is equivalent to the Gini PCA when the variables are Gaussians. Also, it is shown that the generalized Gini PCA may be realized either in the space of the variables or in the space of the observations. In each case, some statistics are proposed to perform some interpretations of the variables and of the observations (absolute and relative contributions). To be precise, an *U*-statistics test is introduced to test for the significance of the correlations between the axes of the new subspace and the variables in order to assess their significance. Monte Carlo simulations are performed in order to show the superiority of the Gini PCA compared with the usual PCA when outlying observations contaminate the data. Finally, with the aid of the well-known cars data, which contain outliers, it is shown that the generalized Gini PCA leads to different results compared with the usual PCA.\nThe outline of the paper is as follows. Section [s2] sets the notations and presents some ℓ2 norm approaches of PCA. Section [s3] reviews the Gini-covariance operator. Section [s4] is devoted to the generalized Gini PCA. Section [s5] focuses on the interpretation of the Gini PCA. Sections [s6] and [s7] present some Monte Carlo simulations and applications, respectively.\nMotivations for the use of Gini PCA\n===================================\nIn this Section, the notations are set. Then, some assumptions are imposed and some ℓ2-norm PCA techniques are reviewed in order to motivate the employ of the Gini PCA.\nNotations and definitions\n-------------------------\nLet N\\* be the set of integers and R [R+ +] the set of [positive] real numbers. Let M be the set of all *N* × *K* matrix ${\\boldsymbol{X}}=[x\\_{ik}]$ that describes *N* observations on *K* dimensions such that *N* ≫ *K*, with elements *x**i**k* ∈ R, and I*n* the *n* × *n* identity matrix. The *N* × 1 vectors representing each variable are expressed as **x**⋅ *k*, for all *k* ∈ {1, …, *K*} and we assume that **x**⋅ *k* ≠ *c***1***N*, with *c* a real constant and **1***N* a *N*-dimensional column vector of ones. The *K* × 1 vectors representing each observation *i* (the transposed *i*th line of ${\\boldsymbol{X}}$) are expressed as **x***i* ⋅, for all *i* ∈ {1, …, *N*}. It is assumed that **x**⋅ *k* is the realization of the random variable *X**k*, with cumulative distribution function *F**k*. The arithmetic mean of each column (line) of the matrix ${\\boldsymbol{X}}$ is given by $\\bar{{\\mathbf{x}}}\\_{\\cdot k}$ ($\\bar{{\\mathbf{x}}}\\_{i\\cdot}$). The cardinal of set *A* is denoted #{*A*}. The ℓ1 norm, for any given real vector **x**, is $\\| \\mathbf x \\|\\_1 = \\sum\\_{k=1}^K|x\\_{\\cdot k}|$, whereas the ℓ2 norm is ∥**x**∥2 = (∑*k* = 1*K**x*⋅ *k*2)1/2.\nThe random variables *X**k* are such that $\\mathbb E[|X\\_k|] < \\infty$ for all *k* ∈ {1, …, *K*}, but no assumption is made on the second moments (that may not exist).\nThis assumption imposes less structure compared with the classical PCA in which the existence of the second moments are necessary, as can be seen in the next subsection.\nVariants of PCA based on the ℓ2 norm\n------------------------------------\nThe classical formulation of the PCA, to obtain the first component, can be obtained by solving\n \n$$\\label{eq:PCA0}\n\\omega\\_1^\\*\\in\\text{argmax}\\left\\lbrace\\text{Var}[{\\boldsymbol{X}}\\omega]\\right\\rbrace\n\\text{ subject to }\\|\\omega\\|\\_2^2=\\omega^{\\top}\\omega=1,$$ \nor equivalently\n \n$$\\label{eq:PCA}\n\\omega\\_1^\\*\\in\\text{argmax}\\left\\lbrace\\omega^{\\top}\\boldsymbol{\\Sigma}\\omega\\right\\rbrace\n\\text{ subject to }\\|\\omega\\|\\_2^2=\\omega^{\\top}\\omega=1,$$ \nwhere *ω* ∈ R*K*, and $\\boldsymbol{\\Sigma}$ is the (symmetric positive semi-definite) *K* × *K* sample covariance matrix. suggest to write \n$$\\omega\\_1^\\*\\in\\text{argmax}\\left\\lbrace\\sum\\_{j=1}^K\\text{Var}[{\\mathbf{x}}\\_{\\cdot, j}]\\cdot\\text{Cor}[{\\mathbf{x}}\\_{\\cdot,j},{\\boldsymbol{X}}\\omega]\\right\\rbrace\n\\text{ subject to }\\|\\omega\\|\\_2^2=\\omega^{\\top}\\omega=1.$$ \n With scaled variables[4](#fn4) (i.e. Var[**x**⋅ , *j*] = 1, ∀*j*)\n \n$$\\label{eq:PCA:norm}\n\\omega\\_1^\\*\\in\\text{argmax}\\left\\lbrace\\sum\\_{j=1}^K \\text{Cor}[{\\mathbf{x}}\\_{\\cdot,j},{\\boldsymbol{X}}\\omega]\\right\\rbrace\n\\text{ subject to }\\|\\omega\\|\\_2^2=\\omega^{\\top}\\omega=1.$$ \nThen a Principal Component Pursuit can start: we consider the ‘residuals’, ${\\boldsymbol{X}}\\_{(1)}={\\boldsymbol{X}}-{\\boldsymbol{X}}\\omega\\_1^\\*\\omega\\_1^{\\*\\top}$, its covariance matrix $\\boldsymbol{\\Sigma}\\_{(1)}$, and we solve \n$$\\omega\\_2^\\*\\in\\text{argmax}\\left\\lbrace\\omega^{\\top}\\boldsymbol{\\Sigma}\\_{(1)}\\omega\\right\\rbrace\n\\text{ subject to }\\|\\omega\\|\\_2^2=\\omega^{\\top}\\omega=1.$$ \n The part ${\\boldsymbol{X}}\\omega\\_1^\\*\\omega\\_1^{\\*\\top}$ is actually a constraint that we add to ensure the orthogonality of the two first components. This problem is equivalent to finding the maxima of $\\text{Var}[{\\boldsymbol{X}}\\omega]$ subject to ∥*ω*∥22 = 1 and *ω* ⊥ *ω*1\\*. This idea is also called Hotelling (or Wielandt) deflation technique. On the *k*-th iteration, we extract the leading eigenvector \n$$\\omega\\_k^\\*\\in\\text{argmax}\\left\\lbrace\\omega^{\\top}\\boldsymbol{\\Sigma}\\_{(k-1)}\\omega\\right\\rbrace\n\\text{ subject to }\\|\\omega\\|\\_2^2=\\omega^{\\top}\\omega=1,$$ \n where $\\boldsymbol{\\Sigma}\\_{(k-1)}=\\boldsymbol{\\Sigma}\\_{(k-2)}-\\omega\\_{k-1}^{\\*}\\omega\\_{k-1}^{\\*\\top}\\boldsymbol{\\Sigma}\\_{(k-1)}\\omega\\_{k-1}^{\\*}\\omega\\_{k-1}^{\\*\\top}$ (see e.g. ). Note that, following and, that it is also possible to write this problem as \n$$\\min\\left\\lbrace\\|{\\boldsymbol{X}}-\\tilde{{\\boldsymbol{X}}}\\|\\_\\star\\right\\rbrace\n\\text{ subject to }\\text{rank}[\\tilde{{\\boldsymbol{X}}}]\\leq k$$ \n where ∥ ⋅ ∥⋆ denotes the nuclear norm of a matrix (*i.e.* the sum of its singular values)[5](#fn5).\nOne extension, introduced in, was to add a constraint based on the cardinality of *ω* (also called ℓ0 norm) corresponding to the number of non-zero coefficients of *ω*. The penalized objective function is then \n$$\\max\\left\\lbrace\\omega^{\\top}\\boldsymbol{\\Sigma}\\omega-\\lambda\\text{card}[\\omega]\\right\\rbrace\n\\text{ subject to }\\|\\omega\\|\\_2^2=\\omega^{\\top}\\omega=1,$$ \n for some *λ* > 0. This is called *sparse PCA*, and can be related to sparse regression, introduced in. But as pointed out in, interpretation is not easy and the components obtained are not orthogonal. considered an extension to nonlinear Principal Manifolds to take into account nonlinearities.\nAnother direction for extensions was to consider Robust Principal Component Analysis. suggested an approach based on the fact that principal component pursuit can be obtained by solving \n$$\\min\\left\\lbrace\\|{\\boldsymbol{X}}-\\tilde{{\\boldsymbol{X}}}\\|\\_\\star+\\lambda\\|\\tilde{{\\boldsymbol{X}}}\\|\\_1\\right\\rbrace.$$ \n But other methods were also considered to obtain Robust PCA. A natural ‘scale-free’ version is obtained by considering a rank matrix instead of ${\\boldsymbol{X}}$. This is also called ‘ordinal’ PCA in the literature, see. The first ‘ordinal’ component is\n \n$$\\omega\\_1^\\*\\in\\text{argmax}\\left\\lbrace\\sum\\_{j=1}^K\\mathcal{R}[{\\mathbf{x}}\\_{\\cdot,j},{\\boldsymbol{X}}\\omega]\\right\\rbrace\n\\text{ subject to }\\|\\omega\\|\\_2^2=\\omega^{\\top}\\omega=1$$ \nwhere R denotes some rank based correlation, *e.g.* Spearman’s rank correlation, as an extention of Equation ([eq:PCA:norm]). So, quite naturally, one possible extension of Equation ([eq:PCA]) would be \n$$\\omega\\_1^\\*\\in\\text{argmax}\\left\\lbrace\\omega^{\\top}\\mathcal{R}[{\\boldsymbol{X}}]\\omega\\right\\rbrace\n\\text{ subject to }\\|\\omega\\|\\_2^2=\\omega^{\\top}\\omega=1$$ \n where $\\mathcal{R}[{\\boldsymbol{X}}]$ denotes Spearman’s rank correlation. In this section, instead of using Pearson’s correlation (as in Equation ([eq:PCA]) when the variables are scaled) or Spearman’s (as in this ordinal PCA), we will consider the multidimensional Gini correlation based on the *h*-covariance operator.\nGeometry of Gini PCA: Gini-Covariance Operators\n===============================================\nThe first PCA was introduced by, projecting $\\boldsymbol{X}$ onto the eigenvectors of its covariance matrix, and observing that the variances of those projections are the corresponding eigenvalues. One of the key property is that ${\\boldsymbol{X}}^\\top{\\boldsymbol{X}}$ is a positive matrix. Most statistical properties of PCAs (see or ) are obtained under Gaussian assumptions. Furthermore, geometric properties can be obtained using the fact that the covariance defines an inner product on the subspace of random variables with finite second moment (up to a translation, *i.e.* we identify any two that differ by a constant).\nWe will discuss in this section the properties of the Gini Covariance operator with the special case of Gaussian random variables, and the property of the Gini correlation matrix that will be used in the next Section for the Gini PCA.\nThe Gini-covariance operator\n----------------------------\nIn this section, $\\boldsymbol{X}=(X\\_1,\\cdots,X\\_K)$ denotes a random vector. The covariance matrix between $\\boldsymbol{X}$ and $\\boldsymbol{Y}$, two random vectors, is defined as the inner product between centered versions of the vectors,\n \n$$\\label{eq:innerprod}\n\\langle \\boldsymbol{X},\\boldsymbol{Y}\\rangle=\\text{Cov}(\\boldsymbol{X},\\boldsymbol{Y})=\\mathbb{E}[(\\boldsymbol{X}-\\mathbb{E}[\\boldsymbol{X}])(\\boldsymbol{Y}-\\mathbb{E}[\\boldsymbol{Y}])^{\\text{\\sffamily T}}].$$ \nHence, it is the matrix where elements are regular covariances between components of the vectors, $\\text{Cov}(\\boldsymbol{X},\\boldsymbol{Y})=[\\text{Cov}(X\\_i,Y\\_j)]$. It is the upper-right block of the covariance matrix of $(\\boldsymbol{X},\\boldsymbol{Y})$. Note that $\\text{Cov}(\\boldsymbol{X},\\boldsymbol{X})$ is the standard variance-covariance matrix of vector $\\boldsymbol{X}$.\nLet $\\boldsymbol{X}=(X\\_1,\\cdots,X\\_{K})$ be collections of *K* identically distributed random variables. Let *h* : R → R denote a non-decreasing function. Let $h(\\boldsymbol{X})$ denote the random vector (*h*(*X*1), ⋯, *h*(*X**K*)), and assume that each component has a finite variance. Then, operator $\\Gamma C\\_{h}(\\boldsymbol{X})=\\emph{Cov}(\\boldsymbol{X},h(\\boldsymbol{X}))$ is called *h*-Gini covariance matrix.\nSince *h* is a non-decreasing mapping, then $\\boldsymbol{X}$ and $h(\\boldsymbol{X})$ are componentwise comonotonic random vectors. Assuming that components of $\\boldsymbol{X}$ are identically distributed is a reasonable assumption in the context of scaled (and centered) PCA, as discussed in footnote [foot3]. Nevertheless, a stronger technical assumption will be necessary: pairwise-exchangeability.\n$\\boldsymbol{X}$ is said to be pairwise-exchangeable if for all pair (*i*, *j*) ∈ {1, ⋯, *K*}2, (*X**i*, *X**j*) is exchangeable, in the sense that $(X\\_i,X\\_j)\\overset{\\mathcal{L}}{=}(X\\_j,X\\_i)$.\nPairwise-exchangeability is a stronger concept than having only one vector with identically distributed components, and a weaker concept than (full) exchangeability. In the Gaussian case where *h*(*X**k*) = Φ(*X**k*) with Φ(*X**k*) being the normal cdf of *X**k* for all *k* = 1, …, *K*, pairwise-exchangeability is equivalent to components identically distributed.\nIf $\\boldsymbol{X}$ is a Gaussian vector with identically distributed components, then $\\boldsymbol{X}$ is pairwise-exchangeable.\nFor simplicity, assume that components of $\\boldsymbol{X}$ are N(0, 1) random variables, then $\\boldsymbol{X}\\sim\\mathcal{N}(\\boldsymbol{0},\\boldsymbol{\\rho})$ where $\\boldsymbol{\\rho}$ is a correlation matrix. In that case \n$$\\left[\n\\begin{matrix}\nX\\_i \\\\\nX\\_j\n\\end{matrix}\n\\right]\\sim\\mathcal{N}\\left(\n\\left[\n\\begin{matrix}\n0 \\\\\n0\n\\end{matrix}\n\\right],\n\\left[\n\\begin{matrix}\n1 & \\rho\\_{ij} \\\\\n\\rho\\_{ji} & 1 \\\\\n\\end{matrix}\n\\right]\n\\right),$$ \n with Pearson correlation *ρ**i**j* = *ρ**j**i*, thus (*X**i*, *X**j*) is exchangeable.\nLet us now introduce the Gini-covariance. introduced the Gini mean difference operator Δ, defined as:\n \n$$\\label{eq:gmd}\n\\Delta(X)=\\mathbb{E}(|X\\_1-X\\_2|)\\text{ where }X\\_1,X\\_2\\sim X,\\text{ and }X\\_1{\\protect\\mathpalette{\\protect\\independenT}{\\perp}}X\\_2,$$ \nfor some random variable *X* (or more specifically for some distribution *F* with *X* ∼ *F*, because this operator is law invariant). One can rewrite: \n$$\\Delta(X)=4\\text{Cov}[X,F(X)]=\\frac{1}{3}\\frac{\\text{Cov}[X,F(X)]}{\\text{Cov}[F(X),F(X)]}$$ \n where the term on the right is interpreted as the slope of the regression curve of the observed variable *X* and its ‘ranks’ (up to a scaling coefficient). Thus, the Gini-covariance is obtained when the function *h* is equal to the cumulative distribution function of the second term, see.\nLet $\\boldsymbol{X}=(X\\_1,\\cdots,X\\_{K})$ be a collection of *K* identically distributed random variables, with cumulative distribution function *F*. Then, the Gini covariance is $\\Gamma C\\_F(\\boldsymbol{X})=\\emph{\\text{Cov}}(\\boldsymbol{X},F(\\boldsymbol{X}))$.\nOn this basis, it is possible to show that the Gini covariance matrix is a positive semi-definite matrix.\nLet *Z* ∼ N(0, 1). If $\\boldsymbol{X}$ represents identically distributed Gaussian random variables, with distribution N(*μ*, *σ*2), then the two following assertions hold:\n*(i)* $\\Gamma C\\_F(\\boldsymbol{X})=\\sigma^{-1} \\emph{Cov}(Z,\\Phi(Z)) \\emph{Var}(\\boldsymbol{X}).$\n*(ii)* $\\Gamma C\\_F(\\boldsymbol{X})$ is a positive-semi definite matrix.\n(i) In the Gaussian case, if *h* is the cumulative distribution function of the *X**k*’s, then Cov(*X**k*, *h*(*X*ℓ)) = *r**σ* ⋅ Cov(*Z*, Φ(*Z*)), where Φ is the normal cdf, see, Chapter 3. Observe that Cov(*X**k*, *h*(*X**k*)) = *σ* ⋅ Cov(*Z*, Φ(*Z*)), if *h* is the cdf of *X**k*. Thus, *λ* : = Cov(*Z*, Φ(*Z*)) yields: \n$$\\Gamma C\\_F((X\\_k,X\\_\\ell))=\\lambda \\left[\n\\begin{matrix}\n\\sigma & \\rho \\sigma \\\\\n\\rho\\sigma & \\sigma \\\\\n\\end{matrix}\n\\right]=\n\\lambda \\sigma \\left[\n\\begin{matrix}\n1 & \\rho \\\\\n\\rho & 1\\\\\n\\end{matrix}\n\\right]=\\frac{\\lambda}{\\sigma}\n\\text{Var}((X\\_k,X\\_\\ell)).$$ \n(ii) We have Cov(*Z*, Φ(*Z*)) ≥ 0, then it follows that *C**F*((*X**k*, *X*ℓ)) ≥ 0: \n$${\\mathbf{x}}^{\\top}C\\_F(\\boldsymbol{X}) {\\mathbf{x}}= {\\mathbf{x}}^{\\top}\\frac{{\\text{Cov}}(Z,\\Phi(Z))}{\\sigma}\\text{Var}(\\boldsymbol{X}) {\\mathbf{x}}\\geq 0,$$ \n which ends the proof.\nNote that $\\Gamma C\\_F(\\boldsymbol{X})=\\text{Cov}(\\boldsymbol{X},-\\overline{F}(\\boldsymbol{X}))=\\Gamma C\\_{-\\overline{F}}(\\boldsymbol{X})$, where $\\overline{F}$ denotes the survival distribution function.\nLet $\\boldsymbol{X}=(X\\_1,\\cdots,X\\_{K})$ be a collection of *K* identically distributed random variables, with survival distribution function $\\overline{F}$. Then, the generalized Gini covariance is $G\\Gamma C\\_{\\nu}(\\boldsymbol{X})=\\Gamma C\\_{-\\overline{F}^{\\nu-1}}(\\boldsymbol{X})=\\text{\\em Cov}(\\boldsymbol{X},-\\overline{F}^{\\nu-1}(\\boldsymbol{X}))$, for *ν* > 1.\nThis operator is related to the one introduced in, called generalized Gini mean difference *G**M**D**ν* operator. More precisely, an estimator of the generalized Gini mean difference is given by: \n$$GMD\\_\\nu({\\mathbf{x}}\\_{\\cdot\\ell},{\\mathbf{x}}\\_{\\cdot k}) := -\\frac{2}{N-1}\\nu {\\text{Cov}}({\\mathbf{x}}\\_{\\cdot\\ell},{\\textsc{\\textbf{r}}}\\_{{\\mathbf{x}}\\_{\\cdot k}}^{\\nu-1}), \\ \\nu > 1,$$ \n where ${\\textsc{\\textbf{r}}}\\_{{\\mathbf{x}}\\_{\\cdot k}} = (R(x\\_{1k}),\\ldots,R(x\\_{nk}))$ is the decumulative rank vector of **x**⋅ *k*, that is, the vector that assigns the smallest value (1) to the greatest observation *x**i**k*, and so on. The rank of observation *i* with respect to variable *k* is: \n$$R(x\\_{ik}) :=\n\\left\\{ \\begin{array}{ll}\nN+1- \\#\\{x \\leq x\\_{ik} \\} & \\text{if no ties} \\\\\nN+1-\\frac{1}{p}\\sum\\_{i=1}^p \\#\\{ x \\leq x\\_{ik} \\} & \\text{if $p$ ties $x\\_{ik}$.}\n\\end{array}\n\\right.$$ \n Hence *G**M**D**ν*(**x**⋅ ℓ, **x**⋅ *k*) is the empirical version of \n$$2\\nu \\Gamma C\\_\\nu(X\\_{\\ell},X\\_{ k}) := -2\\nu{\\text{Cov}}\\big(X\\_\\ell,\\overline{F}\\_k(X\\_k)^{\\nu-1}\\big).$$ \n The index *G**M**D**ν* is a generalized version of the *G**M**D*2 proposed earlier by, and can also be written as: \n$$GMD\\_2(X\\_{k},X\\_{k}) = 4{\\text{Cov}}\\big(X\\_k,F\\_k(X\\_k)\\big) = \\Delta(X\\_k).$$ \n When *k* = ℓ, *G**M**D**ν* represents the variability of the variable **x**⋅ *k* itself. Focus is put on the lower tail of the distribution **x**⋅ *k* whenever *ν* → ∞, the approach is said to be max-min in the sense that *G**M**D**ν* inflates the minimum value of the distribution. On the contrary, whenever *ν* → 0, the approach is said to be max-max, in this case focus is put on the upper tail of the distribution **x**⋅ *k*. As mentioned in, the case *ν* < 1 does not entail simple interpretations, thereby the parameter *ν* is used to be set as *ν* > 1 in empirical applications.[6](#fn6)\nNote that even if *X**k* and *X*ℓ have the same distribution, we might have *G**M**D**ν*(*X**k*, *X*ℓ) ≠ *G**M**D**ν*(*X*ℓ, *X**k*), as shown on the example of Figure [Fig:dessin-echangeable]. In that case E[*X**k**h*(*X*ℓ)] ≠ E[*X*ℓ*h*(*X**k*)] if *h*(2) ≠ 2*h*(1) (this property is nevertheless valid if *h* is linear). We would have *G**M**D**ν*(*X**k*, *X*ℓ) = *G**M**D**ν*(*X*ℓ, *X**k*) when *X**k* and *X*ℓ are exchangeable. But since generally *G**M**D**ν* is not symmetric, we have for **x**⋅ *k* being not a monotonic transformation of **x**⋅ ℓ and *ν* > 1, *G**M**D**ν*(**x**⋅ *k*, **x**⋅ ℓ) ≠ *G**M**D**ν*(**x**⋅ ℓ, **x**⋅ *k*).\n![Joint distribution of a random pair (X_k,X_{\\ell}) such that \\mathbb{E}[X_k h(X_{\\ell})]\\neq \\mathbb{E}[X_{\\ell} h(X_k)], with non-exchangeable components X_k\\overset{\\mathcal{L}}{=}X_{\\ell}.](exchange.png \"fig:\")[Fig:dessin-echangeable]\nGeneralized Gini correlation\n----------------------------\nIn this section, ${\\boldsymbol{X}}$ is a matrix in M. The Gini correlation coefficient (*G*-correlation frown now on), is a normalized *G**M**D**ν* index such that for all *ν* > 1, see, \n$$GC\\_\\nu({\\mathbf{x}}\\_{\\cdot\\ell},{\\mathbf{x}}\\_{\\cdot k}) := \\frac{GMD\\_\\nu({\\mathbf{x}}\\_{\\cdot\\ell},{\\mathbf{x}}\\_{\\cdot k})}{GMD\\_\\nu({\\mathbf{x}}\\_{\\cdot\\ell},{\\mathbf{x}}\\_{\\cdot\\ell})} \\ ; \\ GC\\_\\nu({\\mathbf{x}}\\_{\\cdot k},{\\mathbf{x}}\\_{\\cdot\\ell}) := \\frac{GMD\\_\\nu({\\mathbf{x}}\\_{\\cdot k},{\\mathbf{x}}\\_{\\cdot\\ell})}{GMD\\_\\nu({\\mathbf{x}}\\_{\\cdot k},{\\mathbf{x}}\\_{\\cdot k})},$$ \n with *G**C**ν*(**x**⋅ *k*, **x**⋅ *k*) = 1 and *G**M**D**ν*(**x**⋅ *k*, **x**⋅ *k*) ≠ 0, for all *k*, ℓ = 1, …, *K*. Following, the *G*-correlation is well-suited for the measurement of correlations between non-normal distributions or in the presence of outlying observations in the sample.\n[prop1b] – ***Schechtman and Yitzhaki (2013):***\n*(i)* *G**C**ν*(**x**⋅ ℓ, **x**⋅ *k*) ≤ 1.\n*(ii)* If the variables **x**⋅ ℓ and **x**⋅ *k* are independent, for all *k* ≠ ℓ, then *G**C**ν*(**x**⋅ ℓ, **x**⋅ *k*) = *G**C**ν*(**x**⋅ *k*, **x**⋅ ℓ) = 0.\n*(iii)* For any given monotonic increasing transformation *φ*, *G**C**ν*(**x**⋅ ℓ, *φ*(**x**⋅ *k*)) = *G**C**ν*(**x**⋅ ℓ, **x**⋅ *k*).\n*(iv)* If (**x**⋅ ℓ, **x**⋅ *k*) have a bivariate normal distribution with Pearson correlation *ρ*, then *G**C**ν*(**x**⋅ ℓ, **x**⋅ *k*) = *G**C**ν*(**x**⋅ *k*, **x**⋅ ℓ) = *ρ*.\n*(v)* If **x**⋅ *k* and **x**⋅ ℓ are exchangeable up to a linear transformation, then *G**C**ν*(**x**⋅ ℓ, **x**⋅ *k*) = *G**C**ν*(**x**⋅ *k*, **x**⋅ ℓ).\nWhenever *ν* → 1, the variability of the variables is attenuated so that *G**M**D**ν* tends to zero (even if the variables exhibit a strong variance). The choice of *ν* is interesting to perform generalized Gini PCA with various values of *ν* in order to robustify the results of the PCA, since the standard PCA (based on the variance) is potentially of bad quality if outlying observations drastically affect the sample.\nA *G*-correlation matrix is proposed to analyze the data into a new vector space. Following Property [prop1b] (iv), it is possible to rescale the variables **x**⋅ ℓ thanks to a linear transformation, then the matrix of standardized observation is,\n \n$$\\label{standard}\n{\\boldsymbol{Z}}\\equiv [z\\_{i\\ell}] := \\left[ \\frac{x\\_{i\\ell} - \\bar{{\\mathbf{x}}}\\_{\\cdot \\ell}}{GMD\\_\\nu({\\mathbf{x}}\\_{\\cdot\\ell},{\\mathbf{x}}\\_{\\cdot\\ell})}\\right].$$ \nThe variable *z**i*ℓ is a real number without dimension. The variables **x**⋅ *k* are rescaled such that their Gini variability is equal to unity. Now, we define the *N* × *K* matrix of decumulative centered rank vectors of ${\\boldsymbol{Z}}$, which are the same compared with those of ${\\boldsymbol{X}}$: \n$${\\boldsymbol{R}}\\_{\\mathbf{z}}^c \\equiv [R^c(z\\_{i\\ell})] := [R(z\\_{i\\ell})^{\\nu-1} - \\bar{{\\textsc{\\textbf{r}}}}^{\\nu-1}\\_{{\\mathbf{z}}\\_{\\cdot\\ell}}] = [R(x\\_{i\\ell})^{\\nu-1} - \\bar{{\\textsc{\\textbf{r}}}}\\_{{\\mathbf{x}}\\_{\\cdot\\ell}}^{\\nu-1}].$$ \n Note that the last equality holds since the standardization ([standard]) is a strictly increasing affine transformation.[7](#fn7) The *K* × *K* matrix containing all *G*-correlation indices between all couples of variables **z**⋅ *k* and **z**⋅ ℓ, for all *k*, ℓ = 1, …, *K* is expressed as:\n \n$$\\notag\nGC\\_\\nu({\\boldsymbol{Z}}) := -\\frac{2\\nu}{N(N-1)}{\\boldsymbol{Z}}^{\\top}{\\boldsymbol{R}}\\_{\\mathbf{z}}^c.$$ \nIndeed, if $GMD\\_\\nu({\\boldsymbol{Z}}) \\equiv [GMD\\_\\nu({\\mathbf{z}}\\_{\\cdot k},{\\mathbf{z}}\\_{\\cdot \\ell})]$, then we get the following.\n[prop1] For each standardized matrix ${\\boldsymbol{Z}}$ defined in *([standard])*, the following relations hold:\n \n$$\\begin{aligned}\n\\label{GMD=GC}\n& GMD\\_\\nu({\\boldsymbol{Z}}) = GC\\_\\nu({\\boldsymbol{X}}) = GC\\_\\nu({\\boldsymbol{Z}}).\\\\\n& GMD\\_\\nu({\\mathbf{z}}\\_{\\cdot k},{\\mathbf{z}}\\_{\\cdot k}) = 1, \\ \\forall k=1,\\ldots,K.\\label{GMD=1}\\end{aligned}$$ \nWe have $GMD\\_\\nu({\\boldsymbol{Z}}) \\equiv [GMD\\_\\nu({\\mathbf{z}}\\_{\\cdot k},{\\mathbf{z}}\\_{\\cdot \\ell})]$ being a *K* × *K* matrix. The extra diagonal terms may be rewritten as,\n \n$$\\begin{aligned}\nGMD\\_\\nu({\\mathbf{z}}\\_{\\cdot k},{\\mathbf{z}}\\_{\\cdot \\ell}) &= -\\frac{2}{N-1}\\nu {\\text{Cov}}({\\mathbf{z}}\\_{\\cdot k},{\\textsc{\\textbf{r}}}\\_{{\\mathbf{z}}\\_{\\cdot \\ell}}^{\\nu-1}) \\notag \\\\\n&= -\\frac{2}{N-1}\\nu {\\text{Cov}}\\left( \\frac{{\\mathbf{x}}\\_{\\cdot k} - \\bar{{\\mathbf{x}}}\\_{\\cdot k}\\mathbf{1}\\_N }{GMD\\_\\nu({\\mathbf{x}}\\_{\\cdot k},{\\mathbf{x}}\\_{\\cdot k})},{\\textsc{\\textbf{r}}}\\_{{\\mathbf{z}}\\_{\\cdot \\ell}}^{\\nu-1} \\right) \\notag \\\\\n& = -\\frac{2}{GMD\\_\\nu({\\mathbf{x}}\\_{\\cdot k},{\\mathbf{x}}\\_{\\cdot k})}\\left[ \\frac{\\nu {\\text{Cov}}( {\\mathbf{x}}\\_{\\cdot k}, {\\textsc{\\textbf{r}}}\\_{{\\mathbf{z}}\\_{\\cdot \\ell}}^{\\nu-1})}{N-1} - \\frac{\\nu {\\text{Cov}}( \\bar{{\\mathbf{x}}}\\_{\\cdot k}\\mathbf{1}\\_N,{\\textsc{\\textbf{r}}}\\_{{\\mathbf{z}}\\_{\\cdot \\ell}}^{\\nu-1})}{N-1}\\right] \\notag \\\\\n&= \\frac{GMD\\_\\nu({\\mathbf{x}}\\_{\\cdot k},{\\mathbf{x}}\\_{\\cdot \\ell})}{GMD\\_\\nu({\\mathbf{x}}\\_{\\cdot k},{\\mathbf{x}}\\_{\\cdot k})} = GC\\_\\nu({\\mathbf{x}}\\_{\\cdot k},{\\mathbf{x}}\\_{\\cdot \\ell}).\\notag\\end{aligned}$$ \nFinally, using the same approach as before, we get:\n \n$$\\begin{aligned}\nGMD\\_\\nu({\\mathbf{z}}\\_{\\cdot k},{\\mathbf{z}}\\_{\\cdot k}) &= -\\frac{2}{N-1} \\nu{\\text{Cov}}({\\mathbf{z}}\\_{\\cdot k},{\\textsc{\\textbf{r}}}\\_{{\\mathbf{z}}\\_{\\cdot k}}^{\\nu-1}) \\notag \\\\\n&= \\frac{GMD\\_\\nu({\\mathbf{x}}\\_{\\cdot k},{\\mathbf{x}}\\_{\\cdot k})}{GMD\\_\\nu({\\mathbf{x}}\\_{\\cdot k},{\\mathbf{x}}\\_{\\cdot k})} \\notag \\\\ \n&= GC\\_\\nu({\\mathbf{x}}\\_{\\cdot k},{\\mathbf{x}}\\_{\\cdot k})=1.\\notag\\end{aligned}$$ \nBy Property [prop1] (iv), since ${\\textsc{\\textbf{r}}}\\_{{\\mathbf{x}}\\_{\\cdot k}} = {\\textsc{\\textbf{r}}}\\_{{\\mathbf{z}}\\_{\\cdot k}}$, then *G**C**ν*(**x**⋅ *k*, **x**⋅ ℓ) = *G**C**ν*(**z**⋅ *k*, **z**⋅ ℓ). Thus, \n$$GMD\\_\\nu({\\boldsymbol{Z}}) = -\\frac{2\\nu}{N(N-1)} {\\boldsymbol{Z}}^{\\top}{\\boldsymbol{R}}\\_{\\mathbf{z}}^c = GC\\_\\nu({\\boldsymbol{X}}) = GC\\_\\nu({\\boldsymbol{Z}}),$$ \n which ends the proof.\nFinally, under a normality assumption, the generalized Gini covariance matrix $GC\\_\\nu(\\boldsymbol{X}) \\equiv [GMD\\_\\nu(X\\_{ k},X\\_{\\ell})]$ is shown to be a positive semi-definite matrix.\n[th2] Let *Z* ∼ N(0, 1). If $\\boldsymbol{X}$ represents identically distributed Gaussian random variables, with distribution N(*μ*, *σ*2), then the two following assertions holds:\n*(i)* $GC\\_\\nu(\\boldsymbol{X})= \\sigma^{-1}\\emph{Cov}(Z,\\Phi(Z)) \\emph{Var}(\\boldsymbol{X}).$\n*(ii)* $GC\\_\\nu(\\boldsymbol{X})$ is a positive semi-definite matrix.\nThe first part (i) follows from, Chapter 6. The second part follows directly from (i).\nTheorem [th2] shows that under the normality assumption, the variance is a special case of the Gini methodology. As a consequence, for multivariate normal distributions, it is shown in Section [s4] that Gini PCAs and classical PCA (based on the ℓ2 norm and the covariance matrix) are equivalent.\nGeneralized Gini PCA\n====================\nIn this section, the multidimensional Gini variability of the observations *i* = 1, …, *N*, embodied by the matrix $GC\\_\\nu({\\boldsymbol{Z}})$, is maximized in the R*K*-Euclidean space, *i.e.*, in the set of variables {**z**⋅ 1, …, **z**⋅ *K*}. This allows the observations to be projected onto the new vector space spanned by the eigenvectors of $GC\\_\\nu({\\boldsymbol{Z}})$. Then, the projection of the variables is investigated in the R*N*-Euclidean space induced by $GC\\_\\nu({\\boldsymbol{Z}})$. Both observations and variables are analyzed through the prism of *absolute* and *relative* contributions to propose relevant interpretations of the data in each subspace.\nThe R*K*-Euclidean space\n------------------------\nIt is possible to investigate the projection of the data ${\\boldsymbol{Z}}$ onto the new vector space induced by $GMD\\_\\nu({\\boldsymbol{Z}})$ or alternatively by $GC\\_\\nu({\\boldsymbol{Z}})$ since $GMD\\_\\nu({\\boldsymbol{Z}}) = GC\\_\\nu({\\boldsymbol{Z}})$. Let **f**⋅ *k* be the *k*th principal component, *i.e.* the *k*th axis of the new subspace, such that the *N* × *K* matrix ${\\boldsymbol{F}}$ is defined by ${\\boldsymbol{F}}\\equiv [{\\mathbf{f}}\\_{\\cdot 1},\\ldots,{\\mathbf{f}}\\_{\\cdot K}]$ with ${\\boldsymbol{R}}\\_{\\mathbf{f}}^c \\equiv [{\\textsc{\\textbf{r}}}\\_{c,{\\mathbf{f}}\\_1}^{\\nu-1},\\ldots,{\\textsc{\\textbf{r}}}\\_{c,{\\mathbf{f}}\\_{\\cdot K}}^c]$ its corresponding decumulative centered rank matrix (where each decumulative rank vector is raised to an exponent of *ν* − 1). The *K* × *K* matrix ${\\boldsymbol{B}}\\equiv [{\\mathbf{b}}\\_{\\cdot 1},\\ldots,{\\mathbf{b}}\\_{\\cdot K}]$ is the projector of the observations, with the normalization condition **b**⋅ *k*⊤**b**⋅ *k* = 1, such that ${\\boldsymbol{F}}={\\boldsymbol{Z}}{\\boldsymbol{B}}$. We denote by *λ*⋅ *k* (or 2*μ*⋅ *k*) the eigenvalues of the matrix $[GC\\_\\nu({\\boldsymbol{Z}})+GC\\_\\nu({\\boldsymbol{Z}})^{\\top}]$. Let the basis B : = {**b**⋅ 1, …, **b**⋅ *h*} with *h* ≤ *K* issued from the maximization of the overall Gini variability: \n$$\\max {\\mathbf{b}}\\_{\\cdot k}^{\\top} GC\\_\\nu({\\boldsymbol{Z}}) {\\mathbf{b}}\\_{\\cdot k} \\ \\Longrightarrow \\ [GC\\_\\nu({\\boldsymbol{Z}})+GC\\_\\nu({\\boldsymbol{Z}})^{\\top}] {\\mathbf{b}}\\_{\\cdot k} = 2\\mu\\_{\\cdot k} {\\mathbf{b}}\\_{\\cdot k}, \\ \\forall k=1,\\ldots,K.$$ \n Indeed, from the Lagrangian, \n$$L = {\\mathbf{b}}\\_{\\cdot k}^{\\top} GC\\_\\nu({\\boldsymbol{Z}}) {\\mathbf{b}}\\_{\\cdot k} - \\mu\\_{\\cdot k} [1- {\\mathbf{b}}\\_{\\cdot k}^{\\top}{\\mathbf{b}}\\_{\\cdot k}],$$ \n because of the non-symmetry of $GC\\_\\nu({\\boldsymbol{Z}})$, the eigenvalue equation is, \n$$[GC\\_\\nu({\\boldsymbol{Z}}) + GC\\_\\nu({\\boldsymbol{Z}})^{\\top}]{\\mathbf{b}}\\_{\\cdot k} = 2\\mu\\_{\\cdot k}{\\mathbf{b}}\\_{\\cdot k},$$ \n that is,\n \n$$\\label{eigen-value-eq}\n[GC\\_\\nu({\\boldsymbol{Z}})+GC\\_\\nu({\\boldsymbol{Z}})^{\\top}]{\\mathbf{b}}\\_{\\cdot k} = \\lambda\\_{\\cdot k}{\\mathbf{b}}\\_{\\cdot k}.$$ \nThe new subspace {**f**⋅ 1, …, **f**⋅ *h*} such that *h* ≤ *K* is issued from the maximization of the Gini variability between the observations on each axis **f**⋅ *k*. Although the result of the generalized Gini PCA seems to be close to the classical PCA, some differences exist.\nLet B = {**b**⋅ 1, …, **b**⋅ *h*} with *h* ≤ *K* be the basis issued from the maximization of ${\\mathbf{b}}\\_{\\cdot k}^{\\top} GC\\_\\nu({\\boldsymbol{Z}}) {\\mathbf{b}}\\_{\\cdot k}$ for all *k* = 1, …, *K*, then the following assertions hold:\n*(i)* max*G**M**D**ν*(**f**⋅ *k*, **f**⋅ *k*) = *μ*⋅ *k* for all *k* = 1, …, *K*, if and only if ${\\textsc{\\textbf{r}}}\\_{c,{\\mathbf{f}}\\_{\\cdot k}}^{\\nu-1} = {\\boldsymbol{R}}\\_{\\mathbf{z}}^c {\\mathbf{b}}\\_{\\cdot k}$.\n*(ii)* **b**⋅ *k***b**⋅ *h*⊤ = 0, for all *k* ≠ *h*.\n*(iii)* **b**⋅ *k***b**⋅ *k*⊤ = 1, for all *k* = 1, …, *K*.\n(i) Note that $GMD\\_\\nu({\\mathbf{f}}\\_{\\cdot k},{\\mathbf{f}}\\_{\\cdot k}) = -\\frac{2\\nu}{N(N-1)}({\\mathbf{f}}\\_{\\cdot k} - \\bar{{\\mathbf{f}}})^{\\top}{\\textsc{\\textbf{r}}}\\_{c,{\\mathbf{f}}\\_{\\cdot k}}^{\\nu-1}$, where ${\\textsc{\\textbf{r}}}\\_{c,{\\mathbf{f}}\\_{\\cdot k}}^{\\nu-1}$ is the *k*th column of the centered (decumulative) rank matrix ${\\boldsymbol{R}}\\_{\\mathbf{f}}^c$. Since ${\\mathbf{f}}\\_{\\cdot k} = {\\boldsymbol{Z}}{\\mathbf{b}}\\_{\\cdot k}$ and $\\bar{{\\mathbf{f}}} = (\\bar{{\\mathbf{f}}}\\_{\\cdot 1},\\ldots,\\bar{{\\mathbf{f}}}\\_{\\cdot K}) = \\mathbf{0}$ then:[8](#fn8)\n \n$$\\begin{aligned}\n{\\mathbf{b}}\\_{\\cdot k}^{\\top} GC\\_\\nu({\\boldsymbol{Z}}) {\\mathbf{b}}\\_{\\cdot k} &= -\\frac{2\\nu}{N(N-1)}{\\mathbf{b}}\\_{\\cdot k}^{\\top} {\\boldsymbol{Z}}^{\\top} {\\boldsymbol{R}}\\_{{\\mathbf{z}}}^c {\\mathbf{b}}\\_{\\cdot k} \\\\\n&= -\\frac{2\\nu}{N(N-1)}{\\mathbf{b}}\\_{\\cdot k}^{\\top} {\\boldsymbol{Z}}^{\\top} {\\textsc{\\textbf{r}}}\\_{c,{\\mathbf{f}}\\_{\\cdot k}}^{\\nu-1} \\ \\ \\ \\ (\\text{by} \\ {\\textsc{\\textbf{r}}}\\_{c,{\\mathbf{f}}\\_{\\cdot k}}^{\\nu-1} = {\\boldsymbol{R}}\\_{\\mathbf{z}}^c {\\mathbf{b}}\\_{\\cdot k}) \\notag \\\\\n&= -\\frac{2\\nu}{N(N-1)}{\\mathbf{f}}^{\\top}\\_{\\cdot k} {\\textsc{\\textbf{r}}}\\_{c,{\\mathbf{f}}\\_{\\cdot k}}^{\\nu-1} \\notag \\\\\n&= GMD\\_\\nu({\\mathbf{f}}\\_{\\cdot k},{\\mathbf{f}}\\_{\\cdot k}).\\notag \\end{aligned}$$ \nThen, maximizing the multidimensional variability ${\\mathbf{b}}\\_{\\cdot k}^{\\top} GC\\_\\nu({\\boldsymbol{Z}}) {\\mathbf{b}}\\_{\\cdot k}$ yields from ([eigen-value-eq]):\n \n$$\\begin{aligned}\n{\\mathbf{b}}\\_{\\cdot k}^{\\top} [GC\\_\\nu({\\boldsymbol{Z}})+GC\\_\\nu({\\boldsymbol{Z}})^{\\top}]{\\mathbf{b}}\\_{\\cdot k} &= {\\mathbf{b}}\\_{\\cdot k}^{\\top} \\lambda\\_{\\cdot k}{\\mathbf{b}}\\_{\\cdot k}\\notag \\\\\n\\Longleftrightarrow \\ {\\mathbf{b}}\\_{\\cdot k}^{\\top} GC\\_\\nu({\\boldsymbol{Z}}){\\mathbf{b}}\\_{\\cdot k} + {\\mathbf{b}}\\_{\\cdot k}^{\\top}GC\\_\\nu({\\boldsymbol{Z}})^{\\top} {\\mathbf{b}}\\_{\\cdot k} &= \\lambda\\_{\\cdot k}.\\notag\\end{aligned}$$ \nSince ${\\mathbf{b}}\\_{\\cdot k}^{\\top} GC\\_\\nu({\\boldsymbol{Z}}) {\\mathbf{b}}\\_{\\cdot k} = {\\mathbf{b}}\\_{\\cdot k}^{\\top} GC\\_\\nu({\\boldsymbol{Z}})^{\\top} {\\mathbf{b}}\\_{\\cdot k}$, then \n$${\\mathbf{b}}\\_{\\cdot k}^{\\top}GC\\_\\nu({\\boldsymbol{Z}})^{\\top} {\\mathbf{b}}\\_{\\cdot k} = \\lambda\\_{\\cdot k}/2 = \\mu\\_{\\cdot k},$$ \n and so *G**M**D**ν*(**f**⋅ *k*, **f**⋅ *k*) = *λ*⋅ *k* for all *k* = 1, …, *K*. The results (ii) and (iii) are straightforward.\nDiscussion\n----------\nCondition (i) shows that the maximization of the multidimensional variability (in the Gini sense) ${\\mathbf{b}}\\_{\\cdot k}^{\\top} GC\\_\\nu({\\boldsymbol{Z}}) {\\mathbf{b}}\\_{\\cdot k}$ does not necessarily coincide with the maximization of the variability of the observations projected onto the new axis **f**⋅ *k* embodied by *G**M**D**ν*(**f**⋅ *k*, **f**⋅ *k*). Since in general, the rank of the observations on axis **f**⋅ *k* does not coincide with the projected ranks, that is, \n$${\\textsc{\\textbf{r}}}\\_{c,{\\mathbf{f}}\\_{\\cdot k}}^{\\nu-1} \\neq {\\boldsymbol{R}}\\_{\\mathbf{z}}^c {\\mathbf{b}}\\_{\\cdot k},$$ \n then, \n$$\\max {\\mathbf{b}}\\_{\\cdot k}^{\\top} GC({\\boldsymbol{Z}}) {\\mathbf{b}}\\_{\\cdot k} \\neq GMD\\_\\nu({\\mathbf{f}}\\_{\\cdot k},{\\mathbf{f}}\\_{\\cdot k}).$$ \n In other words, maximizing the quadratic form ${\\mathbf{b}}\\_{\\cdot k}^{\\top} GC({\\boldsymbol{Z}})\\_\\nu {\\mathbf{b}}\\_{\\cdot k}$ does not systematically maximize the overall Gini variability *G**M**D**ν*(**f**⋅ *k*, **f**⋅ *k*). However, it maximizes the following generalized Gini index:\n \n$$\\begin{aligned}\nGGMD\\_\\nu({\\mathbf{f}}\\_{\\cdot k},{\\mathbf{f}}\\_{\\cdot k}) &:= -\\frac{2\\nu}{N(N-1)}{\\mathbf{b}}\\_{\\cdot k}^{\\top} {\\boldsymbol{Z}}^{\\top} {\\boldsymbol{R}}\\_{{\\mathbf{z}}}^c {\\mathbf{b}}\\_{\\cdot k} \\notag \\\\\n& = -\\frac{2\\nu}{N(N-1)} {\\mathbf{f}}\\_{\\cdot k}^{\\top}{\\mathbf{b}}\\_{\\cdot k}^{\\top} ({\\boldsymbol{R}}\\_{{\\mathbf{z}}}^{c})^{\\top}. \\notag\\end{aligned}$$ \nIn the literature on inequality indices, this kind of index is rather known as a generalized Gini index, because of the product between a variable **f**⋅ *k* and a function Ψ of its ranks, $\\Psi ({\\textsc{\\textbf{r}}}\\_{{\\mathbf{f}}\\_{k}}) := {\\mathbf{b}}\\_{\\cdot k}^{\\top} ({\\boldsymbol{R}}\\_{{\\mathbf{z}}}^{c})^{\\top}$, such that: \n$$GGMD\\_\\nu({\\mathbf{f}}\\_{\\cdot k},{\\mathbf{f}}\\_{\\cdot k}) = -\\frac{2\\nu}{N(N-1)}{\\mathbf{f}}\\_{\\cdot k} \\ \\Psi ({\\textsc{\\textbf{r}}}\\_{{\\mathbf{f}}\\_{k}}).$$ \n and subsequently proposes generalized Gini indices with a rank distortion function Ψ that describes the behavior of the decision maker (being either max-min or max-max).[9](#fn9)\nIt is noteworthy that this generalized Gini index of variability is very different from ’s multidimensional Gini index. The author proposes to extract the first eigenvector **e**⋅ 1 of ${\\boldsymbol{X}}^{\\top}{\\boldsymbol{X}}$ and to project the data ${\\boldsymbol{X}}$ such that ${\\mathbf{s}}:={\\boldsymbol{X}}{\\mathbf{e}}\\_{\\cdot 1}$ so that the multidimensional Gini index is $G({\\mathbf{s}}) = {\\mathbf{s}}^{\\top} \\tilde{\\Psi}({\\textsc{\\textbf{r}}}\\_s)$, with ${\\textsc{\\textbf{r}}}\\_s$ the rank vector of **s** and with $\\tilde{\\Psi}$ a function that distorts the ranks. ’s index is derived from the matrix ${\\boldsymbol{X}}^{\\top}{\\boldsymbol{X}}$. To be precise, the maximization of the variance-covariance matrix ${\\boldsymbol{X}}^{\\top}{\\boldsymbol{X}}$ (based on the ℓ2 metric) yields the projection of the data on the first component **f**⋅ 1, which is then employed in the multidimensional Gini index (based on the ℓ1 metric). This approach is legitimated by the fact that *G*(**s**) has some desirable properties linked with the Gini index. However, this Gini index deals with an information issued from the variance, because the vector **s** relies on the maximization of the variance of component **f**⋅ 1. Alternatively, it is possible to make use of the Gini variability, in a first stage, in order to project the data onto a new subspace, and in a second stage, to use the generalized Gini index of the projected data for the interpretations. In such as case, the Gini metric enables outliers to be attenuated. The employ of *G*(**s**) as a result of the variance-covariance maximization may transform the data so that outlying observations would capture an important part of the information (variance) on the first component. This case occurs in the classical PCA. This fact will be proven in the next sections with Monte Carlo simulations. Let us before investigate the employ of the generalized Gini index *G**G**M**D**ν*.\nProperties of *G**G**M**D**ν*\n-----------------------------\nSince the Gini PCA relies on the generalized Gini index *G**G**M**D**ν*, let us explore its properties.\n[prop4.2] Let the eigenvalues of $GC\\_{\\nu}({\\boldsymbol{Z}})+GC\\_{\\nu}({\\boldsymbol{Z}})^{\\top}$ be such that *λ*1 = *μ*1/2 ≥ ⋯ ≥ *λ*⋅ *K* = *μ*⋅ *K*/2. Then,\n*(i)* *G**G**M**D**ν*(**f**⋅ *k*, **f**⋅ *k*) = *G**G**M**D**ν*(**f**⋅ *k*, **f**⋅ ℓ) = *G**G**M**D**ν*(**f**⋅ ℓ, **f**⋅ *k*) = 0, for all ℓ = 1, …, *K*, if and only if, *λ*⋅ *k* = 0.\n*(ii)* max*k* = 1, …, *K**G**G**M**D**ν*(**f**⋅ *k*, **f**⋅ *k*) = *μ*1.\n*(iii)* min*k* = 1, …, *K**G**G**M**D**ν*(**f**⋅ *k*, **f**⋅ *k*) = *μ*⋅ *K*.\n(i) The result comes from the rank-nullity theorem. From the eigenvalue Equation ([eigen-value-eq]), we have: \n$${\\mathbf{b}}\\_{\\cdot k}^{\\top}GC\\_{\\nu}({\\boldsymbol{Z}}) {\\mathbf{b}}\\_{\\cdot k} = \\lambda\\_{\\cdot k}/2 = \\mu\\_{\\cdot k}.$$ \n Let *f* be the linear application issued from the matrix $GC\\_{\\nu}({\\boldsymbol{Z}})$. Whenever *λ*⋅ *k* = 0, two columns (or rows) of $GC\\_{\\nu}({\\boldsymbol{Z}})$ are collinear, then the dimension of the image set of *f* is dim(*f*) = *K* − 1. Hence, **f**⋅ *k* = **0**. Since ${\\mathbf{b}}\\_{\\cdot k}^{\\top}GC\\_{\\nu}({\\boldsymbol{Z}})^{\\top} {\\mathbf{b}}\\_{\\cdot k} = {GGMD}\\_\\nu({\\mathbf{f}}\\_{\\cdot k},{\\mathbf{f}}\\_{\\cdot k})$ for all *k* = 1, …, *K*, then for *λ*⋅ *k* we get: \n$${\\mathbf{b}}\\_{\\cdot k}^{\\top}GC\\_{\\nu}({\\boldsymbol{Z}})^{\\top} {\\mathbf{b}}\\_{\\cdot k} = {GGMD}\\_\\nu({\\mathbf{f}}\\_{\\cdot k},{\\mathbf{f}}\\_{\\cdot k}) =\\lambda\\_{\\cdot k}/2 = \\mu\\_{\\cdot k} = 0.$$ \n On the other hand, since **f**⋅ *k* = **0**, it follows that *G**G**M**D**ν*(**f**⋅ *k*, **f**⋅ ℓ) = 0 for all ℓ = 1, …, *K*. Also, if **f**⋅ *k* = **0** then the centered rank vector ${\\textsc{\\textbf{r}}}^c\\_{{\\mathbf{f}}\\_{\\cdot k}} = \\mathbf{0}$, and so *G**G**M**D**ν*(**f**⋅ ℓ, **f**⋅ *k*) = 0 for all ℓ = 1, …, *K*.\n(ii) The proof comes from the Rayleigh-Ritz identity: \n$$\\lambda\\_{\\max} := \\max \\frac{{\\mathbf{b}}\\_{\\cdot 1}^{\\top}[GC\\_\\nu({\\boldsymbol{Z}})+GC\\_\\nu({\\boldsymbol{Z}})^{\\top}]{\\mathbf{b}}\\_1}{{\\mathbf{b}}\\_1^{\\top}{\\mathbf{b}}\\_1} = \\lambda\\_{1}.$$ \n Since ${\\mathbf{b}}\\_1^{\\top}GC\\_\\nu({\\boldsymbol{Z}}){\\mathbf{b}}\\_{\\cdot 1} = \\lambda\\_1/2$ and because ${\\mathbf{b}}\\_1^{\\top}GC\\_\\nu({\\boldsymbol{Z}}){\\mathbf{b}}\\_1 = GGMD\\_{\\nu}({\\mathbf{f}}\\_1,{\\mathbf{f}}\\_1)$, the result follows.\n(iii) Again, the Rayleigh-Ritz identity yields: \n$$\\lambda\\_{\\min} := \\min \\frac{{\\mathbf{b}}\\_{\\cdot K}^{\\top}[GC\\_\\nu({\\boldsymbol{Z}})+GC\\_\\nu({\\boldsymbol{Z}})^{\\top}]{\\mathbf{b}}\\_{\\cdot K}}{{\\mathbf{b}}\\_{\\cdot K}^{\\top}{\\mathbf{b}}\\_{\\cdot K}} = \\lambda\\_{K}.$$ \n Then, ${\\mathbf{b}}\\_{\\cdot K}^{\\top}GC\\_\\nu({\\boldsymbol{Z}}){\\mathbf{b}}\\_{\\cdot K} = GGMD\\_{\\nu}({\\mathbf{f}}\\_{\\cdot K},{\\mathbf{f}}\\_{\\cdot K}) = \\lambda\\_{\\cdot K}/2$.\nThe index *G**G**M**D**ν*(**f**⋅ *k*, **f**⋅ *k*) represents the variability of the observations projected onto component **f**⋅ *k*. When this variability is null, then the eigenvalue is null (i). In the same time, there is neither co-variability in the Gini sense between **f**⋅ *k* and another axis **f**⋅ ℓ, that is *G**G**M**D**ν*(**f**⋅ *k*, **f**⋅ ℓ) = 0.\nIn the Gaussian case, because the Gini correlation matrix is positive semi-definite, the eigenvalues are non-negative, then *G**G**M**D* is null whenever it reaches its minimum.\nLet $Z \\sim \\mathcal N (0,1)$ and let $\\boldsymbol{X}$ represent identically distributed Gaussian random variables, with distribution $\\mathcal{N}(\\mathbf{0},\\boldsymbol{\\rho})$ such that $\\emph{Var}(X\\_k)=1$ for all *k* = 1, …, *K* and let *γ*1, …, *γ**K* be the eigenvalues of $\\emph{Var}(\\boldsymbol{X})$. Then the following assertions holds:\n*(i)* $\\emph{Tr}[GC\\_\\nu(\\boldsymbol{X})] = \\emph{Cov}(Z,\\Phi(Z))\\emph{Tr}[\\emph{Var}(\\boldsymbol{X})]$.\n*(ii)* $\\mu\\_k = \\emph{Cov}(Z,\\Phi(Z)) \\gamma\\_k$ for all *k* = 1, …, *K*.\n*(iii)* $|GC\\_\\nu(\\boldsymbol{X})| = \\emph{Cov}^K(Z,\\Phi(Z))|\\emph{Var}(\\boldsymbol{X})|$.\n*(iv)* For all *ν* > 1: \n$$\\frac{\\mu\\_k}{\\emph{Tr}[GC\\_\\nu({\\boldsymbol{X}})]} = \\frac{\\gamma\\_k}{\\emph{Tr}[\\emph{Var}({\\boldsymbol{X}})]}, \\ \\forall k=1,\\ldots,K.$$ \nFrom Theorem [th2]: \n$$GC\\_\\nu(\\boldsymbol{X})=\\sigma^{-1} \\text{Cov}(Z,\\Phi(Z)) \\emph{Var}(\\boldsymbol{X}).$$ \n From (Chapter 26), when *Z* ∼ N(0, 1), \n$$\\text{Cov}(Z,\\Phi(Z))=\\frac{1}{2\\sqrt{\\pi}}\\approx 0.2821.$$ \n Then the results follow directly.\nPoint (iv) shows that the eigenvalues of the standard PCA are proportional to those issued from the generalized Gini PCA. Because each eigenvalue (in proportion of the trace) represents the variability (or the quantity of information) inherent to each axis, then both PCA techniques are equivalent when ${\\boldsymbol{X}}$ is Gaussian: \n$$\\frac{\\mu\\_k}{\\text{Tr}[GC\\_\\nu({\\boldsymbol{X}})]} = \\frac{\\gamma\\_k}{\\text{Tr}[\\text{Var}({\\boldsymbol{X}})]}, \\ \\forall k=1,\\ldots,K \\ ; \\ \\forall \\nu > 1.$$ \nThe R*N*-Euclidean space\n------------------------\nIn classical PCA, the duality between R*N* and R*K* enables the eigenvectors and eigenvalues of R*N* to be deduced from those of R*K* and conversely. This duality is not so obvious in the Gini PCA case. Indeed, in R*N* the Gini variability between the observations would be measured by $GC\\_\\nu(\\widetilde{{\\boldsymbol{Z}}}):=\\frac{-2\\nu}{N(N-1)}({\\boldsymbol{R}}\\_{\\mathbf{z}}^{c})^{\\top}{\\boldsymbol{Z}}$, and subsequently the idea would be to derive the eigenvalue equation related to R*N*, \n$$[GC\\_\\nu(\\widetilde{{\\boldsymbol{Z}}})+GC\\_\\nu(\\widetilde{{\\boldsymbol{Z}}})^{\\top}]\\ \\widetilde{{\\mathbf{b}}}\\_{\\cdot k} = \\widetilde{\\lambda}\\_{\\cdot k}\\widetilde{{\\mathbf{b}}}\\_{\\cdot k}.$$ \n The other option is to define a basis of R*N* from a basis already available in R*K*. In particular, the set of principal components {**f**1, …, **f**⋅ *k*} provides by construction a set of normalized and orthogonal vectors. Let us rescale the vectors **f**⋅ *k* such that: \n$$\\widetilde{{\\mathbf{f}}}\\_{\\cdot k} = \\frac{{\\mathbf{f}}\\_{\\cdot k}}{GMD\\_\\nu({\\mathbf{f}}\\_{\\cdot k},{\\mathbf{f}}\\_{\\cdot k})}.$$ \n Then, $\\{\\widetilde{{\\mathbf{f}}}\\_1,\\ldots,\\widetilde{{\\mathbf{f}}}\\_{\\cdot k}\\}$ constitutes an orthonormal basis of R*K* in the Gini sense since $GMD\\_\\nu(\\widetilde{{\\mathbf{f}}}\\_{\\cdot k},\\widetilde{{\\mathbf{f}}}\\_{\\cdot k})=1$. This basis may be used as a projector of the variables **z**⋅ *k* onto R*N*. Let $\\widetilde{{\\boldsymbol{F}}}$ be the *N* × *K* matrix with $\\widetilde{{\\mathbf{f}}}\\_{\\cdot k}$ in columns. The projection of the variables **z**⋅ *k* in R*N* is given by the following Gini correlation matrix: \n$${\\boldsymbol{V}}:= \\frac{-2\\nu}{N(N-1)}\\ \\widetilde{{\\boldsymbol{F}}}^{\\top}{\\boldsymbol{R}}\\_{\\mathbf{z}}^c,$$ \n whereas it is given by $\\frac{1}{N}\\widetilde{{\\boldsymbol{F}}}^{\\top}{\\boldsymbol{Z}}$ in the standard PCA, that is, the matrix of Pearson correlation coefficients between all $\\widetilde{{\\mathbf{f}}}\\_{\\cdot k}$ and **z**⋅ ℓ. The same interpretation is available in the Gini case. The matrix ${\\boldsymbol{V}}$ is normalized in such a way that ${\\boldsymbol{V}}\\equiv [v\\_{k\\ell}]$ are the *G*-correlations indices between $\\widetilde{{\\mathbf{f}}}\\_{\\cdot k}$ and **z**⋅ ℓ. This yields the ability to make easier the interpretation of the variables projected onto the new subspace.\nInterpretations of the Gini PCA\n===============================\nThe analysis of the projections of the observations and of the variables are necessary to provide accurate interpretations. Some criteria have to be designed in order to bring out, in the new subspace, the most significant observations and variables.\nObservations\n------------\nThe absolute contribution of an observation *i* to the variability of a principal component **f**⋅ *k* is: \n$$ACT\\_{ik} = \\frac{f\\_{ik}\\ \\Psi(R(f\\_{ik}))}{GGMD\\_\\nu({\\mathbf{f}}\\_{\\cdot k},{\\mathbf{f}}\\_{\\cdot k})}.$$ \n The absolute contribution of each observation *i* to the generalized Gini Mean Difference of **f**⋅ *k* (*A**C**T**i**k*) is interpreted as a percentage of variability of *G**G**M**D**ν*(**f**⋅ *k*, **f**⋅ *k*), such that ∑*i* = 1*N**A**C**T**i**k* = 1. This provides the most important observations *i* related to component **f**⋅ *k* with respect to the information *G**G**M**D**ν*(**f**⋅ *k*, **f**⋅ *k*). On the other hand, instead of employing the Euclidean distance between one observation *i* and the component **f**⋅ *k*, the Manhattan distance is used. The relative contribution of an observation *i* to component **f**⋅ *k* is then: \n$$RCT\\_{ik} = \\frac{\\left|f\\_{ik}\\right |}{ \\| {\\mathbf{f}}\\_{i\\cdot}\\|\\_1}.$$ \n Remark that the gravity center of {**f**1, …, **f**⋅ *K*} is $\\mathbf{g}:=(\\bar{{\\mathbf{f}}}\\_1,\\ldots,\\bar{{\\mathbf{f}}}\\_{\\cdot K}) =\\mathbf{0}$. The Manhattan distance between observation *i* and **g** is then ∑*k* = 1*K*∣*f**i**k* − 0∣, and so \n$$RCT\\_{ik} = \\frac{\\left|f\\_{ik}\\right |}{\\| {\\mathbf{f}}\\_{i\\cdot} - \\mathbf g\\|\\_1}.$$ \n The relative contribution *R**C**T**i**k* may be interpreted rather as the contribution of dimension *k* to the overall distance between observation *i* and **g**.\nVariables\n---------\nThe most significant variables must be retained for the analysis and the interpretation of the data in the new subspace. It would be possible, in the same manner as in the observations case, to compute absolute and relative contributions from the Gini correlation matrix ${\\boldsymbol{V}}\\equiv [v\\_{k\\ell}]$. Instead, it is possible to test directly for the significance of the elements *v**k*ℓ of ${\\boldsymbol{V}}$ in order to capture the variables that significantly contribute to the Gini variability of components **f**⋅ *k*. Let us denote $\\widetilde{U}\\_{\\ell k}:={\\text{Cov}}({\\mathbf{f}}\\_{\\cdot \\ell},{\\boldsymbol{R}}\\_{{\\mathbf{z}}\\_{\\cdot k}}^c)$ with ${\\boldsymbol{R}}\\_{{\\mathbf{z}}\\_{\\cdot k}}^c$ the (decumulative) centered rank vector of **z**⋅ *k* raised to an exponent of *ν* − 1 and $U\\_{\\cdot \\ell} :={\\text{Cov}}({\\mathbf{f}}\\_{\\cdot \\ell},{\\boldsymbol{R}}^c\\_{{\\mathbf{f}}\\_{\\cdot \\ell}})$. Those two Gini covariances yield the following *U*-statistics:\n \n$$\\notag\nU\\_{\\ell k} = \\frac{\\widetilde{U}\\_{\\ell k}}{U\\_{\\cdot \\ell}} = v\\_{k\\ell}.$$ \nLet *U*ℓ*k*0 be the expectation of *U*ℓ*k*, that is *U*ℓ*k*0 : = E[*U*ℓ*k*]. From, *U*ℓ*k* is an unbiased and consistent estimator of *U*ℓ*k*0. From Theorem 10.4 in, Chapter 10, we asymptotically get that $\\sqrt{N}(U\\_{\\ell k} - U\\_{\\ell k}^0) \\stackrel{a}{\\sim} \\mathcal{N}$. Then, it is possible to test for:\n \n$$\\notag\n\\left\\|\n\\begin{array}{l}\nH\\_0 : U\\_{\\ell k}^0 = 0 \\\\ \nH\\_1 : U\\_{\\ell k}^0 \\neq 0.\n\\end{array}\n\\right.$$ \nLet *σ̂*ℓ*k*2 the Jackknife variance of *U*ℓ*k*, then it is possible to test for the null under the assumption *N* → ∞ as follows:[10](#fn10)\n \n$$\\notag\n\\frac{U\\_{\\ell k}}{\\hat{\\sigma}\\_{\\ell k}} \\stackrel{a}{\\sim} \\mathcal{N}(0,1).$$ \nThe usual PCA enables the variables to be analyzed in the circle of correlation, which outlines the correlations between the variables **z**⋅ *k* and the components **f**⋅ ℓ. In order to make a comparison with the usual PCA, let us rescale the *U*-statistics *U*ℓ*k*. Let **U** be the *K* × *K* matrix such that **U** ≡ [*U*ℓ*k*], and **u**⋅ *k* the *k*-th column of **U**. Then, the absolute contribution of the variable **z**⋅ *k* to the component **f**⋅ ℓ is: \n$$\\widetilde{ACT}\\_{k \\ell} = \\frac{U\\_{\\ell k}}{\\| \\mathbf{u}\\_{\\cdot k}\\|\\_2}.$$ \n The measure $\\widetilde{ACT}\\_{k \\ell}$ yields a graphical tool aiming at comparing the standard PCA with the Gini PCA. In the standard PCA, cos2*θ* (see Figure [circle] below) provides the Pearson correlation coefficient between **f**1 and **z**⋅ *k*. In the Gini PCA, cos2*θ* is the normalized Gini correlation coefficient $\\widetilde{ACT}\\_{k 1}$ thanks to the ℓ2 norm.\n![Circle of correlation](circle.png \"fig:\")[circle]\nIt is worth mentioning that the circle of correlation does not provide the significance of the variables. This significance relies on the statistical test based on the *U*-statistics exposed before. Because $\\widetilde{ACT}$ depends on the ℓ2 metric, it is sensitive to outliers, and as such, the choice of the variables must rely on the test of *U*ℓ*k*0 only.\nMonte Carlo Simulations\n=======================\nIn this Section, it is shown with the aid of Monte Carlo simulations that the usual PCA yields irrelevant results when outlying observations contaminate the data. To be precise, the absolute contributions computed in the standard PCA based on the variance may lead to select outlying observations on the first component in which there is the most important variability (a direct implication of the maximization of the variance). In consequence, the interpretation of the PCA may inflate the role of the first principal components. The Gini PCA dilutes the importance of the outliers to make the interpretations more robust and therefore more relevant.\n[H] *θ* = 1 [ *θ* is the value of the outlier] Mean squared Errors of eigenvalues, *A**C**T* and *R**C**T* [MC]\nThe mean squared errors of the eigenvalues are computed as follows:\n \n$$\\notag\nMSE\\_{\\lambda\\_k} = \\frac{\\sum\\_{i=1}^{1,000}(\\lambda\\_k^{oi} - \\lambda\\_k)^2}{1,000},$$ \nwhere *λ**k**o**i* is the eigenvalue computed with outlying observations in the sample. The MSE of *A**C**T* et *R**C**T* are computed in the same manner.\nWe first investigate the case where the variables are highly correlated in order to gauge the robustness of each technique (Gini for *ν* = 2, 4, 6 and variance). The correlation matrix between the variables is given by:\n \n$$\\notag\n\\boldsymbol{\\rho} = \\left( \\begin{array}{cccc}\n1 & 0.8 & 0.9 & 0.7 \\\\\n0.8 & 1 & 0.8 & 0.75 \\\\\n0.9 & 0.8 & 1 & 0.6 \\\\\n0.7 & 0.75 & 0.6 & 1 \\\\\n \\end{array}\\right)$$ \nAs can be seen in the matrix above, we can expect that all the information be gathered on the first axis because each pair of variables records an important linear correlation. The repartition of the information on each component, that is, each eigenvalue in percentage of the sum of the eigenvalues is the following.\n[h!]\n| c | c || c | c | c | c | Eigenvalues & Gini *ν* = 2 & Gini *ν* = 4 & Gini *ν* = 6 & Variance \n& eigenvalues (%) & 81.65341 & 82.31098 & 82.28372 & 81.11458 \n Axis 1 & & & & & \n& **MSE** & **12.313750** & **12.196975** & **12.221840** & **15.972710** \n& eigenvalues (%) & 10.90079 & 10.47317 & 10.46846 & 11.35471 \n Axis 2 & & & & & \n& **MSE** & **11.478541** & **11.204504** & **10.818344** & **10.688924** \n& eigenvalues (%) & 5.062329 & 4.996538 & 5.088865 & 5.112817 \n Axis 3 & & & & & \n& **MSE** & **2.605312** & **2.608647** & **2.799180** & **4.687323** \n& eigenvalues (%) & 2.383476 & 2.219311 & 2.15896 & 2.417897 \n Axis 4 & & & & & \n& **MSE** & **1.541453** & **1.826055** & **3.068596** & **2.295100** \nThe first axis captures around 82% of the variability of the overall sample (before contamination). Although each PCA method yields the same repartition of the information over the different components before the contamination of the data, it is possible to show that the classical PCA is not robust. For this purpose, let us analyze Figures [fig:3a]-[fig:3d] below that depict the MSE of each observation with respect to the contamination process described in Algorithm 1 above.\n.6![ACT_1](CTA1-fort.png \"fig:\")[fig:3a]\n.6![ACT_2](CTA2-fort.png \"fig:\")[fig:3b]\n.6![RCT_1](CTR1-fort.png \"fig:\")[fig:3c]\n.6![RCT_2](CTR2-fort.png \"fig:\")[fig:3d]\nOn the first axis of Figure [fig:3a], the absolute contribution of each observation (among 500 observations) is not stable because of the contamination of the data, however the Gini PCA performs better. The MSE of the ACTs measured during to the contamination process provides lower values for the Gini index compared with the variance. On the other hand, if we compute the standard deviation of all these MSEs over the two first axis, again the Gini methodology provides lower variations (see Table [tab:2]).\n[h!]\n|c||cccc| &Gini *ν* = 2 & Gini *ν* = 4 & Gini *ν* = 6 & Variance \nAxis 1&6.08 & 6.62 & 7.41 & **12.09** \nAxis 2&4.07 & 5.12 & 13.37 & 2.98 \n[tab:2]\nLet us take now an example with less correlations between the variables in order to get a more equal repartition of the information on the first two axes.\n \n$$\\notag\n\\mathbf{\\rho} = \\left( \\begin{array}{cccc}\n1 & -0.5 & 0.25 & 0.5 \\\\\n-0.5 & 1 & -0.9 & 1 \\\\\n0.25 & -0.9 & 1 & -0.25 \\\\\n0.5 & 0 & -0.25 & 1 \\\\\n \\end{array}\\right)$$ \nThe repartition of the information over the new axes (percentage of each eigenvalue) is given in Table [Tab3]. When the information is less concentrated on the first axis (55% on axis 1 and around 35% on axis 2), the MSE of the eigenvalues after contamination are much more important for the standard PCA compared with the Gini approach (2 to 3 times more important). Although the fourth axis reports an important MSE for the Gini method (*ν* = 6), the eigenvalue percentage is not significant (1.56%).\n[h!]\n| c | c || c | c | c | c | eigenvalues (%) & Gini *ν* = 2 & Gini *ν* = 4 & Gini *ν* = 6 & Variance \n& eigenvalues (%) & 55.3774 & 55.15931 & 54.96172 & 55.08917 \n Axis 1 & & & & & \n& **MSE** & **17.711023** & **14.968196** & **12.745760** & **38.929147** \n& eigenvalues (%) & 35.8385 & 35.86216 & 35.8745 & 36.06118 \n Axis 2 & & & & & \n& **MSE** & **14.012198** & **16.330350** & **18.929923** & **30.948674** \n& eigenvalues (%) & 7.227274 & 7.345319 & 7.527222 & 7.329535 \n Axis 3 & & & & & \n& **MSE** & **4.919686** & **4.897820** & **5.036241** & **6.814252** \n& eigenvalues (%) & 1.556831 & 1.633214 & 1.636561 & 1.520114 \n Axis 4 & & & & & \n& **MSE** & **1.149770** & **7.890184** & **14.047539** & **1.438904** \n[Tab3]\nLet us now have a look on the MSE of the absolute contributions of each observation (*N* = 500) for each PCA technique ([fig:5a]-[fig:5b]). We obtain the same kind of results, with less variability on the second axis. In Figures [fig:5a]-[fig:5b], it is apparent that the classical PCA based on the ℓ2 norm exhibits much more ACT variability (black points). This means that the contamination of the data can lead to the interpretation of some observations as significant (important contribution to the variance of the axis) while they are not (and vice versa). On the other hand, the MSE of the RCTs after contamination of the data, Figures [fig:5c]-[fig:5d], are less spread out for the Gini technique for *ν* = 4 and *ν* = 6, however for *ν* = 2 there is more variability of the MSE compared with the variance. This means that the distance from one observation to an axis may not be reliable (although the interpretation of the data rather depends on the ACTs).\n.6![ACT_1](CTA1-moy.png \"fig:\")[fig:5a]\n.6![ACT_2](CTA2-moy.png \"fig:\")[fig:5b]\n.6![RCT_1](CTR1-moy.png \"fig:\")[fig:5c]\n.6![RCT_2](CTR2-moy.png \"fig:\")[fig:5d]\nThe results of Figures [fig:5a]-[fig:5d] can be synthesized by measuring the standard deviation of the MSE over the ACTs of the 500 observations along the two first axes.\n[h!]\n|c||cccc| &Gini *ν* = 2 & Gini *ν* = 4 & Gini *ν* = 6 & Variance \nAxis 1&7.50 & 7.86 & 8.46 & **11.92** \nAxis 2&0.62& 0.75 & 0.77 & **11.24** \n[tab2]\nAs in the previous example of simulation, Table [tab2] indicates that the PCA based on the variance is less stable about the values of the ACTs that provide the most important observations of the sample. This may lead to irrelevant interpretations.\nApplication on cars data\n========================\nWe propose a simple application with the celebrated cars data (see the Appendix).[11](#fn11) The dataset is particularly interesting since there are highly correlated variables as can be seen in the Pearson correlation matrix given in Table [tab:corr].\n[h!]\n| c || c | c | c | c | c | c | & **capacity** *x*1 & **power** *x*2 & **speed** *x*3 & **weight** *x*4 & **width** *x*5 & **length** *x*6 \n*x*1 & **1.000** & 0.954 & 0.885 & 0.692 & 0.706 & 0.663 \n*x*2 & 0.954 &**1.000** & 0.933 & 0.528 & 0.729 & 0.663 \n*x*3 & 0.885 & 0.933& **1.000** & 0.466 & 0.618 & 0.578 \n*x*4 & 0.692 & 0.528 & 0.466 & **1.000** & 0.477 & 0.794 \n*x*5 & 0.706 & 0.729 & 0.618 & 0.477 & **1.000** & 0.591 \n*x*6 & 0.663 & 0.663 & 0.578 & 0.794 & 0.591 & **1.000** \n[tab:corr]\nAlso, the dataset is composed of some outlying observations (Figure [fig5]): Ferrari enzo (*x*1, *x*2, *x*5), Bentley continental (*x*2), Aston Martin (*x*2), Land Rover discovery (*x*5), Mercedes class S (*x*5), Smart (*x*5, *x*6).\n| | | |\n| --- | --- | --- |\n| capacity *x*1 | power *x*2 | speed *x*3 |\n| image | image | image |\n| weight *x*4 | width *x*5 | length *x*6 |\n| image | image | image |\n[fig5]\nThe overall information (variability) is partitioned over six components (Table [tab6]).\n[h!]\n| c || c | c | c | c | **eigenvalues in %** & Gini *ν* = 2 & Gini *ν* = 4 & Gini *ν* = 6 & Variance \n**Axis 1** & **80.35797** & **83.17172** & **84.84995** &**73.52112** \n**Axis 2** & **12.0761** & **10.58655** & **9.715974** & **14.22349** \n**Axis 3** & 4.132136 & 2.987015 & 3.130199 & **7.26106** \n**Axis 4** & 3.059399 & 2.612411 & 1.519626 & 3.93117 \n**Axis 5** & 0.3332362 & 0.3125735 & 0.2696611 & 0.85727 \n**Axis 6** & 0.04115858 & − 0.3297257 & -0.5145944 & 0.20585 \n**Sum** & **100 %** & **100 %** & **100 %** & **100 %** \n[tab6]\nTwo axes may be chosen to analyze the data. As shown in the previous Section about the simulations, when the data are highly correlated such that two axes are sufficient to project the data, the Gini PCA and the standard PCA yield the same share of information on each axis. However, we can expect some differences for absolute contributions *A**C**T* and relative contributions *R**C**T*.\nThe projection of the data is depicted in Figure [fig6], for each method.\n.6![Gini (\\nu = 2)](Proj_ind_nu2.png \"fig:\")\n.6![Gini (\\nu = 4)](Proj_ind_nu4.png \"fig:\")\n.6![Gini (\\nu = 6)](Proj_ind_nu6.png \"fig:\")\n.6![Variance](Proj_ind_var.png \"fig:\")\n[fig6]\nAs depicted in Figure [fig6], the projection is very similar for each technique. The cars with extraordinary (or very low) abilities are in the same relative position in the four projections: Land Rover Discovery Td5 at the top, Ferrari Enzo at the bottom right, Smart Fortwo coupé at the bottom left. However, when we improve the coefficient of variability *ν* to look for what happens at the tails of the distributions (of the two axes), we see that more cars are distinguishable: Land Rover Defender, Audi TT, BMW Z4, Renault Clio 3.0 V6, Bentley Contiental GT. Consequently, contrary to the case *ν* = 2 or the variance, the projections with *ν* = 4, 6 allow one to find other important observations, which are not outlying observations that contribute to the overall amount of variability. For this purpose, let us first analyze the correlations between the variables and the new axes in order to interpret the results, see Tables [tab7] to [tab10].\nSome slight differences appear between the Gini PCA and the classical one based on the variance. The theoretical Section [s4] indicates that the Gini methodology for *ν* = 2 is equivalent to the variance when the variables are Gaussian. On cars data, we observe this similarity. In each PCA, all variables are correlated with Axis 1 and weight with Axis 2. However, when *ν* increases, the Gini methodology allows outlying observations to be diluted so that some variables may appear to be significant, whereas they are not in the variance case.\n[h!]\n| c | c || l | l | l | l | l | c | **Gini** (*ν* = 2) & capacity & power & speed & weight & width & length \n& correlation & -0.974 & -0.945 & -0.872 & 0.760 & -0.933 & -0.823 \n Axe 1 & & & & & & & \n& ***U*-stat** & **-56.416** & **-25.005** & **-10.055** & **-4.093** & **-24.837** & **-12.626** \n& correlation & -0.032 & -0.241 & -0.405 & 0.510 & 0.183 & -0.379 \n Axe 2 & & & & & & & \n& ***U*-stat** & **-0.112** & **-0.920** & **-1.576** & **2.897** & **0.526** & **1.666** \n[tab7]\n[h!]\n| c | c || l | l | l | l | l | c | **Gini** (*ν* = 4) & capacity & power & speed & weight & width & length \n& correlation & 0.982 & 0.948 & 0.797 & 0.858 & 0.952 & 0.888 \n Axe 1 & & & & & & & \n& ***U*-stat** & **8.990** & **8.758** & **4.805** & **9.657** & **8.517** & **8.182** \n& correlation & -0.021 & 0.207 & 0.516 & -0.279 & -0.147 & -0.246 \n Axe 2 & & & & & & & \n& ***U*-stat** & **-0.095** & **0.817** & **2.299** & **-1.773** & **-0.705** & **-1.200** \n[tab8]\n[h!]\n| c | c || l | l | l | l | l | c | **Gini** (*ν* = 6) & capacity & power & speed & weight & width & length \n& valeurs & -0.781 & -0.759 & -0.598 & -0.730 & -0.755 & -0.701 \n Axe 1 & & & & & & & \n& ***U*-stat** & **-4.036** & **-3.903** & **-3.137** & **-3.125** & **-3.882** & **-3.644** \n& valeurs & 0.019 & -0.170 & -0.570 & 0.153 & 0.125 & 0.218 \n Axe 2 & & & & & & & \n& ***U*-stat** & **0.089** & **-0.734** & **-1.914** & **0.734** & **0.569** & **0.906** \n[tab9]\n| c | c || l | l | l | l | l | c | **Variance** & capacity & power & speed & weight & width & length \n& valeurs & 0.962 & 0.923 & 0.886 & 0.756 & 0.801 & 0.795 \n Axe 1 & & & & & & & \n& ***U*-stat** & **11.802** & **11.322** & **10.866** & **9.282** & **9.825** & **9.752** \n& valeurs & -0.126 & -0.352 & -0.338 & 0.575 & -0.111 & 0.504 \n Axe 2 & & & & & & & \n& ***U*-stat** & **-0.307** & **-0.855** & **-0.821** & **1.396** & **-0.269** & **1.223** \n[tab10]\nTables [tab8] and [tab9] (*ν* = 4, 6) show that Axis 2 is correlated to speed (not weight as in the variance PCA). In this respect the absolute contributions must describe the cars associated with speed on Axis 2. Indeed, the Land Rover discovery, a heavy weight car, is no more available on Axis 2 for the Gini PCA for *ν* = 2, 4, 6 (Figures [fig:8], [fig:9], [fig:10]). Note that the red line in the Figures represents the mean share of the information on each axis, *i.e.* 100%/24 cars = 4.16% of information per car.\n.6![Axis 1 (variance) ](CTA1-var.png \"fig:\")\n.6![Axis 2 (variance) ](CTA2-var.png \"fig:\")\n[fig:7]\n.6![Axis 1 (\\nu=2) ](CTA1_nu2.png \"fig:\")\n.6![Axis 2 (\\nu=2) ](CTA2_nu2.png \"fig:\")\n[fig:8]\n.6![Axis 1 (\\nu=4) ](CTA1_nu4.png \"fig:\")\n.6![Axis 2 (\\nu=4) ](CTA2_nu4.png \"fig:\")\n[fig:9]\n.6![Axis 1 (\\nu=6) ](CTA1_nu6.png \"fig:\")\n.6![Axis 2 (\\nu=6) ](CTA2_nu6.png \"fig:\")\n[fig:10]\nFinally, some cars are not correlated with axis 2 in the standard PCA, see Figures [fig:8]–[fig:10], while this is the case in the Gini PCA. Indeed some cars are now associated with speed: Aston Martin, Bentley Continental GT, Renault Clio 3.0 V6 and Mini 1.6 170. This example of application shows that the use of the Gini metric robust to outliers may involve some serious changes in the interpretation of the results.\nConclusion\n==========\nIn this paper, it has been shown that the geometry of the Gini covariance operator allows one to perform Gini PCA, that is, a robust principal component analysis based on the ℓ1 norm.\nTo be precise, the variance may be replaced by the Gini Mean Difference, which captures the variability of couples of variables based on the rank of the observations in order to attenuate the influence of the outliers. The Gini Mean Difference may be rather interpreted with the aid of the generalized Gini index *G**G**M**D**ν* in the new subspace for a better understanding of the variability of the components, that is, *G**G**M**D**ν* is both a rank-dependent measure of variability in sense and also an eigenvalue of the Gini correlation matrix.\nContrary to many approaches in multidimensional statistics in which the standard variance-covariance matrix is used to project the data onto a new subspace before deriving multidimensional Gini indices (see e.g. ), we propose to employ the Gini correlation indices (see ). This provides the ability to interpret the results with the ℓ1 norm and the use of *U*-statistics to measure the significance of the correlation between the new axes and the variables.\nThis research may open the way on data analysis based on Gini metrics in order to study multivariate correlations with categorical variables or discriminant analyses when outlying observations drastically affect the sample.\n99 Abramowitz, M. & I. Stegun. (1964). *Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables*. National Bureau of Standards Applied Mathematics Series No. 55.\nAnderson, T.W. (1963) Asymptotic Theory for Principal Component Analysis. *The Annals of Mathematical Statistics*, **34**, 122–148.\nBaccini, A., P. Besse & A. de Falguerolles (1996), A *L*1 norm PCA and a heuristic approach, in *Ordinal and Symbolic Data Analysis*, E Didday, Y. Lechevalier and O. Opitz (eds), Springer, 359–368.\nBanerjee, A.K. (2010), A multidimensional Gini index, *Mathematical Social Sciences*, **60**: 87–93.\nCandes, E.J., Xiaodong Li, Yi Ma & John Wright. (2009) Robust Principal Component Analysis?. arXiv:0912.3599.\nCarcea, M. & R. Serfling (2015), A Gini autocovariance function for time series modeling. *Journal of Time Series Analysis* **36**: 817–38.\nDalton, H. 1920. The Measurement of the Inequality of Incomes. *The Economic Journal*, **30**:119, 348–361.\nd’Aspremont, A., L. El Ghaoui, M.I. Jordan, & G. R. G. Lanckriet (2007). A direct formulation for sparse PCA using semidefinite programming. *SIAM Review*, **49**:3, 434–448.\nDecancq, K. & M.-A. Lugo (2013), Weights in Multidimensional Indices of Well-Being: An Overview, *Econometric Reviews*, **32**:1, 7–34.\nDing, C., Zhou, D., He, X. & Zha, H. (2006). *R*1-PCA: rotational invariant L1-norm principal component analysis for robust subspace factorization. *ICML ’06 Proceedings of the 23rd international conference on Machine learning*, 281–288\nEckart, C. & G. Young (1936) The approximation of one matrix by another of lower rank. *Psychometrika*,**1**, 211–218.\nFlury & Riedwyl (1988). Multivariate Statistics: A Practical Approach. Chapman & Hall\nFurman, E. & R. Zitikis (2017), Beyond the Pearson Correlation: Heavy-Tailed Risks, Weighted Gini Correlations, and A Gini-Type Weighted Insurance Pricing Model, *ASTIN Bulletin: The Journal of the International Actuarial Association*, **47(03)**: 919-942.\nGajdos, T. and J. Weymark (2005), Multidimensional generalized Gini indices, *Economic Theory*, **26**:3, 471-496.\nGini, C. (1912), Variabilità e mutabilità, Memori di Metodologia Statistica, Vol. 1, Variabilità e Concentrazione. Libreria Eredi Virgilio Veschi, Rome, 211–382.\nGiorgi, G.M. (2013), Back to the future: some considerations on Shlomo Yitzhaki and Edna Schechtman’s book “The Gini Methodology: A Primer on a Statistical Methodology”, *Metron*, **71(2)**: 189-195.\nGorban, A.N., B. Kegl, D.C. Wunsch, & A. Zinovyev (Eds.) (2007) Principal Manifolds for Data Visualisation and Dimension Reduction. LNCSE 58, Springer Verlag.\nHotelling, H. (1933) Analysis of a complex of statistical variables into principal components. *Journal of Educational Psychology*, **24**, 417–441, 1933.\nKorhonen, P. & Siljamäki, A. (1998). Ordinal principal component analysis theory and an application. *Computational Statistics & Data Analysis*, **26**:4, 411–424.\nList, C. (1999), Multidimensional Inequality Measurement: A Proposal, *Working paper*, Nuffield College.\nMackey, L. (2009) Deflation methods for sparse PCA. *Advances in Neural Information Processing Systems*, **21**: 1017–1024.\nMardia, K, Kent, J. & Bibby, J. (1979). Multivariate Analysis. Academic Press, London.\nOlkin, Ingram, and Shlomo Yitzhaki. 1992. Gini regression analysis. *International Statistical Review* **60**: 185–96.\nPearson, K. (1901), On Lines and Planes of Closest Fit to System of Points in Space, *Philosophical Magazine*, **2**: 559–572.\nSaad, Y. (1998). Projection and deflation methods for partial pole assignment in linear state feedback.*IEEE Trans. Automat. Contr.*, **33**: 290–297.\nSchechtman, E. & S. Yitzhaki (1987), A Measure of Association Based on Gini’s Mean Difference, *Communications in Statistics: A*, **16**: 207–231.\nSchechtman, E. & S. Yitzhaki (2003), A family of correlation coefficients based on the extended Gini index, *Journal of Economic Inequality*, **1**:2, 129–146.\nShelef, A. (2016), A Gini-based unit root test. *Computational Statistics & Data Analysis*, **100**: 763–772.\nShelef, A., and E. Schechtman (2011), A Gini-based methodology for identifying and analyzing time series with non-normal innovations. *SSNR Electronic Journal* July: 1–26.\nTibshirani, R. (1996) Regression shrinkage and selection via the LASSO. *Journal of the Royal statistical society, series B*, **58**:1, 267–288.\n Yaari, M.E. (1987), The Dual Theory of Choice Under Risk, *Econometrica*, **55**: 99–115.\n Yaari, M.E. (1988), A Controversial Proposal Concerning Inequality Measurement, *Journal of Economic Theory*, **44**: 381–397.\nYitzhaki, S. (1991), Calculating Jackknife Variance Estimators for Parameters of the Gini Method. *Journal of Business and Economic Statistics*, **9**: 235–239.\nYitzhaki, S. (2003), Gini’s Mean difference: a superior measure of variability for non-normal distributions, *Metron*, **LXI(2)**: 285-316.\nYitzhaki, S. & Olkin, I. (1991). Concentration indices and concentration curves. *Institute of Mathematical Statistics Lecture Notes*, **19**: 380–392.\nYitzhaki, S. & E. Schechtman (2013), *The Gini Methodology. A Primer on a Statistical Methodology*, Springer.\nZou, H., Hastie, T. & R. Tibshirani (2006), Sparse Principal Component Analysis, *Journal of Computational and Graphical Statistics*, **15**:2, 265-286.\nAppendix\n========\n[h]\n|c ||c c c c c c| cars& capacity *x*1 &power *x*2 &speed *x*3 &weight *x*4 &width *x*5 & length *x*6 \nCitroën C2 1.1 Base &1124 &61 &158 &932 &1659 &3666 \nSmart Fortwo Coupé& 698 &52 &135 &730 &1515 &2500 \nMini 1.6 170 &1598 &170 &218 &1215 &1690 &3625 \nNissan Micra 1.2 65 &1240 &65 &154 &965 &1660 &3715 \nRenault Clio 3.0 V6 &2946 &255 &245 &1400 &1810 &3812 \nAudi A3 1.9 TDI &1896 &105 &187 &1295 &1765 &4203 \nPeugeot 307 1.4 HDI 70 &1398 &70 &160 &1179 &1746 &4202 \nPeugeot 407 3.0 V6 BVA &2946 &211 &229 &1640 &1811 &4676 \nMercedes Classe C 270 CDI &2685 &170 &230 &1600 &1728 &4528 \nBMW 530d& 2993 &218 &245 &1595 &1846 &4841 \nJaguar S-Type 2.7 V6 Bi-Turbo& 2720 &207 &230 &1722 &1818 &4905 \nBMW 745i &4398 &333 &250 &1870 &1902 &5029 \nMercedes Classe S 400 CDI &3966 &260 &250 &1915 &2092 &5038 \nCitroën C3 Pluriel 1.6i &1587 &110 &185 &1177 &1700 &3934 \nBMW Z4 2.5i &2494 &192 &235 &1260 &1781 &4091 \nAudi TT 1.8T 180 &1781 &180 &228 &1280 &1764 &4041 \nAston Martin Vanquish &5935 &460 &306 &1835 &1923 &4665 \nBentley Continental GT &5998 &560 &318 &2385 &1918 &4804 \nFerrari Enzo &5998 &660 &350 &1365 &2650 &4700 \nRenault Scenic 1.9 dCi 120 &1870 &120 &188 &1430 &1805 &4259 \nVolkswagen Touran 1.9 TDI 105 &1896 &105 &180 &1498 &1794 &4391 \nLand Rover Defender Td5 &2495 &122 &135 &1695 &1790 &3883 \nLand Rover Discovery Td5 &2495 &138 &157 &2175 &2190 &4705 \nNissan X-Trail 2.2 dCi &2184 &136 &180 &1520 &1765 &4455 \n---\n1. Université de Nîmes – e-mail: stephane.mussard@unimes.fr, Research fellow Grédi University of Sherbrooke, and Liser Luxembourg.[↩](#fnref1)\n2. See for an overview of the “Gini methodology”.[↩](#fnref2)\n3. See for the link between variability (risk) measures and the Gini correlation index.[↩](#fnref3)\n4. In most cases, PCA is performed on scaled (and centered) variables, otherwise variables with large scales might alter interpretations. Thus, it will make sense, later on, to assume that components of ${\\boldsymbol{X}}$ have identical distributions. At least the first two moments will be equal.[foot3][↩](#fnref4)\n5. but other norms have also been considered in statistical literature, such as the Froebenius norm in the Eckart-Young theorem, or the maximum of singular values – also called 2-(induced)-norm.[↩](#fnref5)\n6. In risk analysis *ν* ∈ (0, 1) denotes risk lover decision makers (max-max approach), whereas *ν* > 1 stands for risk averse decision makers, and *ν* → ∞ extreme risk aversion (max-min approach).[↩](#fnref6)\n7. By definition *G**M**D**ν*(**x**⋅ ℓ, **x**⋅ ℓ) ≥ 0 for all ℓ = 1, …, *K*. As we impose that **x**⋅ ℓ ≠ *c***1***N*, the condition becomes *G**M**D**ν*(**x**⋅ ℓ, **x**⋅ ℓ) > 0.[↩](#fnref7)\n8. We have: \n$$\\bar{\\mathbf{f}}\\_{\\cdot k} = 1/N\\sum\\_{i=1}^N f\\_{ik} = 1/N \\left[ \\sum\\_{i=1}^N {\\mathbf{z}}^{\\top}\\_{i\\cdot} \\mathbf{b}\\_{\\cdot k} \\right] = 1/N\\left[ \\sum\\_{i=1}^N {\\mathbf{z}}^{\\top}\\_{i\\cdot} \\right]\\mathbf{b}\\_{\\cdot k} = \\mathbf{0}.$$ \n[↩](#fnref8)\n9. Strictly speaking and suggests probability distortion functions Ψ : [0, 1] → [0, 1], which does not necessarily coincide to our case.[↩](#fnref9)\n10. As indicated by, the efficient Jackknife method may be used to find the variance of any *U*-statistics.[↩](#fnref10)\n11. An R markdown for Gini PCA is available: \nData from Michel Tenenhaus’s website (see also the Appendix): \n[↩](#fnref11)\n"},"data_id":{"kind":"string","value":"arxiv_0184775"}}}],"truncated":false,"partial":true},"paginationData":{"pageIndex":461,"numItemsPerPage":100,"numTotalItems":46503,"offset":46100,"length":100}},"jwt":"eyJhbGciOiJFZERTQSIsImtpZCI6IjVHZDBvd0g5MTM2eDZjc1FvbE1zcktNWUZoRVFoUm5PVVVybEpjOUhaUEEifQ.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc4NjE0MjEwMCwianRpIjoiZjhjZmVmZjMtZTk2ZC00NTg4LWFmNDYtYWNlZTQ0MzkxNjRlIiwic3ViIjoiL2RhdGFzZXRzL0x5dW4wOTEyL0xvbmdBQkMiLCJleHAiOjE3ODYxNDU3MDAsImlzcyI6Imh0dHBzOi8vaHVnZ2luZ2ZhY2UuY28ifQ.G0jnbuHDsna5G6k-6SeyNbNMnopnsY0fezAOOuRnlz9skx1-ian_PwmFOW0AeSfrxEtGFGPqUMvZhHf8DugTAg","displayUrls":true,"splitSizeSummaries":[{"config":"default","split":"train","numRows":46503,"numBytesParquet":2327876865}]},"discussionsStats":{"closed":0,"open":2,"total":2},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
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Near and full quasi-optimality of finite element approximations of stationary second-order mean field games
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### xx/xx/2024
We establish *a priori* error bounds for monotone stabilized finite element discretizatio... | arxiv_0184308 |
The growth of disks and bulges during hierarchical galaxy formation. I: fast evolution *vs* secular processes
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We present a theoretical model for the evolution of mass, angular momentum and size of galaxy disks ... | arxiv_0184312 |
However, these integrals are still computable that can yield relatively good numerical results. And as for the principal value integral, we consider
$$\label{eq:20}
I(s)=\lim\_{\epsilon\to 0}\Bigg\{\int\_{0}^{\frac{1}{2}-\epsilon}\frac{1}{t^s}\frac{\zeta^{\prime}}{\zeta}(\frac{1}{2}+t)dt+\int\_{\frac{1}{2}+\epsilon... | arxiv_0184313 |
THE CHIME FAST RADIO BURST PROJECT: SYSTEM OVERVIEW
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The Canadian Hydrogen Intensity Mapping Experiment (CHIME) is a novel transit radio telescope operating across the 400–800-MHz band. CHIME is comprised of four 20-m × 100-m semi-cylindrical paraboloid reflectors, e... | arxiv_0184320 |
Geometric Analysis of Reductions from Schlesinger Transformations to Difference Painlevé Equations
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We present two examples of reductions from the evolution equations describing discrete Schlesinger transformations of Fuch... | arxiv_0184322 |
& 12.1 & 14.0 & 4.2 & 4.3 & 0.6 & 0.7 & 1.7 & 2.0 & 2.4 & 2.7 & & & Y
I17220\_O2 & [-140, -102] & [ -86, 0] & 11.6 & 10.2 & 38.5 & 60.7 & 80.1 & 286.0 & 3.5 & 1.7 & 3.3 & 6.0 & 10.8 & 35.7 & 18.7 & 139.2 & & & Y
I17220\_O3 & [-146, -102] & [ -86, -62] & 32.8 & 1.4 & 114.8 & 4.0 & 249.7 & 6.8 & 3.0 & 5.6 &... | arxiv_0184326 |
FedHM: Efficient Federated Learning for Heterogeneous Models via Low-rank Factorization
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One underlying assumption of recent federated learning (FL) paradigms is that all local models usually share the same network architecture and s... | arxiv_0184327 |
and Petrovskii S (2009), Excitable population dynamics, biological control failure, and spatiotemporal pattern formation in a model ecosystem. Bulletin of Mathematical Biology,71, 863-887.
Morozov A and Petrovskii S (2013), Feeding on Multiple Sources: Towards a Universal Parameterization of the Functional Response of... | arxiv_0184333 |
Toward an accurate strongly coupled many-body theory within the equation-of-motion framework
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Non-perturbative aspects of the quantum many-body problem are revisited, discussed and advanced in the equation of motion framework. W... | arxiv_0184338 |
Crossed Andreev reflection in spin-polarized chiral edge states due to the Meissner effect
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We consider a hybrid quantum Hall-superconductor system, where a superconducting finger with oblique profile is wedged into a two-dimensio... | arxiv_0184346 |
6 (1.4) & 58.0 (2.4) & 3 & 6.6 (5.5, 7.9) & 34.0 (32.6, 35.5) & candidate
24 & WD 0637+477 & 36.2 (0.7) & 39.3 (2.1) & 3 & 3.0 (1.7, 4.6) & 30.9 (30.6, 31.2) & candidate
25 & WD 0641+438 & 41.8 (1.1) & 71.2 (3.1) & 3 & 19.9 (17.9, 21.9) & 46.2 (43.8, 48.6) & non-member
26 & WD 0743+442 & 33.6 (0.6) & 38.1 (1.7) &... | arxiv_0184349 |
� is swept from 0 to *π*, while in frequency they traverse in opposite directions from above to below *ω**m*. For future reference, it needs to be mentioned that *ϕ*ℓ in any case affects both the frequency and the loss of these modes.
Around the EP, *μ* ∼ *μ*EP we see in Fig. [fig3](b) a drastic change from the previou... | arxiv_0184351 |
Entanglement in the Quantum Hall Matrix Model
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Characterizing the entanglement of matrix degrees of freedom is essential for understanding the holographic emergence of spacetime. The Quantum Hall Matrix Model is a gauged *U*(*N*) matrix quantum mechanics with two matrices wh... | arxiv_0184353 |
*c*2) in addition to the private bit-pipe *x*3 over which the secret message is sent.
2. Use part of the public bit-pipe to distill the sources and generate a secret key.
3. Use the remainder of the public bit-pipe to send a secret message by using the secret key just generated as a one-time pad.
In the simple example ... | arxiv_0184360 |
Pseudogap and singlet formation in cuprate and organic superconductors
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The pseudogap phase occurring in cuprate and organic superconductors is analyzed based on the dynamical cluster approximation (DCA) approach to the Hubbard model. In this method ... | arxiv_0184364 |
Clifford algebra and the projective model of Elliptic spaces
============================================================
I apply the algebraic framework developed in to study geometry of elliptic spaces in 1, 2, and 3 dimensions. The background material on projectivised Clifford algebras and their application to Cayl... | arxiv_0184370 |
, approximately $\sim 4\times 10^6 \, \rm yr$, decrease only by a modest amount (roughly less than a factor of 10), i.e. the new abundances are $\rm X\_H=10^{-19} - 10^{-18}$ and $\rm X\_{He}=10^{-7}$.
General Remarks
---------------
To conclude, the experiment and arguments presented above suggest that traces of hydro... | arxiv_0184384 |
=1
Abstract
We develop an analytical framework and numerical approach to obtain the coefficient of self-diffusivity for the transport of a rarefied gas in channels in the limit of large Knudsen number. This framework provides a method for determining the influence of channel surface microstructure on the value of diffu... | arxiv_0184387 |
often important implications for the solution: a naked singularity may appear in the IR. Singularities are in general not allowed in saddle-point solutions, and typically they should be cloaked by regular horizons. The presence of an horizon reflects an effective coarse-graining in the underlying theory, and typically... | arxiv_0184407 |
Fast Landmark Localizationwith 3D Component Reconstruction and CNN for Cross-Pose Recognition
=============================================================================================
Two approaches are proposed for cross-pose face recognition, one is based on the 3D reconstruction of facial components and the othe... | arxiv_0184416 |
Vanishing capillarity limit of the Navier-Stokes-Korteweg system in one dimension with degenerate viscosity coefficient and discontinuous initial density
=========================================================================================================================================================
In the first... | arxiv_0184419 |
Weight-Preserving Bijections Between Integer Partitions and a Class of Alternating Sign Trapezoids
==================================================================================================
We construct weight-preserving bijections between column strict shifted plane partitions with one row and alternating sign... | arxiv_0184425 |
b] [table:email-features-short]
|l|c|c|lcc & & &
& FS & PT & & &
[c]@l@ Lexical Features (# of words, # of characters,
Function words, Tokens, Regular expressions, etc.)
& M & M & & &
Style metrics (number of paragraphs in the email, Yule metric, etc.)& S & S & &&
[c]@l@ Topic in the body
& S & S & &&
[c]@... | arxiv_0184431 |
First-principles quantum corrections for carrier correlationsin double-layer two-dimensional heterostructures
=============================================================================================================
We present systematic ab initio calculations of the charge carrier correlations between adjacent lay... | arxiv_0184436 |
Energy Flow Networks: Deep Sets for Particle Jets
=================================================
Introduction
============
Collisions at accelerators like the Large Hadron Collider (LHC) produce multitudes of particles. Particles are the fundamental objects of interest in collider physics and provide an interface be... | arxiv_0184443 |
to population P2, and a larger dispersion for population P2 compared to population P3, a trend that is in keeping with the more centrally concentrated distribution of population P3 reported by . The observed dispersion profiles decrease with increasing radius within the range of our data.
A clear rotation signature is... | arxiv_0184444 |
Spontaneous chiral symmetry breaking in holographic soft wall models
====================================================================
Introduction
============
The investigation of spontaneous symmetry breaking (SSB) is a very interesting subject, essential in condensed matter and particle physics contexts. It is r... | arxiv_0184452 |
K\_{\text{max}}}
\text{d}K \int\_{-1}^{1} \text{d}\mu \, K^2 \tilde{F}\_{\rm{a}\rm{b}}(K, \mu, z).$$
Here *V* is the survey volume, *K*min and *K*max are the minimum and maximum moduli of the modes probed, and we already integrated over the azimuthal direction, supposing no dependence from it in the integrand.
For o... | arxiv_0184454 |
refers to the *j*-th state (action) in process *i*’s local history An event is a tuple ⟨*s*, *α*, *s*ʹ⟩ consisting of a state, an action, and a state. The *j*th event in process *i*’s history is *e**i**j* denoting ⟨*s**i**j* − 1, *α**i**j*, *s**i**j*⟩.
An asynchronous system consists of the following sets.
1. A set *P... | arxiv_0184459 |
Sample-Efficient Sparse Phase Retrieval via Stochastic Alternating Minimization
===============================================================================
In this work we propose a nonconvex two-stage tochastic lternating inimizing (SAM) method for sparse phase retrieval. The proposed algorithm is guaranteed to ha... | arxiv_0184460 |
well suited to this investigation.
Data: Cassini Spacecraft Precision Ranging
==========================================
The range between the Cassini spacecraft and ground stations on Earth can be determined from the communication time between the ground stations and the spacecraft. In particular, the range *R* is gi... | arxiv_0184461 |
Panchromatic spectral energy distributions of *Herschel* sources [1](#fn1)
==========================================================================
### Received 06/12/2012; accepted 16/01/2013
Introduction
============
The advent of wide-field surveys on the one hand, and deep pencil-beam observation of selected blan... | arxiv_0184469 |
x,y)& =\epsilon^{-1}K^{\rm geo}\_{21}(k,\ell), &\quad K^\epsilon\_{22}(x,y) &= \epsilon^{2n}K^{\rm geo}\_{22}(k,\ell).\end{aligned}$$
Then the following result is straightforward and does not require complicated asymptotic analysis.
[prop:ConvGeom] Uniformly for *x*, *y* in a bounded set of R+, we have
lim*ε* → 0*... | arxiv_0184470 |
Pseudogap phenomena in ultracold atomic Fermi gases
===================================================
The pairing and superfluid phenomena in a two-component ultracold atomic Fermi gas is an analogue of Cooper pairing and superconductivity in an electron system, in particular, the high *T**c* superconductors. Owing t... | arxiv_0184471 |
A Hilbert manifold structure on the refined Teichmüller space of bordered Riemann surfaces
==========================================================================================
We consider bordered Riemann surfaces which are biholomorphic to compact Riemann surfaces of genus *g* with *n* regions biholomorphic to t... | arxiv_0184472 |
On the detection of TeV *γ*-rays from GRB with km3 neutrino telescopes — I. Muon event rate from single GRBs
============================================================================================================
[firstpage]
This is a preliminary study to examine the prospect of detecting TeV photons from *γ*-ray ... | arxiv_0184480 |
to be about 0.18 fm− 3 ( − 22.9MeV) with Φʹ ≈ 2.70 and *g* ≈ − 0.817, introducing a relative improvement about 10% (14%) on *ρ*0 (*E*0(*ρ*0)) compared to the QCDSR-2 prediction (0.2 fm− 3 and − 26.7MeV). On the other hand, with a larger Φʹ ≈ 4.00 and correspondingly *g* ≈ − 0.552, the saturation density (the corres... | arxiv_0184482 |
, showing that surface barrier for vortex entry is significantly reduced with a N layer in contact to the surface of the superconductor.
In nanostructures, multiquanta vortices can nucleate under appropriate conditions, as well as vortex-antivortex pairs. This can significantly alter vortex core shapes. In principle ST... | arxiv_0184486 |
On the *M**t*/*M**t*/*K**t* + *M**t* queue in heavy traffic
===========================================================
The focus of this paper is on the asymptotics of large-time numbers of customers in time-periodic Markovian many-server queues with customer abandonment in heavy traffic. Limit theorems are obtained f... | arxiv_0184491 |
matched with each other. This is less related to our use of the notion of “sampling dynamics,” which is in line with the literature cited above.[↩](#fnref9)
10. See Appendix [sub-standard-definitions] for the formal definitions of Lyapunov and asymptotic stability. Lyapunov stability requires that populations starting... | arxiv_0184493 |
Illumination strategies for intensity-only imaging
==================================================
We propose a new strategy for narrow band, active array imaging of localized scatterers when only the intensities are recorded and measured at the array. We consider a homogeneous medium so that wave propagation is ful... | arxiv_0184497 |
On the origin of hard X-ray emissions from the behind-the-limb flare on 2014 September 1
========================================================================================
The origin of hard X-rays and *γ*-rays emitted from the solar atmosphere during occulted solar flares is still debated. The hard X-ray emissio... | arxiv_0184503 |
Y}\rightarrow\mathcal{Y}$. We now set
$$\boldsymbol{G}^{n}=\underbrace{\underset{n-times}{\boldsymbol{G}\circ\boldsymbol{G\circ\cdots\circ G}}}.$$
We show that for *n* sufficiently large $\boldsymbol{G}^{n}$ is a contraction. We first notice that
$$\left\Vert \boldsymbol{G}g(t)-\boldsymbol{G}g(t)\right\Vert \_{... | arxiv_0184505 |
*IEEE Transactions on Software Engineering* **12**, 662–675.
D.L. Eager, E.D. Lazokwska and J. Zahorjan (1986). A comparison of receiver-initiated and sender-initiated adaptive load sharing. *Performance Evaluation Review* **6**, 53–68.
D.L. Eager, E.D. Lazokwska and J. Zahorjan (1988). The limited performance benefit... | arxiv_0184509 |
Universal Reverse Information Projections and Optimal `E`-statistics
====================================================================
Information projections have found important applications in probability theory, statistics, and related areas. In the field of hypothesis testing in particular, the reverse informat... | arxiv_0184519 |
[tab:xsecfit] with m_{{W_R}}=1900 GeV. The region compatible with data at 68% (90%) CL is shown with a black solid (dotted) line. The white dashed line denotes the theoretical limit \kappa > s_W / c_W \simeq 0.55. The star represents the best fit point. Left: fit including the ATLAS leptonic tb search that systematical... | arxiv_0184523 |
Multiple components in the molecular outflow of the red supergiant NML Cyg
==========================================================================
### Accepted 2021 November 04. Received 2021 November 04; in original form 2021 September 14
[firstpage]
Despite their large impact on stellar and galactic evolution, the... | arxiv_0184525 |
T\_H},$$
where one plugs in the correct flat space Hagedorn temperature for the type of string (bosonic, type II or heterotic) one is considering.
Usually, the Hagedorn temperature is determined when the circumference of the singly wound string around the thermal circle becomes too small. In formulas
$$\text{cir... | arxiv_0184539 |
Solitons in Peyrard–Bishop model of DNA and the Renormalization group method
============================================================================
We investigate solitons in Peyrard–Bishop model of DNA molecule using Renormalization group methods which provide a systematic way for perturbation analysis. Small am... | arxiv_0184540 |
Multi-finger binary search trees
================================
We study multi-finger binary search trees (BSTs), a far-reaching extension of the classical BST model, with connections to the well-studied *k*-server problem. Finger search is a popular technique for speeding up BST operations when a query sequence has ... | arxiv_0184542 |
performed their measurements at elevated temperatures using various high laser powers. Considering their laser power $P\_{\rm abs}$ and the focus size *r*0 in Eq. (Appendix), the associated temperatures may be estimated to be as high as 650K. The huge heating Δ*T* implies an inhomogeneous broadening of the phonon lin... | arxiv_0184550 |
Wide-band spectroscopy of the Compton thick Seyfert 2 galaxy Mrk 3 with Suzaku
==============================================================================
We obtained a wide-band spectrum of the Compton-thick Seyfert 2 galaxy Mrk 3 with Suzaku. The observed spectrum was clearly resolved into weak, soft power-law emi... | arxiv_0184566 |
and *y* ∈ *Y**D* are located on a same line.
Clearly $K\_\eta={\bf C}\_X\cap H\_\eta$ is the polytope generated by *X*. Let *V**η* be the vertex set of *K**η*. Clearly *V**η* ⊂ *X*. Similarly we define *K**η*ʹ and *V**η*ʹ.
For any *x* ∈ *V**η*, there is a point *y* ∈ *V**η*ʹ such that *x* and *y* are located on a same... | arxiv_0184570 |
Higher Quantum Theory
=====================
### July 19, 2012
Introduction
============
Overview
--------
### The formalism
This paper presents a new formalism for describing flows of quantum and classical information. The key physical insight, which is by no means new, is that classical information can be encoded in c... | arxiv_0184573 |
Modeling Dense Star Clusters in the Milky Way and Beyond with the Cluster Monte Carlo Code
==========================================================================================
We describe the public release of the Cluster Monte Carlo Code (`CMC`) a parallel, star-by-star *N*-body code for modeling dense star clus... | arxiv_0184583 |
**Slow and fast minimal speed traveling waves of the FKPP equation with chemotaxis**
====================================================================================
We examine a general model for the Fisher-KPP (FKPP) equation with nonlocal advection. The main interpretation of this model is as describing a diffus... | arxiv_0184589 |
716** 30 (2012).
G. Degrassi *et al.*, JHEP 1208 (2012), 098; D. Buttazzo *et al.*, JHEP 1312, 089 (2013). J. F. Gunion and H. E. Haber, Phys. Rev. D **67**, 075019 (2003)
I. Masina and A. Notari A., Phys. Rev. D **85** 123506 (2012); Isidori G. *et al.*, Phys. Rev. D **77** 025034 (2008); F. L. Bezrukov and M. Shaposh... | arxiv_0184590 |
Unified characterizations of minuscule Kac–Moody representations built from colored posets[1](#fn1)
===================================================================================================
### June 17, 2019
1.1
R.M. Green described structural properties that “doubly infinite” colored posets should possess so... | arxiv_0184594 |
Flocking with discrete symmetry: the 2d Active Ising Model
==========================================================
We study in detail the active Ising model, a stochastic lattice gas where collective motion emerges from the spontaneous breaking of a discrete symmetry. On a 2d lattice, active particles undergo a diff... | arxiv_0184597 |
Maximal chain descent orders
============================
This paper introduces a partial order on the maximal chains of any finite bounded poset *P* which has a CL-labeling *λ*. We call this the maximal chain descent order induced by *λ*, denoted *P**λ*(2). As a first example, letting *P* be the Boolean lattice and *λ... | arxiv_0184600 |
and 3FGL J2032.2+4126 shows that, below 1 GeV, the extrapolated flux from FGL J2032.2+4128e would be stronger than 3FGL J2032.2+4126. This is clearly not the case as it has not been detected in lower energy analyses. It is likely, therefore, that at low energies some of the emission from 3FGL J2032.2+4126 and potentia... | arxiv_0184617 |
potential, we have obtained exact solution of the relativistic (2+1) dimensional Dirac-Landau problem. The exact solutions are given by orthogonal Romanovski polynomials and only finite number of possible bound state solutions are permitted. The eigenenergy spectrum is modified with negative shift as compared to the s... | arxiv_0184625 |
one of the many reported subgroups of $Pm\bar{3}m$ due to structural distortions or cation orderings, which themselves can be classified as transforming as irreps of $Pm\bar{3}m$. However, from a materials design perspective, it is most convenient to always list these couplings with respect to the aristotypical symmet... | arxiv_0184638 |
Which galaxy property is the best gauge of the oxygen abundance?
================================================================
We present an extensive exploration of the impact of 29 physical parameters in the oxygen abundance for a sample of 299 star-forming galaxies extracted from the extended CALIFA sample. We co... | arxiv_0184639 |
UAV Swarm-Enabled Aerial CoMP: A Physical Layer Security Perspective
====================================================================
Unlike aerial base station enabled by a single unmanned aerial vehicle (UAV), aerial coordinated multiple points (CoMP) can be enabled by a UAV swarm. In this case, the management of... | arxiv_0184641 |
Time reversal dualities for some random forests
===============================================
We consider a random forest $\mathcal F^\*$, defined as a sequence of i.i.d. birth-death (BD) trees, each started at time 0 from a single ancestor, stopped at the first tree having survived up to a fixed time *T*. We denote ... | arxiv_0184647 |
Random Multi-Hopper Model. Super-Fast Random Walks on Graphs
============================================================
We develop a model for a random walker with long-range hops on general graphs. This random multi-hopper jumps from a node to any other node in the graph with a probability that decays as a function ... | arxiv_0184655 |
Continuous-Time Quantum Walks:Models for Coherent Transport on Complex Networks
===============================================================================
This paper reviews recent advances in continuous-time quantum walks (CTQW) and their application to transport in various systems. The introduction gives a brief... | arxiv_0184666 |
Δ*P*/*P* = 0.1 and *P* ∝ *t*− 1/2, then this leads to an age uncertainty of 20 per cent if only a single measurement of *P* is available.
The assumption of convergence to a unique I-sequence is approximate. Convergence takes longer at lower masses and so the older and more massive the star, the better this approximati... | arxiv_0184669 |
Sublinearly Morse geodesics in CAT(0) spaces: lower divergence and hyperplane characterization
==============================================================================================
We introduce the notion of *κ*-lower divergence for geodesic rays in CAT(0) spaces. Building on work of Charney and Sultan we give... | arxiv_0184674 |
Toroidal magnetic fields in type II superconducting neutron stars
=================================================================
We determine constraints on the form of axisymmetric toroidal magnetic fields dictated by hydrostatic balance in a type II superconducting neutron star with a barotropic equation of state.... | arxiv_0184682 |
ion rates around $\sim{\rm few}\times10^{-7}$ M⊙ yr− 1. It has been argued that this is the regime allowing the most efficient build-up of mass of the WD. Due to the energy released in the hydrogen fusion, the WD becomes a powerful source of soft X-ray emission, the so-called supersoft X-ray source (SSS; ; ). At larger... | arxiv_0184684 |
**Demographics of young stars and their protoplanetary disks: lessons learned on disk evolutionand its connection to planet formation**
=======================================================================================================================================
= 11pt = 1.5cm = 1.5cm Since Protostars and Plan... | arxiv_0184685 |
then ${\gamma\_t}(G) \le \frac{2}{5}n$.
**Proof.** Let *G* be a quadrilateral-free graph of order *n* with *δ*(*G*) ≥ 4 and let *H**G* be the ONH of *G*. Then, each edge of *H**G* has size at least 4. Since *G* is contains no 4-cycle, the hypergraph *H**G* contains no overlapping edges and is therefore linear. Let *H*... | arxiv_0184688 |
| |
| --- |
| Department of Statistics⋆, and |
| Department of Electrical Engineering and Computer Sciences‡ |
| UC Berkeley, Berkeley, CA |
| |
| |
| --- |
| Department of Electrical Engineering and Computer Sciences† |
| Department of Mathematics‡ |
| Massachusetts Institute of Technology, Cambridge, MA |
We prese... | arxiv_0184689 |
Alternative experimental ways to access entropy production
==========================================================
[1](#fn1)
[2](#fn2)
We theoretically derive and experimentally compare several different ways to access entropy production in a quantum process under feedback control. We focus on a bipartite quantum sy... | arxiv_0184693 |
of non-Gaussianities in the CMB maps .
The impact of most of the aforementioned effects on the CMB lensing analysis with WMAP data has been investigated by . The fact that they found a negligible contamination level is encouraging. However, such a result could change when one considers the higher resolution, better se... | arxiv_0184696 |
Hybrid Discontinuous Galerkin methods with relaxed H(div)-conformity for incompressible flows. Part I[3](#fn3)
==============================================================================================================
We propose a new discretization method for the Stokes equations. The method is an improved version... | arxiv_0184699 |
The Spatial Distribution of Type Ia Supernovae within Host Galaxies
===================================================================
We study how type Ia supernovae (SNeIa) are spatially distributed within their host galaxies, using data taken from the Sloan Digital Sky Survey (SDSS). This paper specifically tests t... | arxiv_0184706 |
* which contributes to the integral ([gammaone]) is $\sim {\rm log} \left(
\frac{C}{s\_{2}s\_{3}}\right) $ where *C* is a numerical constant of the order one. This is why we expect ([gammaone]) to contain only the simple logarithms ${\rm log}(s\_{2})$, ${\rm log}(s\_{3})$. This is a natural result for the one-instanton... | arxiv_0184707 |
Algebraic Families of Galois Representations and Potentially Semi-Stable Pseudodeformation Rings
================================================================================================
We construct and study the moduli of continuous representations of a profinite group with integral *p*-adic coefficients. We p... | arxiv_0184715 |
ST1]
[YbPtBiST2]
[YbPtBiST3]
[YbPtBiSH1]
[YbPtBiSH2]
[YbPtBiTEPPhase]
[YbPtBiPhase1]
[YbPtBiPhase2]
[YbPtBiPhase3]
[YbPtBiKW]
[YbPtBiRsimulation]
[YbPtBiKW1]
[YbPtBiGlobal]
Magnetic field tuned quantum criticality of heavy fermion system YbPtBi
==============================================================... | arxiv_0184716 |
Exact Master Equation and General Non-Markovian Dynamicsin Open Quantum Systems
===============================================================================
Investigations of quantum and mesoscopic thermodynamics force one to answer two fundamental questions associated with the foundations of statistical mechanics: ... | arxiv_0184717 |
700 ± 54 & 1601 ± 224
NGC5548 & 4354 ± 25 & 12402 ± 111 & 3900 ± 266 & 12539 ± 1927 [table:Hbetawidth]
ANALYSIS AND RESULTS
====================
Testing the Assumptions of SE BH Mass Estimators
------------------------------------------------
Single-epoch $M\_{\rm BH}$ estimates are based on the “virial” assumption ... | arxiv_0184719 |
process responsible for internal logging of information not necessarily contained in feedback through the standard hub connection. This allows detailed information from various processes to be archived.
Instrument Control Computer
---------------------------
The ICS resides in a 19-inch computer rack adjacent to the i... | arxiv_0184725 |
all, we note that at *e**B* = 0, these term will vanish. With the increase of the magnetic field, the eB-dependent vacuum term also increases and the contribution of Π*β* is more than Π*α*.
Having obtained the real and imaginary parts of the self energy, we now proceed to evaluate the in-medium spectral functions of *... | arxiv_0184729 |
Spin-orbit torques and spin Hall magnetoresistance generated by twin-free and amorphous Bi0.9Sb0.1 topological insulator films
==============================================================================================================================
Dr. F. Binda, S. Fedel, Dr. S. F. Alvarado, Dr. P. Noël, Prof. P. ... | arxiv_0184731 |
clash is met, this should also involve a special choice of boundary conditions or there must be a linearization instability of AdS vacuum in the theory.[↩](#fnref6)
7. This corresponds to the sign choice *ε**t**x**y* = − 1.[↩](#fnref7)
Causality in 3D Massive Gravity Theories
========================================... | arxiv_0184734 |
areas of optical imaging, they are generally difficult to implement in practice. Hence, the development of algorithms for signal recovery from magnitude measurements is still a very active field of research. Existing methods for phase retrieval rely on all kinds of a priori information about the signal, such as positi... | arxiv_0184742 |
Automated Refactoring of Legacy JavaScript Code to ES6 Modules
==============================================================
The JavaScript language did not specify, until ECMAScript 6 (ES6), native features for streamlining encapsulation and modularity. Developer community filled the gap with a proliferation of desig... | arxiv_0184743 |
m*\*(1 − max*u* ∈ U*φ**u*)− 1, which can provide significant computational benefits.
Thus, we obtain a three-strep approach to compute an approximate solution to the network monitoring problem and evaluate its optimality gap. Next, we further study our solution for problem instances with particular characteristics.
Spe... | arxiv_0184745 |
Three-boson bound states in Minkowski space with contact interactions
=====================================================================
The structure of the three-boson bound state in Minkowski space is studied for a model with contact interaction. The Faddeev-Bethe-Salpeter equation is solved both in Minkowski and... | arxiv_0184754 |
addtoresetequationsection
NIKHEF-2015-027
CPHT-RR025.0815
**Dyonic ISO(7) supergravity and the duality hierarchy**
Adolfo Guarino1 and Oscar Varela2, 3
*1Nikhef Theory Group, Science Park 105, 1098 XG Amsterdam, The Netherlands*
*2 Center for the Fundamental Laws of Nature,
Harvard University, Cambridge, MA 021... | arxiv_0184761 |
**Abstract**
This book uses the modern theory of artificial intelligence (AI) to understand human suffering or mental pain. Both humans and sophisticated AI agents process information about the world in order to achieve goals and obtain rewards, which is why AI can be used as a model of the human brain and mind. This b... | arxiv_0184763 |
Synchrotron emission from virial shocks around stacked OVRO-LWA galaxy clusters
===============================================================================
### Accepted —. Received —; in original —
[firstpage]
01 0.2in
Galaxy clusters accrete mass through large scale, strong, structure-formation shocks. Such a viri... | arxiv_0184766 |
As a simple example of cross-model fairness consider the classification scenario depicted in Figure [fig:lineramodelmultiplicity] shown in Section [sec:background] where three distinct predictors from the family of linear classifiers achieve 100% *accuracy* with respect to the displayed *validation set*. While models ... | arxiv_0184767 |
Chandra observations of outflows from PSR J1509–5850
====================================================
PSR J1509–5850 is a middle-aged pulsar with the period *P* ≈ 89 ms, spin-down power $\dot{E}=5.1\times 10^{35}$ erg s− 1, at a distance of about 3.8 kpc. We report on deep *Chandra X-ray Observatory* observations o... | arxiv_0184770 |
Order algebras: a quantitative model of interaction
===================================================
### July 7, 2011
A quantitative model of concurrent interaction is introduced. The basic objects are linear combinations of partial order relations, acted upon by a group of permutations that represents potential non... | arxiv_0184772 |
A. 2010, ApJ, 712, 294 Evans, N. J., II 2008, Pathways Through an Eclectic Universe, 390, 52 Fraternali, F. & Binney, J. J. 2008, MNRAS, 386, 935 Frisch U. 1995, Turbulence – The Legacy of A.N. Kolmogorov, Cambridge University press Gil de Paz, A., et al. 2007,, 173, 185 Gisler, G.R. 1979, ApJ, 228, 385 Gomez G.C. & C... | arxiv_0184773 |
*6
*x*1 & **1.000** & 0.954 & 0.885 & 0.692 & 0.706 & 0.663
*x*2 & 0.954 &**1.000** & 0.933 & 0.528 & 0.729 & 0.663
*x*3 & 0.885 & 0.933& **1.000** & 0.466 & 0.618 & 0.578
*x*4 & 0.692 & 0.528 & 0.466 & **1.000** & 0.477 & 0.794
*x*5 & 0.706 & 0.729 & 0.618 & 0.477 & **1.000** & 0.591
*x*6 & 0.663 & 0.663 &... | arxiv_0184775 |
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