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The transitive path hypothesis can be justified if our knowledge about a system is limited to pairwise interactions. Thus, we should view a network model as a maximum entropy model of paths |
in a complex system, much like a mean-field model can be seen as a maximum entropy model for the topology of direct interactions. From a statistical point of view, this interpretation of a network model corresponds to the assumption that the links [MATH] that contribute to a path [MATH] are statistically independent or... |
, temporal patterns in trade relations , navigation paths of humans in information networks , and traces of dynamical processes in networked systems |
. Similar to how the surge of data on direct interactions has questioned mean-field models, these studies of actual paths in complex systems show that standard network models can blur our understanding of critical nodes and functional modules |
After briefly introducing the higher-order network model framework, this progress article provides examples where higher-order network models offer new and critical insights into the systems that they represent by asking: How do path structures in complex systems affect the applicability of popular network science meth... |
Higher-order models of interactions in complex systems Owing to the different systems and data under study, the growing community working with higher-order network models has taken a multi-pronged approach: One stream of works builds on the observation that many systems exhibit many-body interactions , which require ge... |
Examples include triangles that are known to be fundamental building blocks of social networks , cliques in scientific coauthorship networks |
, feed-forward loop network motifs in biochemical transcription networks , and spatial coexistence relations between species in an ecosystem |
, as well as trigenic interactions in gene regulatory networks Another line of research acknowledges that the links in many complex systems can be of multiple types, which calls for multi-layer generalisations of network models |
Examples include multi-modal transportation systems , interdependent layers of power and communication infrastructures , and multi-layer financial networks |
And finally, the increasing quantity of high-resolution time-series data can be used to infer temporal sequences or paths of interactions between more than two components in a system. Examples include data on scholarly citation networks |
, time-stamped social networks , and patient pathways in hospital networks , as well as human trajectories in transportation and information networks |
Despite differences in motivation and mathematical underpinning, these approaches have one aspect in common: They account for the fact that standard network models are too simple to explain the complex paths of influence captured in the growing volume of rich data on biological, technical, economic, and social systems.... |
Focusing on pairwise interactions, a standard network model represents the link topology of the underlying system (Fig. b). This representation discards information on how links contribute to paths, implicitly implying that nodes can indirectly influence each other via transitive paths that traverse nodes in a memoryle... |
Figure d illustrates this idea with a second-order model that accounts for the topology of paths of length two. In the spirit of higher-order Markov chain models, this model can be represented with a memory network |
, where state nodes represent states in a second-order state space and links encode possible transitions between states. Depending on the topology of paths, each of the five physical nodes |
[MATH] [MATH] [MATH] [MATH] , and [MATH] , which typically are the object of interest in the real world, has one or more state nodes. These state nodes enable efficient higher-order network models of paths: A path described by a Markovian model on the state nodes, which directs the path from one state node to the next ... |
Non-Markovian paths and community detection Community detection is an umbrella term for a large number of algorithms that group nodes into modules to simplify and highlight essential structures in the network topology. Since higher-order representations can capture more complex forms of interactions, community detectio... |
. Independent of where the flows come from, they continue to another journal proportionally to the number of citations to that journal among all articles published in the multidisciplinary journal. As a consequence, citation flows from different fields mix and move in a non-realistic way from one field to another. For ... |
In contrast, a second-order Markov representation of citations flows, which takes into account where citations come from, captures the fact that most citation flows coming to Nature from one field return to the same field (Fig. b,d). For example, when going from a first- to a second-order Markov representation, the rel... |
. Moreover, the citation flows that do not return move in more realistic ways: Figures b and d illustrate how citation flows from the Journal of Microbiology and the Journal of Bacteriology in microbiology mostly return to either journal and, similarly, how citation flows from Plant Cell and Plant Physiology in plant s... |
Non-Markovian paths and node centralities The development of algorithms to identify important nodes is one of the major success stories in network science. They help us to locate critical elements in networked infrastructures, identify influential actors in social systems, or find relevant pages in the World Wide Web. ... |
These methods assume that node centrality can be characterised based on the topology of pairwise interactions between system components. However, in social systems, information is shared via specific paths that are influenced by context, preferences, and time and that cannot be understood merely based on the network to... |
To truly capture the importance of nodes, we must go beyond network models and account for the complex structure of paths in high-dimensional, time-resolved data. An example is shown in Figure , which is based on time-stamped social interactions between the developers in a major Open Source Software community. The netw... |
But is this a good estimate for the actual importance of developers? We can answer this question by inferring causal paths in the underlying time-series data That is, we consider which paths actually exist based on the chronological ordering and timing of time-stamped interactions. In a nutshell, for a sequence of two ... |
[MATH] only exists if [MATH] occurs before [MATH] Hence, time-stamped network data allow us to calculate causal path statistics that may or may not be consistent with the assumption of transitive, Markovian paths that are implicitly made when using a network model |
In the example shown in Fig. , a calculation of betweenness centralities based on actual shortest causal paths considerably shifts the importance of nodes. The alluvial diagrams in Fig. a and b visualise these differences, revealing that the shortest causal paths passing through node 32 are considerably more constraine... |
As a result, node 32 is less central than we would assume, based on the network topology. In contrast, node 26, which ranks among the least central nodes from a topology perspective, turns out to be the most important node in terms of causal paths in the temporal interaction sequence. |
Higher-order models open new ways to address these limitations of existing centrality measures. We can, for instance, generalise path-based centrality measures like betweenness or closeness to higher-dimensional De Bruijn graphs, which like memory networks have nodes that represent sequences of length [MATH] and links ... |
Similarly, spectral measures such as PageRank or eigenvector centrality can be redefined based on eigenvectors of linear operators derived from De Bruijn graphs or memory networks |
Such novel measures help us to quantitatively assess the true importance of elements in a complex system, considering a system’s network topology as well as temporal patterns in non-Markovian paths. |
Non-Markovian paths and dynamical processes Along with enabling us to reason about topological features such as community structures or node centralities, network science has improved our understanding of how a system’s topology influences dynamical processes, and thus a system’s function Much of this research is based... |
While it has provided powerful tools to relate the structure and dynamics of complex systems, the assumption of transitive, Markovian paths implicitly introduced by the application of algebraic methods is not justified in many real systems. Figure a illustrates an example of such a complex system, which shows a network... |
These patterns considerably change the evolution of dynamical processes in the system and limit what a system’s topology can tell us about the robustness and efficiency of real transportation networks. |
Higher-order models help us to overcome these limitations of standard network methods. We can, for instance, generalise Laplacian and transition matrices to higher-dimensional De Bruijn graph models |
that capture a system’s causal topology , which is due to the non-Markovian characteristics of paths. Such higher-order network representations are the basis by which we can apply methods from the study of dynamical systems, such as eigendecompositions, spectral analysis or stability theory, to systems with non-Markovi... |
Perspectives To explain the properties of complex systems, it is important to understand how a system’s components influence each other. Network science provides powerful computational and analytical tools to address this challenge based on network abstractions of direct, pairwise interactions. This remarkably simple a... |
An important epistemological challenge is to find new ways to infer optimal models of complex systems, given high-dimensional data. Referring to Ockham’s razor, such models should be maximally parsimonious. That is, we want to limit our assumptions to enable generalisable statements that go beyond the concrete system u... |
, but little is known about how this challenge can be solved for other types of models and data. The fact that the size of higher-order models can grow quickly as model dimensionality increases introduces both statistical and scalability challenges that must be addressed. Reliable inference of models with high orders p... |
and multi-order models , as well as model order reduction techniques , to generate computationally tractable models that neither under- nor overfit the data. |
In addition to these methodological issues, the study of non-Markovian paths also foreshadows a new class of higher-order generative network models similar to, for example, the stochastic block model |
, the Watts-Strogatz model or the Barabási-Albert model In network science, such generative models play a crucial role both in the detection of structural patterns in networks and in the identification of maximally simple mechanisms from which they emerge. Little is known about the mechanisms by which similar non-Marko... |
Finally, there are broad perspectives for research on the relationship between higher-order network models and dynamical processes. While most research has focused on linear diffusive processes, some works have considered the impact of non-Markovian paths on metapopulation models, akin to reaction-diffusion on networks |
The properties of general non-linear processes with non-Markovian interaction paths are mostly unexplored. These include coupled dynamical systems, for example, oscillators, on higher-order networks, the identification of conditions that allow local and non-local patterns to emerge when standard tools such as master st... |
, and the use of higher-order models to study network control Higher-order modelling techniques allow us to leverage existing network methods, extending them toward optimal models in order to explain the interaction topology in complex systems. Such models and techniques not only create new opportunities for interdisci... |
# Source: arxiv 1806.06236 # Title: Divisibility in *N and $βN$ # Sections: all # Downloaded: 2026-03-03T02:35:34.758918+00:00 Divisibility in [MATH] and [MATH] |
Boris Šobot Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad, Trg Dositeja Obradovića 4, 21000 Novi Sad, Serbia |
e-mail: sobot@dmi.uns.ac.rs ORCID: 0000-0002-4848-0678 Abstract The paper first covers several properties of the extension of the divisibility relation to a set [MATH] of nonstandard integers, including an analogue of the basic theorem of arithmetic. After that, a connection is established with the divisibility in the ... |
2010 Mathematics Subject Classification : 11U10, 03H15, 54D35, 54D80 Key words and phrases : divisibility, nonstandard integer, Stone-Čech compactification, ultrafilter |
In four different ways to extend the divisibility relation on [MATH] to the Stone-Čech compactification [MATH] were introduced. One of them, the relation [MATH] , seems to be the most fruitful for investigation, and some of its properties were extracted in |
. In this paper we prove more about it, using mostly the connection between [MATH] and nonstandard extensions [MATH] of [MATH] The first section covers most of the facts needed for the rest of the paper about nonstandard extensions, [MATH] and the connection between them. In the second section we single out some facts ... |
Introduction Nonstandard methods. In the course of the last 60 years many approaches to nonstandard extensions have been developed. One general idea is to work with a particular construction of nonstandard universe, most frequently an ultrapower or an ultralimit. The other is to give an axiomatic development of the not... |
contains an overview of various historically relevant axiomatic systems. In this paper we mostly follow the Robinson-Zakon superstructure approach as exposed in Henson’s chapter |
, a text highly recommended for mathematicians new to the subject. Let [MATH] be a set; we assume that elements of [MATH] are atoms: none of them contains as an element any of the others. Let [MATH] [MATH] for [MATH] and [MATH] . The rank of [MATH] is the smallest [MATH] such that [MATH] [MATH] is called a superstructu... |
Let [MATH] be a superstructure. Its nonstandard extension is a pair [MATH] , where [MATH] is a superstructure with the set of atoms [MATH] and [MATH] is a rank-preserving function such that [MATH] and satisfying the following principle. |
The Transfer Principle. For every bounded formula [MATH] and every [MATH] [MATH] [MATH] holds in [MATH] if and only if [MATH] holds in [MATH] |
(A first-order formula is bounded if all its quantifiers are bounded, i.e. of the form [MATH] or [MATH] . The free variables that appear in [MATH] are exactly objects [MATH] from [MATH] and in [MATH] they are replaced with their star-counterparts. The atomic subformulas in [MATH] are of the form [MATH] for some [MATH] ... |
We may abuse the notation and call [MATH] the extension of [MATH] , or even call [MATH] the extension of [MATH] . We restrict ourselves to nonstandard arithmetic, i.e. extensions of the set [MATH] of natural numbers (including zero). Objects of the form [MATH] for [MATH] are called standard . For each [MATH] , the elem... |
To every [MATH] -ary relation [MATH] on [MATH] corresponds a [MATH] -ary relation [MATH] on [MATH] ; the same holds for relations of a higher rank of the superstructure hierarchy (such as relations on subsets of [MATH] ). To every operation [MATH] corresponds a [MATH] -ary operation [MATH] on [MATH] . We will frequentl... |
, Remark 5.1). We learn about properties of such relations and functions mainly from the transfer principle. For example, we will use without mention facts such as [MATH] for [MATH] |
An object [MATH] is called internal if it is an element of a standard set [MATH] for [MATH] . Thus all atoms [MATH] are internal. Elements of internal sets are also internal. |
Proposition 1.1 (The Internal Definition Principle) For any formula [MATH] , any [MATH] and any internal objects [MATH] , the set |
[EQUATION] is internal. The important thing to remember is that the quantifiers in the Transfer Principle range only over internal objects. For example, by transfer it is easy to obtain the following. |
Proposition 1.2 Every bounded internal subset of [MATH] has the greatest element. An important type of nonstandard extension is an enlargement . We call a binary relation [MATH] in [MATH] |
concurrent if for every finitely many elements [MATH] of its domain there is [MATH] such that [MATH] for [MATH] . A nonstandard extension [MATH] is an enlargement if for every concurrent relation [MATH] in [MATH] there is [MATH] in the extension such that [MATH] for all [MATH] in the domain of [MATH] . Such extensions ... |
, Theorem 7.12. The Stone-Čech compatification. The set of all ultrafilters on [MATH] is denoted by [MATH] . For each [MATH] the principal ultrafilter [MATH] is identified with [MATH] . A topology can be defined on [MATH] so that it becomes the maximal compactification of the discrete space on [MATH] . This means that ... |
Only eventually constant sequences in [MATH] are convergent in the usual sense. Hence convergence via ultrafilters is often used: if [MATH] are ultrafilters, [MATH] if, for every [MATH] [MATH] . More on these limits can be found in |
, section 3.5. For every [MATH] the family [MATH] is an ultrafilter; we denote this ultrafilter by [MATH] . Thus a function [MATH] is obtained. For example, for [MATH] |
[MATH] is the corresponding principal ultrafilter. In general, [MATH] is not 1-1 (unless [MATH] is obtained as ultrapower by a Hausdorff ultrafilter, see |
). [MATH] is onto if [MATH] is an enlargement. Fact 1.3 (a) For every function [MATH] and every [MATH] [MATH] (b) For every [MATH] and every [MATH] [MATH] |
Proof. (a) is , Lemma 1. (b) Let [MATH] . Then, in [MATH] [MATH] . By transfer, for every [MATH] [MATH] [MATH] Many aspects of the connection and the similarities between [MATH] and [MATH] were investigated in |
and The sets [MATH] for [MATH] are called monads ; they were investigated in a more general context in . By , Theorem 3.1(a), monad of every nonprincipal ultrafilter in an enlargement has the same cardinality as [MATH] itself. |
By , Theorem 2.10, [MATH] for every [MATH] and every [MATH] . We make a short digression to provide more information about this in the following lemma. |
Lemma 1.4 (a) If [MATH] , then the following conditions are equivalent: (i) [MATH] ; (ii) [MATH] ; (iii) [MATH] (b) If [MATH] and [MATH] is such that [MATH] , then [MATH] |
Proof. (a) If [MATH] , then for every [MATH] we have [MATH] (Lemma 1.3 (b)), so [MATH] . (ii) [MATH] (iii) is obvious, and if [MATH] , then [MATH] , so [MATH] |
(b) If [MATH] then, by Lemma 1.3 (a), for every [MATH] [MATH] , so [MATH] On the other hand, if [MATH] , by (a) the condition [MATH] implies [MATH] , so there is [MATH] such that [MATH] . If we denote [MATH] , then [MATH] and [MATH] [MATH] |
Notation. Throughout the paper, [MATH] denotes the set of natural numbers (including zero) and [MATH] denotes the set of (standard) prime numbers. |
If [MATH] is maximal such that [MATH] , we write [MATH] . Analogous notation will be used in [MATH] (see Lemma 2.4 (b)). For [MATH] let us denote [MATH] . Analogously, [MATH] for [MATH] |
The elements of [MATH] will be denoted by small letters [MATH] with [MATH] reserved for primes. The notation for ultrafilters will differ from that in |
and ; they will be denoted by [MATH] , again with [MATH] reserved for prime ultrafilters. For [MATH] we denote [MATH] [MATH] [MATH] and [MATH] . If [MATH] [MATH] . Also, [MATH] [MATH] (for [MATH] ) and [MATH] |
Divisibility in [MATH] In the author defined four relations on [MATH] extending divisibility in [MATH] . The one that most attention was given to is [MATH] , further investigated in |
[EQUATION] We recapitulate some of the basic properties of this relation. It is not antisymmetric, so we think of it as an order on the equivalence classes [MATH] of the relation defined by: [MATH] . For such a class we denote [MATH] |
An ultrafilter [MATH] is prime (for [MATH] ) if it is divisible only by [MATH] and itself. By , Theorem 2.3, an ultrafilter [MATH] is prime if and only if [MATH] |
In we described the lower part of the [MATH] -hierarchy, more precisely the first [MATH] -many levels. [MATH] is antisymmetric within these lower levels ( |
, Lemma 5.13). On the second level [MATH] (directly above prime ultrafilters) there are three types of ultrafilters: (1) those of the form [MATH] [MATH] for some prime [MATH] |
(2) those containing [MATH] for some prime [MATH] and (3) ultrafilters containing [MATH] for some two distinct prime ultrafilters [MATH] and [MATH] |
The ultrafilters of the third type are divisible by exactly two primes, and those of the first two types have only one prime divisor (”counted” twice). In a similar way, each ultrafilter on the [MATH] -th level [MATH] of the hierarchy has exactly [MATH] ”ingredients”, not necessarilly distinct, with powers of primes [M... |
, Theorem 5.5. The divisibility relation on [MATH] In this section we recall some number-theoretic properties of the extension [MATH] of the divisibility relation [MATH] on [MATH] . Note that the notions of greatest common divisor, least common multiplier and mutually prime numbers transfer directly from [MATH] to [MAT... |
Definition 2.1 [MATH] is prime if it is divisible only by [MATH] and itself. Clearly, ” [MATH] is prime” can be written as [MATH] |
Lemma 2.2 For every [MATH] [MATH] is prime if and only if [MATH] Proof. The formula [MATH] [MATH] is prime” [MATH] holds in [MATH] so, by transfer, its counterpart [MATH] holds in [MATH] [MATH] |
The next lemma also follows directly from the Transfer Principle. Lemma 2.3 (a) For all [MATH] [MATH] and [MATH] [MATH] if and only if [MATH] for some [MATH] |
(b) For all [MATH] [MATH] and [MATH] [MATH] if and only if [MATH] In particular, [MATH] is divisible by [MATH] if and only if [MATH] . Our next lemma lists several other properties of [MATH] that mostly follow directly from the Transfer Principle. |
Lemma 2.4 (a) [MATH] (b) For every [MATH] and every [MATH] there is maximal [MATH] such that [MATH] (c) If [MATH] have the same sets of divisors of the form [MATH] [MATH] [MATH] ), then [MATH] |
Proof. (a) Let [MATH] be the function mapping every [MATH] to the [MATH] -th prime number; so [MATH] [MATH] etc. Then the formula [MATH] [MATH] is prime” [MATH] holds in [MATH] , so by transfer each [MATH] is prime in [MATH] [MATH] is a bijection, so [MATH] is a bijection too. Finally, by Lemma 1.3 (b), [MATH] |
(b) We have [MATH] , so the same holds in [MATH] (c) Since [MATH] holds, it follows that for all [MATH] [MATH] implies [MATH] [MATH] |
[MATH] is not well-ordered, so infinite sums and products can not be defined in the usual way, by induction. However, using transfer we can bypass this, using an idea described in |
, Remark 5.8. The next theorem is an extension of the fundamental theorem of arithmetic. Within it [MATH] is the function enumerating all primes (defined in the proof of Lemma 2.4 (a)). Note that, since a sequence [MATH] is a set of ordered pairs, we have [MATH] |
Theorem 2.5 (a) For every [MATH] and every internal sequence [MATH] there is unique [MATH] such that [MATH] for [MATH] and [MATH] for [MATH] ; we denote such element by [MATH] |
(b) Every [MATH] can be uniquely represented as [MATH] for some [MATH] and some internal sequence [MATH] such that [MATH] Proof. (a) Let, as usual, ” [MATH] ” denote the formula: [MATH] . Let ” [MATH] ” denote the formula |
[EQUATION] In [MATH] we have: [EQUATION] [EQUATION] By transfer, the same holds in [MATH] . Now, if we are given [MATH] and an internal sequence [MATH] , by the Internal Definition Principle the sequence [MATH] defined by [MATH] is also internal. ( LABEL:proizvod ) now produces a sequence [MATH] such that [MATH] |
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