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[MATH] Then [MATH] is represented as [MATH] and by setting [MATH] and [MATH] we have [EQUATION] [MATH] For [MATH] , define [MATH] by [MATH] Then [MATH] is represented as |
[MATH] and [EQUATION] Example 4.3 Let [MATH] be a primitive [MATH] th root of unity. For a braiding [MATH] of [MATH] we set [MATH] . Then |
[EQUATION] It follows that all pairs of [MATH] [MATH] [MATH] [MATH] [MATH] are not braided Morita equivalent. The braided Morita equivalence classes of [MATH] |
In this section we compute the automorphism group of the [MATH] -dimensional Kac-Paljutkin algebra [MATH] , and determine its braided Morita equivalence classes. |
Since the finite group [MATH] is isomorphic to the dihedral group [MATH] of order [MATH] it follows that [MATH] is isomorphic to the group algebra [MATH] as an algebra. An algebra isomorphism |
[MATH] is given by [MATH] The induced Hopf algebra structure of [MATH] that [MATH] is a Hopf algebra map is as follows [EQUATION] |
where [MATH] They are central orthogonal idempotents, and satisfy [MATH] Via the map [MATH] we identify [MATH] Then [EQUATION] By ( 3.3 ) we see that the group-like elements of [MATH] are given by |
[EQUATION] We set [MATH] and [MATH] Let [MATH] be a Hopf algebra automorphism on [MATH] Then we see that [EQUATION] Now, we write [MATH] as |
[MATH] [MATH] Then, [EQUATION] [EQUATION] By solving the above equations, [MATH] is one of the following. (i) [MATH] (ii) [MATH] |
where [MATH] is satisfied for all cases. In these [MATH] , we search [MATH] so that [MATH] Then we see that [MATH] is identical on [MATH] , or coincides with [MATH] on [MATH] defined by [MATH] Furthermore, it can be shown that |
[MATH] is satisfied if and only if [MATH] for [MATH] , and [MATH] for [MATH] In this way we have: Lemma 5.1 If [MATH] is a Hopf algebra automorphism on [MATH] , then |
[MATH] is one of the Hopf algebra automorphisms [MATH] , where [MATH] are defined by [MATH] Therefore, the group [MATH] of the Hopf algebra automorphisms is |
[MATH] Since [MATH] [MATH] it follows that [MATH] , and this implies the following result. Corollary 5.2 As braided Hopf algebras [MATH] |
[MATH] In particular, there are isomorphisms [MATH] and [MATH] as [MATH] -linear braided monoidal categories. ∎ By Corollary 5.2 and Example 4.3 we have: |
Theorem 5.3 Let [MATH] be an algebraically closed field whose characteristic is not [MATH] For two braidings [MATH] of the [MATH] -dimensional Kac-Paljutkin algebra [MATH] over |
[MATH] the braided Hopf algebras [MATH] and [MATH] are braided Morita equivalent if and only if one of the following is satisfied: |
[MATH] [MATH] [MATH] [MATH] [MATH] [MATH] Therefore, there are exactly [MATH] braided Morita equivalence classes for [MATH] Appendix: List of corrigenda in |
with correct statements. [MATH] p.333 in the abstract and p.334, l.6–7; the following sentence should be deleted: As a consequence, we see that such a Hopf algebra has a coribbon |
structure if and only if it is of Kac-Paljutkin type (see Theorem 5). [MATH] p.339, the statements of Theorem 5 should be changed as follows. |
Theorem 5 (1) The set of coribbon elements of the braided Hopf algebra [MATH] is [MATH] (2) The set of coribbon elements of the braided Hopf algebra [MATH] is [MATH] |
Here, [MATH] are the elements of [MATH] , that are determined by the condition (iii) in Definition 3 and the equations [MATH] [MATH] |
p.340, the conclusion part of Lemma 8 (1) should be changed as follows: Then, [MATH] induces a coribbon element of the bialgebra [MATH] if and only if [MATH] |
[MATH] p.340, the symbols [MATH] should be replaced by [MATH] , respectively, and ( 3.1 ) should be added. [MATH] p.341, the parts from the fourth line to the 16th line should be modified as follows: |
If [MATH] is even, then [EQUATION] If [MATH] is odd, then [EQUATION] Hence, if [MATH] is even, then [EQUATION] and if [MATH] is odd, then |
[EQUATION] Thus, we have [EQUATION] It follows that [MATH] [MATH] p.342, the equations in the 14th and 16th lines should be modified as follows, respectively: |
[MATH] [MATH] [MATH] p.342, the equation “ [MATH] ” in the 19th line should be modified as [MATH] ” and the equation in the 21st line should be modified as [MATH] |
[MATH] p.342, the equations in the 25th and 27th lines should be modified as follows, respectively: [MATH] [MATH] [MATH] p.342, in the fourth line from the bottom the sentence [MATH] ,” should be modified as [MATH] as a braided bialgebra,” |
[MATH] p.343, l.9; the statement “By Lemma 8, it follows that [MATH] and [MATH] .” should be corrected as follows: By Lemma 8, it follows that [MATH] Since [MATH] the condition [MATH] implies [MATH] |
[MATH] p.343, the part from the 17th line to the 18th line should be modified below: Therefore, by Lemma 8, it follows that [MATH] Since [MATH] the condition [MATH] implies [MATH] |
# Source: arxiv 1806.03134 # Title: On Minimal Sets to Destroy the $k$-Core in Random Networks # Sections: all # Downloaded: 2026-03-03T05:15:22.200121+00:00 |
On Minimal Sets to Destroy the [MATH] -Core in Random Networks Abstract We study the problem of finding the smallest set of nodes in a network whose removal results in an empty [MATH] -core; where the [MATH] -core is the sub-network obtained after the iterative removal of all nodes of degree smaller than [MATH] . This ... |
Introduction Threshold models are a common approach to model collective dynamical processes on networks. On a daily basis, we face examples such as the spreading of epidemics, opinions and decisions in social networks and biological systems. Questions of practical importance are often related to optimal policies to con... |
The contribution of this paper is related to a widely studied model for dynamics on a network: the threshold model Granovetter ( 1978 , also known as bootstrapping percolation in physics Chalupa et al. 1979 . The network is represented by a graph [MATH] , the nodes of the graph can be either in an active or inactive st... |
In graph theory, the [MATH] -core of a graph is defined as the largest induced subgraph of [MATH] with minimum degree at least [MATH] . This is equivalent to the set of nodes that are left after repeatedly removing all nodes of degree smaller than [MATH] The significance of the [MATH] -core of a graph for the above pro... |
Most of the existing studies of the minimal contagious set problem are algorithmic works in which algorithms are proposed and heuristically tested against other algorithms on real and synthetic networks; see e.g. Altarelli et al. 2013a ); Braunstein et al. 2016 ); Zdeborová et al. 2016 ); Sen et al. 2017 for some recen... |
To review briefly the theoretical works most related to our contribution we start with the special case of [MATH] that has been studied more thoroughly than [MATH] . The choice of [MATH] leads to the removal of the [MATH] -core, i.e. removal of all cycles, and is therefore referred to as the decycling problems. This ca... |
The case of contagious sets with [MATH] is studied less broadly, but the state-of-the-art is similar to the decycling problem. Rigorous upper bounds stem from analysis of greedy algorithms Coja-Oghlan et al. 2015 . The problem has been studied very thoroughly via the cavity method on random regular graphs in Guggiola a... |
I.1 Summary of our contribution This work is inspired by the very simple decycling algorithm corehd proposed in Zdeborová et al. 2016 . In numerical experiments the authors of Zdeborová et al. 2016 found that corehd is close in performance to the message passing of Braunstein et al. 2016 , which is so far the best perf... |
In this work, we observe that the dynamics of the corehd algorithm is amenable to rigorous analysis and performance characterization for random graphs drawn from the configuration model with any bounded degree distribution. We show that it is possible to find a deterministic approximation of the macroscopic dynamics of... |
Our analysis applies not only to random regular graphs, but also to random graphs from the configuration model defined by a degree distribution. The basic theory requires that the degree distribution is bounded (i.e. all the degrees are smaller than some large constant independent of the size of the graph). But, the mo... |
The following results are presented. Exact analysis of the corehd algorithm. We show that the corehd algorithm (generalized to [MATH] -core removal) translates to a random process on the degree distribution of the graph [MATH] . We track this random process by derivation of the associated continuous limit. This is done... |
New upper bounds on the size of the minimal contagious sets in random graphs. The stopping time of the before-mentioned ODEs is related to the number of nodes that were removed from the [MATH] core during the process. Thus providing upper bounds on the expected minimal size of the contagious set of [MATH] . A numerical... |
Improved heuristic algorithm. Based on intuition we gained analyzing the corehd algorithm we propose it’s extension that further improves the performance. In this new algorithm, instead of first removing high degree nodes, we first remove nodes according to the decision rule [MATH] . On graphs with bounded degree this ... |
The paper is organized in two main parts. The first part in section II is devoted to the analysis of the generalized corehd algorithm and comparison of the resulting upper bounds with existing results. In the second part in section III we introduce the new algorithm called the weak-neighbor (that we do not study analyt... |
II The analysis of the corehd algorithm In Algorithm we outline the corehd algorithm of Zdeborová et al. 2016 , generalized from [MATH] to generic [MATH] . The algorithm provides us with a contagious set of nodes [MATH] such that after their removal the resulting graph has an empty [MATH] -core. Consequently the size o... |
Data: [MATH] Result: A set of nodes [MATH] whose removal makes the [MATH] -core vanish Function: core [MATH] [MATH] ) returns the [MATH] -core of the graph [MATH] |
Initialize: [MATH] [MATH] while [MATH] do [MATH] // [MATH] is the set of nodes in [MATH] [MATH] [MATH] [MATH] [MATH] end while Algorithm 1 |
Generalized corehd algorithm. With a proper book-keeping for the set [MATH] and dynamic updating of the [MATH] -core the running time of corehd on graphs with bounded degree is [MATH] . The algorithm can be implemented such that in each iteration exactly one node is removed: if a node of degree smaller than [MATH] is p... |
II.1 Reduction into a random process on the degree distribution In the next several sections, we derive closed-form deterministic equations for the macroscopic behavior of the corehd algorithm in the limit of large random graphs taken from the configuration model. This is possible because, when the corehd algorithm is ... |
In particular, the behaviour of the corehd procedure averaged over the graph [MATH] can be described explicitly in terms of the following process involving colored balls in an urn Pfister ( 2014 At time step [MATH] there will be [MATH] balls in the urn, each of which carries a color [MATH] , with [MATH] and [MATH] bein... |
In a first step, called “removal” (line 3-6 in Alg. ), one ball is drawn among the [MATH] balls of maximum degree (color [MATH] ) uniformly at random. Next [MATH] balls [MATH] are drawn with colors following the excess degree distribution of the graph. The excess degree distribution [MATH] of a graph of degree distribu... |
One difficulty, when analyzing the above process, is to chose the right observables. In the previous paragraph the nodes were used as observables. However, equally, one might consider the process in terms of the edges of the graph. As outlined in the previous paragraph, it is important to keep track of the excess degre... |
To reinterpret the above process in terms of half-edges let [MATH] be the total number of half-edges that are connected to nodes of degree [MATH] at the current iteration. Furthermore, we distinguish nodes of degree smaller than [MATH] from all the others. To do so we adapt our index notation in what follows and identi... |
The two steps (trimming and removal) can be described in terms of half-edges as follows. We start with the “removal” step (line 3-6 in Alg. ). It can be recast in the following rule |
[EQUATION] where the vector [MATH] is a random vector that has zeros everywhere except in one of the [MATH] directions, in which it carries a one. The probability that [MATH] is pointing in direction at iteration [MATH] is given by the excess degree distribution [MATH] for [MATH] When a node of degree [MATH] is removed... |
[EQUATION] Analogously, the “trimming” step (line 7 in Alg. ) can be cast in the following update rule where step [MATH] removes a single half-edge of degree [MATH] and subsequently step [MATH] trims away the dangling cavity half-edge |
[EQUATION] where again the position [MATH] where to place the one in the random variable [MATH] is chosen from the current excess degree distribution [MATH] for [MATH] |
The advantage of working with a representation in terms of half-edges is that we do not need to distinguish the different edges of color “ [MATH] ”. Further [MATH] is deterministic because each column of ( ) sums to the same constant. During the removal step, eq. ( ), we remove [MATH] half-edges and in one iteration of... |
This alternating process is also related to low-complexity algorithms for solving [MATH] -SAT problems Achlioptas ( 2000 2001 ); Cocco and Monasson ( 2001 These [MATH] -SAT solution methods alternate between guessing variables, which may create new unit clauses, and trimming unit clauses via unit clause propagation. Du... |
II.2 Taking the average over randomness As the equations stand in ( ) and ( ) they define a random process that behaves just as Alg. on a random graph [MATH] , but with [MATH] and the stopping time of the trimming step implicitly containing all randomness. In terms of the urn with balls representing half-edges, the ran... |
[EQUATION] Performing the average over the randomness per se only yields the average behaviour of the algorithm. In section II.4 it is shown that the stochastic process concentrates around its average in the continuous limit. |
Next the combination of steps [MATH] and [MATH] in eq. ( ) for the “removal” and eq. ( ) for the “trimming” is considered. In terms of half-edges we remove [MATH] half-edges in one iteration of ( ) and [MATH] half-edges in one iteration of ( ). In order to write the average of the removal step, we recall that the proba... |
[EQUATION] where [MATH] represents the identity matrix. In the above estimate, we use [MATH] intermediate steps to transition from [MATH] , that is the removal of a whole degree [MATH] node. We assume that [MATH] is [MATH] . It then follows that the coefficients [MATH] and [MATH] are [MATH] . The last line follows from... |
The average removal step can now be written as [EQUATION] with the effective average drift matrix [EQUATION] where the matrix [MATH] has all entries in the last row equal to [MATH] and zeros everywhere else, such that [MATH] for a non-negative, normalized vector [MATH] Similarly, taking the average in one trimming time... |
[EQUATION] For the trimming step the effective drift is simply [EQUATION] where now [MATH] has all its entries in the first row equal to [MATH] and zeros everywhere else. |
We emphasize that the two processes ( ) and ( 10 ), while acting on the same vector, are separate processes and the latter ( 10 ) needs be repeated until the stopping condition [MATH] is hit. Note also, that in the trimming process, one iteration [MATH] indicates the deletion of a single edge, while it indicates the de... |
II.3 Operator and Continuous Limits As discussed at the end of Sec. II.1 , a key observation, by virtue of which we can proceed, is that [MATH] is deterministic (and hence equal to its average) during both the removal and trimming steps. This is due to the structure of [MATH] and [MATH] that have columns sums independe... |
[EQUATION] The only randomness occurs in the stopping time of the trimming step. In this section the transition to the continuous time-variable [MATH] is performed. To that end we define the scaled process |
[EQUATION] and presume that the derivative [MATH] is equal to its expected change. Here [MATH] stands for the initial number of vertices in the graph. |
Before proceeding to the analysis of corehd , let us first describe the solution for the two processes (removal and trimming) as if they were running separately. Let us indicate the removal process ( ) and trimming processes ( 10 ) with subscripts [MATH] and [MATH] respectively. It then follows from ( ) and ( 10 ) that... |
[EQUATION] Owing to the deterministic nature of the drift terms [MATH] we have [EQUATION] and the above differential equation can be solved explicitly as |
[EQUATION] We have thus obtained an analytic description of each of the two separate processes ( ) and ( 10 ). Note that this implies that we can analytically predict the expected value of the random process in which all nodes of degree [MATH] are removed from a graph successively until none remains and then all nodes ... |
Crucially the running time of the trimming process depends on the final state of the removal process, i.e. the differential equations become nonlinear in [MATH] . As a consequence, they can no longer be brought into a simple analytically solvable form (at least as far as we were able to tell). To derive the differentia... |
[EQUATION] where we defined the propagator [EQUATION] In what follows we will be considering the removal and trimming processes to belong to one and the same process and therefore [MATH] will no longer be carrying subscripts. Upon combination a full step in the combined process in terms of the operators then reads |
[EQUATION] Note that one infinitesimal time step is the continuous equivalent of the removal of one degree [MATH] node, together with the resulting cascade of degree [MATH] nodes. It is for that reason that the final continuous time after which the [MATH] -core vanishes will be directly related to the size of the set [... |
[EQUATION] Requiring this condition in eq. ( 17 ) we get from an expansion to linear order in [MATH] that [EQUATION] Once again, we recall that [MATH] denotes the first component of the vector [MATH] We can now use this equation to eliminate the dependence on [MATH] in the combined operator [MATH] Using ( 16 ) and keep... |
[EQUATION] which leads us to the following differential equation [EQUATION] The nonlinearity [MATH] is directly linked to the trimming time and defined as |
[EQUATION] To obtain the last equality in ( 22 ) we used the trimming condition, i.e. set [MATH] The initial conditions are such that the process starts from the [MATH] -core of the original graph. This is achieved by solving ( 15 ), with [MATH] , for arbitrary initial degree distribution [MATH] (without bounded maximu... |
[EQUATION] II.4 Rigorous Analysis A rigorous analysis of the [MATH] -core peeling process for Erdős-Rényi graphs is presented in Pittel et al. 1996 This analysis is based on the Wormald approach Wormald ( 1995 but the presentation in Pittel et al. 1996 is more complicated because it derives an exact formula for the thr... |
The peeling process in corehd alternates between deleting maximum-degree nodes and degree [MATH] edges and this introduces a similar problem for the Wormald method. In particular, the corehd peeling schedule typically reduces the fraction of maximum-degree nodes to zero at some point and then the maximum degree jumps d... |
For corehd , one solution is to use weighted preferences where the removal of degree [MATH] edges is most preferred, then removal of degree- [MATH] nodes, then degree [MATH] nodes, and so on. In this case, the drift equation remains Lipschitz continuous if the weights are finite but the model dynamics only approximate ... |
The key argument is that the [MATH] -core peeling step (after each maximum-degree node removal) does not last too long or affect too many edges in the graph. A very similar argument (dubbed the Lazy-Server Lemma) is used in the analysis of low-complexity algorithms for solving [MATH] -SAT problems Achlioptas ( 2000 200... |
Lemma 1 For some [MATH] , suppose [MATH] satisfies [MATH] and [MATH] Consider the corehd process where a maximum-degree node is removed and then the trimming operation continues until there are no edges with degree less than [MATH] (see ( )). Let the random variable [MATH] denote the total number of trimming steps, whi... |
[EQUATION] Proof. See Appendix A.1 Lemma 2 Let [MATH] be the solution to the operator-limit differential equation ( 21 ) at time [MATH] starting from [MATH] Assume, for some [MATH] , there is a [MATH] such that |
[MATH] and [MATH] for all [MATH] Then, there is [MATH] such that [EQUATION] Proof. See Appendix A.2 Theorem 1 The multistage corehd process converges, with high probability as [MATH] , to the piecewise solution of the operator-limit differential equation. |
Sketch of Proof. The first step is recalling that the standard [MATH] -core peeling algorithm results in graph distributed according to the configuration model with a degree distribution that, with high probability as [MATH] , converges to the solution of the standard [MATH] -core differential equation Pittel et al. 19... |
II.5 Evaluating the results Here we clarify how the upper bound is extracted from the equations previously derived. Note that the nonlinearity ( 22 ) exhibits a singularity when |
[EQUATION] that is, when the gain (r.h.s.) and loss (l.h.s.) terms in the trimming process are equal. This can be either trivially true when no more nodes are left, [MATH] , or it corresponds to an infinite trimming time. The latter is precisely the point where the size of the [MATH] -core jumps downward discontinuousl... |
[MATH] of the differential process ( 21 ). By construction the stopping time [MATH] provides the size of the set [MATH] that contains all the nodes the corehd algorithm removed to break up the [MATH] -core. It hence also provides an upper bound on the size of the minimal contagious set, i.e. the smallest such set that ... |
Note that [MATH] (for an infinitesimally small [MATH] ) gives the size of the [MATH] -core, right before it disappears. For all the cases investigated in this paper we found that solving eqs. ( 23 25 ) for [MATH] yields a continuous disappearance of the [MATH] -core, and for [MATH] the stopping criteria yield discontin... |
Data: Initial degree distribution [MATH] [MATH] Result: The relative size of set of removed nodes, [MATH] Initialize half-edges [MATH] |
Compute distribution of half-edges in the [MATH] -core [MATH] with [MATH] such that [MATH] and [MATH] defined in ( 11 ); Set [MATH] the degree associated to the last non-zero component of [MATH] [MATH] |
while [MATH] do Solve [EQUATION] with initial condition [MATH] . Until either [MATH] or [MATH] // [MATH] and [MATH] defined by ( 11 ) and ( ) with [MATH] set to the current largest degree |
Update [MATH] if [MATH] or [MATH] then stop [MATH] true; else 10 [MATH] 11 [MATH] 12 13 end if 14 15 end while Algorithm 2 Analysis of corehd . Recall that indices are [MATH] are referring to the first component of a vector, [MATH] to the second and so forth until the last component [MATH] |
In order to solve the above set of ODEs numerically, we first use equation ( 15 ) to trim away nodes of color [MATH] , i.e. reduce the graph to it’s [MATH] -core. Then we use equation ( 20 ) recursively, until the last component [MATH] is zero. Subsequently we reduce [MATH] by removing its last component, send [MATH] ,... |
Example: two-cores on three regular random graphs. For a simple example of how to extract the upper bound consider the following case. We have [MATH] and [MATH] and we set [MATH] , then the differential equation in ( 21 ) becomes |
[EQUATION] with initial condition [MATH] because there are [MATH] half-edges, all connected to nodes of degree [MATH] initially. The equations are readily solved |
[EQUATION] According to ( 26 ) the stopping time is [MATH] , i.e. [MATH] , which suggests that the decycling number ( [MATH] -core) for cubic random graphs is bounded by [MATH] . In accordance with Theorem 1.1 in Bau et al. 2002 this bound coincides with the actual decycling number. For [MATH] the lower and upper bound... |
Finding a closed form expression for the generic case is more involved and we did not manage to do it. However, very reliable numerical resolution is possible. The simplest approach to the differential equations is to work directly with ( 20 ) as indicated in Algorithm |
II.6 corehd analyzed and compared with existing results In this section we evaluate the upper bound on minimal contagious set obtained by our analysis of corehd . In Figure we compare the fraction of nodes of a given degree that are in the graph during the corehd procedure obtained from solving the differential equatio... |
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