convergence towards an erdős renyi graph structure in network contraction processes in a highly influential paper twenty years ago barabasi and albert science 286 509 1999 scieas0036 807510 1126 science 286 5439 509 showed that networks undergoing generic growth processes with preferential attachment evolve towards scale free structures in any finite system the growth eventually stalls and is likely to be followed by a phase of network contraction due to node failures attacks or epidemics using the master equation formulation and computer simulations we analyze the structural evolution of networks subjected to contraction processes via random preferential and propagating node deletions we show that the contracting networks converge towards an erd s renyi network structure whose mean degree continues to decrease as the contraction proceeds this is manifested by the convergence of the degree distribution towards a poisson distribution and the loss of degree degree correlations

reduced fusion systems over 2 groups of small order we prove when s is a 2 group of order at most 29 that each reduced fusion system over s is the fusion system of a finite simple group and is tame it then follows that each saturated fusion system over a 2 group of order at most 29 is realizable what is most interesting about this result is the method of proof we show that among 2 groups with order in this range the ones which can be sylow 2 subgroups of finite simple groups are almost completely determined by criteria based on bender s classification of groups with strongly 2 embedded subgroups

reduced fusion systems over p groups with abelian subgroup of index p ii abstract for an odd prime p we look at simple fusion systems over a finite nonabelian p group s which has an abelian subgroup a of index p when s has more than one such subgroup we reduce this to a case already studied by ruiz and viruel when a is the unique abelian subgroup of index p in s and is not essential equivalently is not radical in the fusion system we give a complete list of all possibilities which can occur this includes several families of exotic fusion systems including some which have proper strongly closed subgroups

fusion systems over a sylow p subgroup of text g _2 p g 2 p for s a sylow p subgroup of the group text g 2 p for p odd up to isomorphism of fusion systems we determine all saturated fusion systems mathcal f on s with o p mathcal f 1 for p ne 7 all such fusion systems are realized by finite groups whereas for p 7 there are 29 saturated fusion systems of which 27 are exotic

a shooting method for fully implicit index 2 differential algebraic equations a shooting method for two point boundary value problems bvps for fully implicit index 1 and index 2 differential algebraic equations daes is presented a combination of the shooting equations with a method of the calculation of consistent initial values leads to a system of nonlinear algebraic equations with a nonsingular jacobian matrix examples are given

fundamental difference between superblockers and superspreaders in networks two important problems regarding spreading phenomena in complex topologies are the optimal selection of node sets either to minimize or maximize the extent of outbreaks both problems are nontrivial when a small fraction of the nodes in the network can be used to achieve the desired goal the minimization problem is equivalent to a structural optimization the superblockers i e the nodes that should be removed from the network to minimize the size of outbreaks are those nodes that make connected components as small as possible superspreaders are instead the nodes such that if chosen as initiators they maximize the average size of outbreaks the identity of superspreaders is expected to depend not just on the topology but also on the specific dynamics considered recently it has been conjectured that the two optimization problems might be equivalent in the sense that superblockers act also as superspreaders in spite of its potential groundbreaking importance no empirical study has been performed to validate this conjecture in this paper we perform an extensive analysis over a large set of real world networks to test the similarity between sets of superblockers and of superspreaders we show that the two optimization problems are not equivalent superblockers do not act as optimal spreaders

spin glass model for the c dismantling problem c dismantling cd problem aims at finding the minimum vertex set d of a graph g v e after removing which the remaining graph will break into connected components with the size not larger than c in this paper we introduce a spin glass model with c 1 integer value states into the cd problem and then study the properties of this spin glass model by the belief propagation bp equations under the replica symmetry ansatz we give the lower bound rho c of the relative size of d with finite c on regular random graphs and erdos renyi random graphs we find rho c will decrease gradually with growing c and it converges to rho infty as c to infty the cd problem is called dismantling problem when c is a small finite fraction of v therefore rho infty is also the lower bound of the dismantling problem when v to infty to reduce the computation complexity of the bp equations taking the knowledge of the probability of a random selected vertex belonging to a remaining connected component with the size a the original bp equations can be simplified to one with only three states when c to infty the simplified bp equations are very similar to the bp equations of the feedback vertex set spin glass model h j zhou eur phys j b 86 455 2013 at last we develop two practical belief propagation guide decimation algorithms based on the original bp equations cd bpd and the simplified bp equations scd bpd to solve the cd problem on a certain graph our bpd algorithms and two other state of art heuristic algorithms are applied on various random graphs and some real world networks computation results show that the cd bpd is the best in all tested algorithms in the case of small c but considering the performance and computation consumption we recommend using scd bpd for the network with small clustering coefficient when c is large

saturated fusion systems over 2 groups we develop methods for listing for a given 2 group s all nonconstrained centerfree saturated fusion systems over s these are the saturated fusion systems which could potentially include minimal examples of exotic fusion systems fusion systems not arising from any finite group to test our methods we carry out this program over four concrete examples two of order 2 7 and two of order 2 10 our long term goal is to make a wider more systematic search for exotic fusion systems over 2 groups of small order

nonbacktracking expansion of finite graphs message passing equations yield a sharp percolation transition in finite graphs as an artifact of the locally treelike approximation for an arbitrary finite connected undirected graph we construct an infinite tree having the same local structural properties as this finite graph when observed by a nonbacktracking walker formally excluding the boundary this infinite tree is a generalization of the bethe lattice we indicate an infinite locally treelike random network whose local structure is exactly given by this infinite tree message passing equations for various cooperative models on this construction are the same as for the original finite graph but here they provide the exact solutions of the corresponding cooperative problems these solutions are good approximations to observables for the models on the original graph when it is sufficiently large and not strongly correlated we show how to express these solutions in the critical region in terms of the principal eigenvector components of the nonbacktracking matrix as representative examples we formulate the problems of the random and optimal destruction of a connected graph in terms of our construction the nonbacktracking expansion we analyze the limitations and the accuracy of the message passing algorithms for different classes of networks and compare the complexity of the message passing calculations to that of direct numerical simulations notably in a range of important cases simulations turn out to be more efficient computationally than the message passing

statistical analysis of articulation points in configuration model networks an articulation point ap in a network is a node whose deletion would split the network component on which it resides into two or more components aps are vulnerable spots that play an important role in network collapse processes which may result from node failures attacks or epidemics therefore the abundance and properties of aps affect the resilience of the network to these collapse scenarios we present analytical results for the statistical properties of aps in configuration model networks in order to quantify their abundance we calculate the probability p i in rm ap that a random node i in a configuration model network with p k k is an ap we also obtain the conditional probability p i in rm ap k that a random node of degree k is an ap and find that high degree nodes are more likely to be aps than low degree nodes using bayes theorem we obtain the conditional degree distribution p k k rm ap over the set of aps and compare it to p k k we propose a new centrality measure based on aps each node can be characterized by its articulation rank r which is the number of components that would be added to the network upon deletion of that node for nodes which are not aps the articulation rank is r 0 while for aps r ge 1 we obtain a closed form expression for the distribution of articulation ranks p r r configuration model networks often exhibit a coexistence between a giant component and finite components to examine the distinct properties of aps on the giant and on the finite components we calculate the probabilities presented above separately for the giant and the finite components we apply these results to ensembles of configuration model networks with a poisson exponential and power law degree distributions the implications of these results are discussed in the context of common attack scenarios and network dismantling processes

identifying optimal targets of network attack by belief propagation for a network formed by nodes and undirected links between pairs of nodes the network optimal attack problem aims at deleting a minimum number of target nodes to break the network down into many small components this problem is intrinsically related to the feedback vertex set problem that was successfully tackled by spin glass theory and an associated belief propagation guided decimation bpd algorithm zhou eur phys j b 86 455 2013 in the present work we apply the bpd algorithm which has approximately linear time complexity to the network optimal attack problem and demonstrate that it has much better performance than a recently proposed collective information algorithm morone and makse nature 524 65 2015 for different types of random networks and real world network instances the bpd guided attack scheme often induces an abrupt collapse of the whole network which may make it very difficult to defend

network dismantling abstract we study the network dismantling problem which consists of determining a minimal set of vertices in which removal leaves the network broken into connected components of subextensive size for a large class of random graphs this problem is tightly connected to the decycling problem the removal of vertices leaving the graph acyclic exploiting this connection and recent works on epidemic spreading we present precise predictions for the minimal size of a dismantling set in a large random graph with a prescribed light tailed degree distribution building on the statistical mechanics perspective we propose a three stage min sum algorithm for efficiently dismantling networks including heavy tailed ones for which the dismantling and decycling problems are not equivalent we also provide additional insights into the dismantling problem concluding that it is an intrinsically collective problem and that optimal dismantling sets cannot be viewed as a collection of individually well performing nodes

rare events and discontinuous percolation transitions percolation theory characterizing the robustness of a network has applications ranging from biology to epidemic spreading and complex infrastructures percolation theory however only concerns the average response of a network to random damage of its nodes while in real finite networks fluctuations around this average behavior are observable consequently for finite networks there is an urgent need to evaluate the risk of collapse in response to rare configurations of the initial damage here we build a large deviation theory of percolation characterizing the response of a sparse network to rare events this general theory includes the second order phase transition observed typically for random configurations of the initial damage but reveals also discontinuous transitions corresponding to rare configurations of the initial damage for which the size of the giant component is suppressed

minimal contagious sets in random regular graphs the bootstrap percolation or threshold model is a dynamic process modelling the propagation of an epidemic on a graph where inactive vertices become active if their number of active neighbours reach some threshold we study an optimization problem related to it namely the determination of the minimal number of active sites in an initial configuration that leads to the activation of the whole graph under this dynamics with and without a constraint on the time needed for the complete activation this problem encompasses in special cases many extremal characteristics of graphs like their independence decycling or domination number and can also be seen as a packing problem of repulsive particles we use the cavity method including the effects of replica symmetry breaking an heuristic technique of statistical mechanics many predictions of which have been confirmed rigorously in the recent years we have obtained in this way several quantitative conjectures on the size of minimal contagious sets in large random regular graphs the most striking being that 5 regular random graph with a threshold of activation of 3 resp 6 regular with threshold 4 have contagious sets containing a fraction 1 6 resp 1 4 of the total number of vertices equivalently these numbers are the minimal fraction of vertices that have to be removed from a 5 regular resp 6 regular random graph to destroy its 3 core we also investigated survey propagation like algorithmic procedures for solving this optimization problem on single instances of random regular graphs

extensions of p local finite groups a p local finite group consists of a finite p group s together with a pair of categories which encode conjugacy relations among subgroups of s and which are modelled on the fusion in a sylow p subgroup of a finite group it contains enough information to define a classifying space which has many of the same properties as p completed classifying spaces of finite groups in this paper we study and classify extensions of p local finite groups and also compute the fundamental group of the classifying space of a p local finite group

subgroup families controlling p local finite groups a p local finite group consists of a finite p group s together with a pair of categories which encode conjugacy relations among subgroups of s and which are modelled on the fusion in a sylow p subgroup of a finite group it contains enough information to define a classifying space which has many of the same properties as p completed classifying spaces of finite groups in this paper we examine which subgroups control this structure more precisely we prove that the question of whether an abstract fusion system mathcal f over a finite p group s is saturated can be determined by just looking at smaller classes of subgroups of s we also prove that the homotopy type of the classifying space of a given p local finite group is independent of the family of subgroups used to define it in the sense that it remains unchanged when that family ranges from the set of mathcal f centric mathcal f radical subgroups at a minimum to the set of mathcal f quasicentric subgroups at a maximum finally we look at constrained fusion systems analogous to p constrained finite groups and prove that they in fact all arise from groups

saturated fusion systems over mathcal a _2 groups in this paper we study the saturated fusion systems mathcal f over mathcal a 2 groups s of order greater than p 4 this study requires the analysis of the possible saturated fusion systems in terms of the mathcal f automorphisms of the possible mathcal f centric and mathcal f radical subgroups also for each case in the classification either s is resistant or mathcal f is constrained the classification identifies a large family of resistant groups complementing the results on resistant groups found by various authors such as green minh stancu and diaz ruiz viruel moreover the classification provides some interesting examples of saturated fusion systems which have an mathcal f essential abelian subgroup of index p

the homotopy theory of fusion systems the main goal of this paper is to identify and study a certain class of spaces which in many ways behave like p completed classifying spaces of finite groups these spaces occur as the classifying spaces of certain algebraic objects which we call p local finite groups a p local finite group consists roughly speaking of a finite p group s and fusion data on subgroups of s encoded in a way explained below our starting point is our earlier paper blo on p completed classifying spaces of finite groups together with the axiomatic treatment by lluis puig pu pu2 of systems of fusion among subgroups of a given p group the p completion of a space x is a space xp which isolates the properties of x at the prime p and more precisely the properties which determine its mod p cohomology for example a map of spaces x f y induces a homotopy equivalence xp y p if and only if f induces an isomorphism in mod p cohomology and h xp fp h x fp in favorable cases if x is p good when g is a finite group the p completion bgp of its classifying space encodes many of the properties of g at p for example not only the mod p cohomology of bg but also the sylow p subgroup of g together with all fusion among its subgroups are determined up to isomorphism by the homotopy type of bgp our goal here is to give a direct link between p local structures and homotopy types which arise from them this theory tries to make explicit the essence of what it means to be the p completed classifying space of a finite group and at the same time yields new spaces which are not of this type but which still enjoy most of the properties a space of the form bgp would have we hope that the ideas presented here will have further applications and generalizations in algebraic topology but this theory also fits well with certain aspects of modular representation theory in particular it may give a way of constructing classifying spaces for blocks in the group ring of a finite group over an algebraically closed field of characteristic p a saturated fusion system f over a p group s consists of a set homf p q of monomorphisms for each pair of subgroups p q s which form a category under composition include all monomorphisms induced by conjugation in s and satisfy certain other axioms formulated by puig definitions 1 1 and 1 2 below in particular these axioms are satisfied by the conjugacy homomorphisms in a finite group we refer to pu and pu2 for more details of puig s work on saturated

the classification of p local finite groups over the extraspecial group of order p 3 and exponent p in this paper we classify the p local finite groups over p1 2 the extraspecial group of order p3 and exponent p for odd p this study reduces to the classification of the saturated fusion systems over p1 2 which will be characterized by the outer automorphism group the number of open image in new window radical subgroups and the automorphism group of each nontrivial open image in new window radical subgroup as part of this classification we obtain three new exotic 7 local finite groups