clustering in complex networks i general formalism we develop a full theoretical approach to clustering in complex networks a key concept is introduced the edge multiplicity that measures the number of triangles passing through an edge this quantity extends the clustering coefficient in that it involves the properties of two and not just one vertices the formalism is completed with the definition of a three vertex correlation function which is the fundamental quantity describing the properties of clustered networks the formalism suggests different metrics that are able to thoroughly characterize transitive relations a rigorous analysis of several real networks which makes use of this formalism and the metrics is also provided it is also found that clustered networks can be classified into two main groups the weak and the strong transitivity classes in the first class edge multiplicity is small with triangles being disjoint in the second class edge multiplicity is high and so triangles share many edges as we shall see in the following paper the class a network belongs to has strong implications in its percolation properties

belief propagation algorithm and the ising model on networks with arbitrary distributions of motifs we generalize the belief propagation algorithm to sparse random networks with arbitrary distributions of motifs triangles loops etc each vertex in these networks belongs to a given set of motifs generalization of the configuration model these networks can be treated as sparse uncorrelated hypergraphs in which hyperedges represent motifs here a hypergraph is a generalization of a graph where a hyperedge can connect any number of vertices these uncorrelated hypergraphs are treelike hypertrees which crucially simplifies the problem and allows us to apply the belief propagation algorithm to these loopy networks with arbitrary motifs as natural examples we consider motifs in the form of finite loops and cliques we apply the belief propagation algorithm to the ferromagnetic ising model with pairwise interactions on the resulting random networks and obtain an exact solution of this model we find an exact critical temperature of the ferromagnetic phase transition and demonstrate that with increasing the clustering coefficient and the loop size the critical temperature increases compared to ordinary treelike complex networks however weak clustering does not change the critical behavior qualitatively our solution also gives the birth point of the giant connected component in these loopy networks

percolation and epidemics in random clustered networks the social networks that infectious diseases spread along are typically clustered because of the close relation between percolation and epidemic spread the behavior of percolation in such networks gives insight into infectious disease dynamics a number of authors have studied percolation or epidemics in clustered networks but the networks often contain preferential contacts in high degree nodes we introduce a class of random clustered networks and a class of random unclustered networks with the same preferential mixing percolation in the clustered networks reduces the component sizes and increases the epidemic threshold compared to the unclustered networks

loop calculus in statistical physics and information science considering a discrete and finite statistical model of a general position we introduce an exact expression for the partition function in terms of a finite series the leading term in the series is the bethe peierls belief propagation bp contribution the rest are expressed as loop contributions on the factor graph and calculated directly using the bp solution the series unveils a small parameter that often makes the bp approximation so successful applications of the loop calculus in statistical physics and information science are discussed

ferromagnetic transition in a simple variant of the ising model on multiplex networks abstract multiplex networks consist of a fixed set of nodes connected by several sets of edges which are generated separately and correspond to different networks layers here a simple variant of the ising model on multiplex networks with two layers is considered with spins located in the nodes and edges corresponding to ferromagnetic interactions between them critical temperatures for the ferromagnetic transition are evaluated for the layers in the form of random erdos renyi graphs or heterogeneous scale free networks using the mean field approximation and the replica method from the replica symmetric solution both methods require the use of different partial magnetizations associated with different layers of the multiplex network and yield qualitatively similar results if the layers are strongly heterogeneous the critical temperature differs noticeably from that for the ising model on a network being a superposition of the two layers evaluated in the mean field approximation neglecting the effect of the underlying multiplex structure on the correlations between the degrees of nodes the critical temperature evaluated from the replica symmetric solution depends sensitively on the correlations between the degrees of nodes in different layers and shows satisfactory quantitative agreement with that obtained from monte carlo simulations the critical behavior of the magnetization for the model with strongly heterogeneous layers can depend on the distributions of the degrees of nodes and is then determined by the properties of the most heterogeneous layer

enhancing performance and robustness of ilu preconditioners by blocking and selective transposition incomplete factorization is one of the most effective general purpose preconditioning methods for krylov subspace solvers for large sparse systems of linear equations two techniques for enhancing the robustness and performance of incomplete lu factorization for sparse unsymmetric systems are described a block incomplete factorization algorithm based on the crout variation of lu factorization is presented the algorithm is suitable for incorporating threshold based dropping as well as unrestricted partial pivoting and it overcomes several limitations of existing incomplete lu factorization algorithms with and without blocking it is shown that blocking has a three pronged impact it speeds up the computation of incomplete factors and the solution of the associated triangular systems it permits denser and more robust factors to be computed economically and it permits a trade off with the restart parameter of gmres to further improve the overall speed and robustness a highly effective heuristic for imp

cavity analysis on the robustness of random networks against targeted attacks influences of degree degree correlations we developed a scheme for evaluating the size of the largest connected subnetwork giant component in random networks and the percolation threshold when sites nodes and or bonds edges are removed from the networks based on the cavity method of statistical mechanics of disordered systems we apply our scheme particularly to random networks of bimodal degree distribution two peak networks which have been proposed in earlier studies as robust networks against random failures of site and or targeted random degree dependent attacks on sites our analysis indicates that the correlations among degrees affect a network s robustness against targeted attacks on sites or bonds nontrivially depending on details of network configurations

algorithm for counting large directed loops we derive a belief propagation algorithm for counting large loops in a directed network we evaluate the distribution of the number of small loops in a directed random network with given degree sequence we apply the algorithm to a few characteristic directed networks of various network sizes and loop structures and compare the algorithm with exhaustive counting results when possible the algorithm is adequate in estimating loop counts for large directed networks and can be used to compare the loop structure of directed networks and their randomized counterparts

random graphs containing arbitrary distributions of subgraphs traditional random graph models of networks generate networks that are locally treelike meaning that all local neighborhoods take the form of trees in this respect such models are highly unrealistic most real networks having strongly nontreelike neighborhoods that contain short loops cliques or other biconnected subgraphs in this paper we propose and analyze a class of random graph models that incorporates general subgraphs allowing for nontreelike neighborhoods while still remaining solvable for many fundamental network properties among other things we give solutions for the size of the giant component the position of the phase transition at which the giant component appears and percolation properties for both site and bond percolation on networks generated by the model

percolation on correlated networks we reconsider the problem of percolation on an equilibrium random network with degree degree correlations between nearest neighboring vertices focusing on critical singularities at a percolation threshold we obtain criteria for degree degree correlations to be irrelevant for critical singularities we present examples of networks in which assortative and disassortative mixing leads to unusual percolation properties and new critical exponents

statistical mechanical iterative algorithms on complex networks the ising models have been applied for various problems on information sciences social sciences and so on in many cases solving these problems corresponds to minimizing the bethe free energy to minimize the bethe free energy a statistical mechanical iterative algorithm is often used we study the statistical mechanical iterative algorithm on complex networks to investigate effects of heterogeneous structures on the iterative algorithm we introduce an iterative algorithm based on information of heterogeneity of complex networks in which higher degree nodes are likely to be updated more frequently than lower degree ones numerical experiments clarified that the usage of the information of heterogeneity affects the algorithm in barab asi and albert networks but does not influence that in erd os and r enyi networks it is revealed that information of the whole system propagates rapidly through such high degree nodes in the case of barab asi albert s scale free networks

stable propagation algorithm for the minimization of the bethe free energy we propose a new propagation algorithm for the minimization of the bethe free energy for a generic lattice model with pair interactions the algorithm turns out to be more stable than belief propagation as it reaches a fixed point also for highly frustrated systems such as spin glasses and faster than the provably convergent double loop algorithms

properties of highly clustered networks we propose and solve exactly a model of a network that has both a tunable degree distribution and a tunable clustering coefficient among other things our results indicate that increased clustering leads to a decrease in the size of the giant component of the network we also study susceptible infective recovered type epidemic processes within the model and find that clustering decreases the size of epidemics but also decreases the epidemic threshold making it easier for diseases to spread in addition clustering causes epidemics to saturate sooner meaning that they infect a near maximal fraction of the network for quite low transmission rates

ising model on networks with an arbitrary distribution of connections we find the exact critical temperature t c of the nearest neighbor ferromagnetic ising model on an equilibrium random graph with an arbitrary degree distribution p k we observe an anomalous behavior of the magnetization magnetic susceptibility and specific heat when p k is fat tailed or loosely speaking when the fourth moment of the distribution diverges in infinite networks when the second moment becomes divergent t c approaches infinity the phase transition is of infinite order and size effect is anomalously strong

acoustic diffraction by a fluid loaded membrane corner the complete solution is derived for an acoustic plane wave incident on a membrane which is stretched over a line constraint forming a wedge shaped fluid region of interior angle 2 the exact two dimensional solution is developed using malyuzhinets functions and integral transforms explicit formulae are presented for the acoustic and structural diffraction coefficients for 2 and numerical results are given for the frequency and angular dependence of the membrane wave excitation coefficients for identical membranes on either side of the vertex

cluster variation method in statistical physics and probabilistic graphical models the cluster variation method cvm is a hierarchy of approximate variational techniques for discrete ising like models in equilibrium statistical mechanics improving on the mean field approximation and the bethe peierls approximation which can be regarded as the lowest level of the cvm in recent years it has been applied both in statistical physics and to inference and optimization problems formulated in terms of probabilistic graphical models the foundations of the cvm are briefly reviewed and the relations with similar techniques are discussed the main properties of the method are considered with emphasis on its exactness for particular models and on its asymptotic properties the problem of the minimization of the variational free energy which arises in the cvm is also addressed and recent results about both provably convergent and message passing algorithms are discussed

entropy of network ensembles in this paper we generalize the concept of random networks to describe network ensembles with nontrivial features by a statistical mechanics approach this framework is able to describe undirected and directed network ensembles as well as weighted network ensembles these networks might have nontrivial community structure or in the case of networks embedded in a given space they might have a link probability with a nontrivial dependence on the distance between the nodes these ensembles are characterized by their entropy which evaluates the cardinality of networks in the ensemble in particular in this paper we define and evaluate the structural entropy i e the entropy of the ensembles of undirected uncorrelated simple networks with given degree sequence we stress the apparent paradox that scale free degree distributions are characterized by having small structural entropy while they are so widely encountered in natural social and technological complex systems we propose a solution to the paradox by proving that scale free degree distributions are the most likely degree distribution with the corresponding value of the structural entropy finally the general framework we present in this paper is able to describe microcanonical ensembles of networks as well as canonical or hidden variable network ensembles with significant implications for the formulation of network constructing algorithms

hierarchical coefficient of a multifractal based network abstract the hierarchical property for a general class of networks stands for a power law relation between clustering coefficient c c and connectivity k c c k this relation is empirically verified in several biologic and social networks as well as in random and deterministic network models in special for hierarchical networks in this work we show that the hierarchical property is also present in a lucena network to create a lucena network we use the dual of a multifractal lattice ml the vertices are the sites of the ml and links are established between neighbouring lattices therefore this network is space filling and planar besides a lucena network shows a scale free distribution of connectivity we deduce a relation for the maximal local clustering coefficient c c i max of a vertex i in a planar graph this condition expresses that the number of links among neighbour n of a vertex i is equal to its connectivity k i that means n k i the lucena network fulfils the condition n k i independent of k i and the anisotropy of ml in addition c c max implies the threshold 1 for the hierarchical property for any scale free planar network

belief propagation in networks of spiking neurons from a theoretical point of view statistical inference is an attractive model of brain operation however it is unclear how to implement these inferential processes in neuronal networks we offer a solution to this problem by showing in detailed simulations how the belief propagation algorithm on a factor graph can be embedded in a network of spiking neurons we use pools of spiking neurons as the function nodes of the factor graph each pool gathers messages in the form of population activities from its input nodes and combines them through its network dynamics each of the various output messages to be transmitted over the edges of the graph is computed by a group of readout neurons that feed in their respective destination pools we use this approach to implement two examples of factor graphs the first example drawn from coding theory models the transmission of signals through an unreliable channel and demonstrates the principles and generality of our network approach the second more applied example is of a psychophysical mechanism in which visual cues are used to resolve hypotheses about the interpretation of an object s shape and illumination these two examples and also a statistical analysis demonstrate good agreement between the performance of our networks and the direct numerical evaluation of belief propagation

clustering in complex networks ii percolation properties the percolation properties of clustered networks are analyzed in detail in the case of weak clustering we present an analytical approach that allows us to find the critical threshold and the size of the giant component numerical simulations confirm the accuracy of our results in more general terms we show that weak clustering hinders the onset of the giant component whereas strong clustering favors its appearance this is a direct consequence of the differences in the k core structure of the networks which are found to be totally different depending on the level of clustering an empirical analysis of a real social network confirms our predictions