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 tree-like (hypertrees), which crucially simplify the problem and allow 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 on the resulting random networks. We obtain an exact solution of this model on networks with finite loops or cliques as motifs. 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 tree-like complex networks. Our solution also gives the birth point of the giant connected component in these loopy networks.Comment: 9 pages, 4 figure

Critical phenomena in complex networks

The combination of the compactness of networks, featuring small diameters, and their complex architectures results in a variety of critical effects dramatically different from those in cooperative systems on lattices. In the last few years, researchers have made important steps toward understanding the qualitatively new critical phenomena in complex networks. We review the results, concepts, and methods of this rapidly developing field. Here we mostly consider two closely related classes of these critical phenomena, namely structural phase transitions in the network architectures and transitions in cooperative models on networks as substrates. We also discuss systems where a network and interacting agents on it influence each other. We overview a wide range of critical phenomena in equilibrium and growing networks including the birth of the giant connected component, percolation, k-core percolation, phenomena near epidemic thresholds, condensation transitions, critical phenomena in spin models placed on networks, synchronization, and self-organized criticality effects in interacting systems on networks. We also discuss strong finite size effects in these systems and highlight open problems and perspectives.Comment: Review article, 79 pages, 43 figures, 1 table, 508 references, extende

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 BA networks, but does not influence that in ER networks. It is revealed that information of the whole system propagates rapidly through such high-degree nodes in the case of Barab{\'a}si-Albert's scale-free networks.Comment: 7 pages, 6 figure

Correlation in Complex Networks

Network representations are now ubiquitous across science. Indeed, they are the natural representation for complex systems—systems composed of large numbers of interacting components. Occasionally systems can be well represented by simple, regular networks, such as lattices. Usually, however, the networks themselves are complex—highly structured but with no obvious repeating pattern. In this thesis I examine the effects of correlation and interdependence on network phenomena, from three different perspectives. First, I consider patterns of mixing within networks. Nodes within a network frequently have more connections to others that are similar to themselves than to those that are dissimilar. However, nodes can (and do) display significant heterogeneity in mixing behavior—not all nodes behave identically. This heterogeneity manifests as correlations between individuals' connections. I show how to identify and characterize such patterns, and how this correlation can be used for practical tasks such as imputation. Second, I look at the effects of correlation on the structure of networks. If edges within a relational data set are correlated with each other, and if we construct a network from this data, then several of the properties commonly associated with real-world complex networks naturally emerge, namely heavy-tailed degree distributions, large numbers of triangles, short path lengths, and large connected components. Third, I develop a family of technical tools for calculations about networks. If you are using a network representation, there's a good chance you wish to calculate something about the network—for example, what will happen when a disease spreads across it. An important family of techniques for network calculations assume that the networks are free of short loops, which means that the neighbors of any given node are conditionally independent. However, real-world networks are clustered and clumpy, and this structure undermines the assumption of conditional independence. I consider a prescription to deal with this issue, opening up the possibility for many more analyses of realistic and real-world data.PHDPhysicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/163269/1/gcant_1.pd