101,806 research outputs found
Inducing Effect on the Percolation Transition in Complex Networks
Percolation theory concerns the emergence of connected clusters that
percolate through a networked system. Previous studies ignored the effect that
a node outside the percolating cluster may actively induce its inside
neighbours to exit the percolating cluster. Here we study this inducing effect
on the classical site percolation and K-core percolation, showing that the
inducing effect always causes a discontinuous percolation transition. We
precisely predict the percolation threshold and core size for uncorrelated
random networks with arbitrary degree distributions. For low-dimensional
lattices the percolation threshold fluctuates considerably over realizations,
yet we can still predict the core size once the percolation occurs. The core
sizes of real-world networks can also be well predicted using degree
distribution as the only input. Our work therefore provides a theoretical
framework for quantitatively understanding discontinuous breakdown phenomena in
various complex systems.Comment: Main text and appendices. Title has been change
Distance, dissimilarity index, and network community structure
We address the question of finding the community structure of a complex
network. In an earlier effort [H. Zhou, {\em Phys. Rev. E} (2003)], the concept
of network random walking is introduced and a distance measure defined. Here we
calculate, based on this distance measure, the dissimilarity index between
nearest-neighboring vertices of a network and design an algorithm to partition
these vertices into communities that are hierarchically organized. Each
community is characterized by an upper and a lower dissimilarity threshold. The
algorithm is applied to several artificial and real-world networks, and
excellent results are obtained. In the case of artificially generated random
modular networks, this method outperforms the algorithm based on the concept of
edge betweenness centrality. For yeast's protein-protein interaction network,
we are able to identify many clusters that have well defined biological
functions.Comment: 10 pages, 7 figures, REVTeX4 forma
Heterogeneous micro-structure of percolation in sparse networks
We examine the heterogeneous responses of individual nodes in sparse networks to the random removal of a fraction of edges. Using the message-passing formulation of percolation, we discover considerable variation across the network in the probability of a particular node to remain part of the giant component, and in the expected size of small clusters containing that node. In the vicinity of the percolation threshold, weakly non-linear analysis reveals that node-to-node heterogeneity is captured by the recently introduced notion of non-backtracking centrality. We supplement these results for fixed finite networks by a population dynamics approach to analyse random graph models in the infinite system size limit, also providing closed-form approximations for the large mean degree limit of Erdős-Rényi random graphs. Interpreted in terms of the application of percolation to real-world processes, our results shed light on the heterogeneous exposure of different nodes to cascading failures, epidemic spread, and information flow
Scale-Free Networks Emerging from Weighted Random Graphs
We study Erd\"{o}s-R\'enyi random graphs with random weights associated with
each link. We generate a new ``Supernode network'' by merging all nodes
connected by links having weights below the percolation threshold (percolation
clusters) into a single node. We show that this network is scale-free, i.e.,
the degree distribution is with . Our
results imply that the minimum spanning tree (MST) in random graphs is composed
of percolation clusters, which are interconnected by a set of links that create
a scale-free tree with . We show that optimization causes the
percolation threshold to emerge spontaneously, thus creating naturally a
scale-free ``supernode network''. We discuss the possibility that this
phenomenon is related to the evolution of several real world scale-free
networks
Statistical mechanics of complex networks
Complex networks describe a wide range of systems in nature and society, much
quoted examples including the cell, a network of chemicals linked by chemical
reactions, or the Internet, a network of routers and computers connected by
physical links. While traditionally these systems were modeled as random
graphs, it is increasingly recognized that the topology and evolution of real
networks is governed by robust organizing principles. Here we review the recent
advances in the field of complex networks, focusing on the statistical
mechanics of network topology and dynamics. After reviewing the empirical data
that motivated the recent interest in networks, we discuss the main models and
analytical tools, covering random graphs, small-world and scale-free networks,
as well as the interplay between topology and the network's robustness against
failures and attacks.Comment: 54 pages, submitted to Reviews of Modern Physic
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