Finding community structure in networks using the eigenvectors of matrices
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A modularity matrix plays a role in community detection similar to that played by the graph Laplacian in graph partitioning calculations, and a spectral measure of bipartite structure in networks and a centrality measure that identifies vertices that occupy central positions within the communities to which they belong are proposed.Abstract:
We consider the problem of detecting communities or modules in networks, groups of vertices with a higher-than-average density of edges connecting them. Previous work indicates that a robust approach to this problem is the maximization of the benefit function known as ``modularity'' over possible divisions of a network. Here we show that this maximization process can be written in terms of the eigenspectrum of a matrix we call the modularity matrix, which plays a role in community detection similar to that played by the graph Laplacian in graph partitioning calculations. This result leads us to a number of possible algorithms for detecting community structure, as well as several other results, including a spectral measure of bipartite structure in networks and a centrality measure that identifies vertices that occupy central positions within the communities to which they belong. The algorithms and measures proposed are illustrated with applications to a variety of real-world complex networks.read more
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References
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Journal ArticleDOI
Fast algorithm for detecting community structure in networks.
TL;DR: An algorithm is described which gives excellent results when tested on both computer-generated and real-world networks and is much faster, typically thousands of times faster, than previous algorithms.
Journal ArticleDOI
An efficient heuristic procedure for partitioning graphs
Brian W. Kernighan,Shou-De Lin +1 more
TL;DR: A heuristic method for partitioning arbitrary graphs which is both effective in finding optimal partitions, and fast enough to be practical in solving large problems is presented.
Book
Matrix Analysis and Applied Linear Algebra
TL;DR: The author presents Perron-Frobenius theory of nonnegative matrices Index, a theory of matrices that combines linear equations, vector spaces, and matrix algebra with insights into eigenvalues and Eigenvectors.
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Assortative mixing in networks.
Mark Newman,Mark Newman +1 more
TL;DR: This work proposes a model of an assortatively mixed network and finds that networks percolate more easily if they are assortative and that they are also more robust to vertex removal.