Finding community structure in networks using the eigenvectors of matrices

M. E. J. Newman
Phys. Rev. E 74, 036104 – Published 11 September 2006

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.

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  • Received 19 May 2006

DOI:https://doi.org/10.1103/PhysRevE.74.036104

©2006 American Physical Society

Authors & Affiliations

M. E. J. Newman

  • Department of Physics and Center for the Study of Complex Systems, University of Michigan, Ann Arbor, Michigan 48109, USA

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Issue

Vol. 74, Iss. 3 — September 2006

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