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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 themaximization of the benefit function known as "modularity" over possibledivisions of a network. Here we show that this maximization process can bewritten in terms of the eigenspectrum of a matrix we call the modularitymatrix, which plays a role in community detection similar to that played by thegraph Laplacian in graph partitioning calculations. This result leads us to anumber of possible algorithms for detecting community structure, as well asseveral other results, including a spectral measure of bipartite structure innetworks and a new centrality measure that identifies those vertices thatoccupy central positions within the communities to which they belong. Thealgorithms and measures proposed are illustrated with applications to a varietyof real-world complex networks.Comment: 22 pages, 8 figures, minor corrections in this versio

Finding community structure in networks using the eigenvectors of matrices