Finding community structure in networks using the eigenvectors of matrices

TLDR

Community detection in networks seeks to identify modules of densely connected vertices, and prior work shows that maximizing modularity is a robust approach. The study demonstrates that modularity maximization can be expressed via the eigenspectrum of a modularity matrix, analogous to the graph Laplacian in partitioning. The authors derive algorithms and measures from the modularity matrix eigenspectrum and illustrate them on diverse real‑world networks. The approach yields algorithms for community detection, a spectral bipartite structure measure, and a centrality metric highlighting vertices central to their communities.

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.