Finding community structure in networks using the eigenvectors of matrices

TL;DR

This paper reframes network community detection as a spectral optimization problem using the modularity matrix B = A − P, enabling efficient, scalable algorithms that rival existing methods. By expressing modularity as Q = (1/4m) s^T B s and extending to multiway partitions with S, it reveals fundamental links between community structure and eigenvectors, including the roles of positive and negative eigenvalues. The work introduces practical leading-eigenvector, vector-partitioning, and refinement algorithms, discusses implementation details for sparse networks, and demonstrates additional uses such as detecting bipartite structure and defining a measure of community centrality. Through theoretical insight and real-network applications, it advances both the methodology and understanding of how communities arise in complex systems.

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 new centrality measure that identifies those 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.

Figures (8)

• Figure 1: (a) The mesh network of Bern et al.BEG90. (b) The best division into equal-sized parts found by the spectral partitioning algorithm based on the Laplacian matrix.
• Figure 2: The dolphin social network of Lusseau et al.Lusseau03a. The dashed curve represents the division into two equally sized parts found by a standard spectral partitioning calculation (Section \ref{['specpart']}). The solid curve represents the division found by the modularity-based method of this section. And the squares and circles represent the actual division of the network observed when the dolphin community split into two as a result of the departure of a keystone individual. (The individual who departed is represented by the triangle.)
• Figure 3: The network of political books described in the text. Vertex colors range from blue to red to represent the values of the corresponding elements of the leading eigenvector of the modularity matrix.
• Figure 4: A plot of the vertex vectors $\mathbf{r}_i$ for a small network with $p=2$. The dotted line represents one of the $n$ possible topologically distinct cut planes.
• Figure 5: Division by the method of optimal modularity of a simple network consisting of eight vertices in a line. (a) The optimal division into just two parts separates the network symmetrically into two groups of four vertices each. (b) The optimal division into any number of parts divides the network into three groups as shown here.