What Is a Network Community?

Abstract

In this study, we introduce a novel quality function for a network community, which we refer to as the communitude. The communitude has a strong statistical background. Specifically, it measures the Z-score of a subset of vertices S with respect to the fraction of the number of edges within the subgraph induced by S. Due to the null model of a random graph used in the definition, our quality function focuses not only on the inside of the subgraph but also on the cut edges, unlike some quality functions for extracting dense subgraphs. To evaluate the detection ability of our quality function, we address the communitude maximization problem and its variants for realistic scenarios. For the problems, we propose a two-phase heuristic algorithm together with some modified versions. In the first phase, it repeatedly removes the vertex with the smallest degree, and then obtains the subgraph with maximum communitude over the iterations. In the second phase, the algorithm improves the obtained solution using a simple local search heuristic. This algorithm runs in linear time when the number of iterations is fixed to a constant; thus, it is applicable to massive graphs. Computational experiments using both synthetic graphs and real-world networks demonstrate the validity and reliability of the proposed quality function and algorithms.