Bipartite communities via spectral partitioning

Abstract

In this paper we are interested in finding communities with bipartite structure. A bipartite community is a pair of disjoint vertex sets S, $$S'$$ such that the number of edges with one endpoint in S and the other endpoint in $$S'$$ is “significantly more than expected.” This additional structure is natural to some applications of community detection. In fact, using other terminology, they have already been used to study correlation networks, social networks, and two distinct biological networks. In 2012 two groups independently [(1) Lee, Oveis Gharan, and Trevisan and (2) Louis, Raghavendra, Tetali, and Vempala] used higher eigenvalues of the normalized Laplacian to find an approximate solution to the k-sparse-cuts problem. In 2015 Liu generalized spectral methods for finding k communities to find k bipartite communities. Our approach improves the bounds on bipartite conductance (measure of strength of a bipartite community) found by Liu and also implies improvements to the original spectral methods by Lee et al. and Louis et al. We also highlight experimental results found when applying a practical algorithm derived from our theoretical results to three distinct real-world networks.