Detectability threshold of the spectral method for graph partitioning

Abstract

Graph partitioning, or community detection, is an important tool for investigating the structures embedded in real data. The spectral method is a major algorithm for graph partitioning and is also analytically tractable. In order to analyze the performance of the spectral method, we consider a regular graph of two loosely connected clusters, each of which consists of a random graph, i.e., a random graph with a planted partition. Since we focus on the bisection of regular random graphs, whether the unnormalized Laplacian, the normalized Laplacian, or the modularity matrix is used does not make a difference. Using the replica method, which is often used in the field of spin-glass theory, we estimate the so-called detectability threshold; that is, the threshold above which the partition obtained by the method is completely uncorrelated with the planted partition.