Graph spectra and the detectability of community structure in networks

TL;DR

This paper analyzes the spectral properties of networks with community structure, revealing a phase transition that determines when spectral methods can successfully detect communities, and shows spectral modularity maximization is optimal in certain regimes.

Contribution

It introduces a spectral analysis framework for community detection, identifying a phase transition and proving the optimality of spectral modularity maximization.

Findings

Spectral methods detect communities above a certain threshold.

A phase transition separates detectable and undetectable regimes.

Spectral modularity maximization is proven optimal in the detectable regime.

Abstract

We study networks that display community structure -- groups of nodes within which connections are unusually dense. Using methods from random matrix theory, we calculate the spectra of such networks in the limit of large size, and hence demonstrate the presence of a phase transition in matrix methods for community detection, such as the popular modularity maximization method. The transition separates a regime in which such methods successfully detect the community structure from one in which the structure is present but is not detected. By comparing these results with recent analyses of maximum-likelihood methods we are able to show that spectral modularity maximization is an optimal detection method in the sense that no other method will succeed in the regime where the modularity method fails.