Modularity of Erd\H{o}s-R\'enyi random graphs

TL;DR

This paper analyzes the behavior of the modularity measure in Erdős-Rényi random graphs, revealing phase transitions and confirming a conjecture about its order in different regimes, with implications for community detection algorithms.

Contribution

It provides a rigorous analysis of the modularity of Erdős-Rényi graphs, including phase transition thresholds and validation of a 2006 conjecture on its order.

Findings

Modularity approaches 1 for sparse graphs with np up to 1

Modularity scales as (np)^{-1/2} when np ≥ 1

Modularity is robust to small edge modifications

Abstract

For a given graph $G$, each partition of the vertices has a modularity score, with higher values indicating that the partition better captures community structure in $G$. The modularity $q^*(G)$ of the graph $G$ is defined to be the maximum over all vertex partitions of the modularity score, and satisfies $0\leq q^*(G) < 1$. Modularity is at the heart of the most popular algorithms for community detection. We investigate the behaviour of the modularity of the Erd\H{o}s-R\'enyi random graph $G_{n,p}$ with $n$ vertices and edge-probability $p$. Two key findings are that the modularity is $1+o(1)$ with high probability (whp) for $np$ up to $1+o(1)$ and no further; and when $np \geq 1$ and $p$ is bounded below 1, it has order $(np)^{-1/2}$ whp, in accord with a conjecture by Reichardt and Bornholdt in 2006. We also show that the modularity of a graph is robust to changes in a few edges, in contrast to the sensitivity of optimal vertex partitions.