Modularity of Erdős‐Rényi random graphs

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 ≤ 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 Erdős‐Rényi 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 ≥ 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.