Modularity of Erdos-Rényi random graphs

© Colin McDiarmid and Fiona Skerman; licensed under Creative Commons License CC-BY. For a given graph G, modularity gives a score to each vertex partition, with higher values taken to indicate that the partition better captures community structure in G. The modularity q∗(G) (where 0 ≤ q∗(G) ≤ 1) of the graph G is defined to be the maximum over all vertex partitions of the modularity value. Given the prominence of modularity in community detection, it is an important graph parameter to understand mathematically. For the Erdos-Rényi random graph Gn,pwith n vertices and edge-probability p, the likely modularity has three distinct phases. For np ≤ 1 + o(1) the modularity is 1 + o(1) with high probability (whp), and for np → 1 the modularity is o(1) whp. Between these regions the modularity is non-trivial: for constants 1 < c0 ≤ c1 there exists δ > 0 such that when c0 ≤ np ≤ c1 we have δ < q∗(G) < 1 - δ whp. For this critical region, we show that whp q∗(Gn, p) has order (np)-1/2, in accord with a conjecture by Reichardt and Bornholdt in 2006 (and disproving another conjecture from the physics literature).