Modularity of Erdös-Rényi Random Graphs

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 Erdös-Rényi random graph G_{n,p} with 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 --> infty the modularity is o(1) whp. Between these regions the modularity is non-trivial: for constants 1 < c_0 <= c_1 there exists delta>0 such that when c_0 <= np <= c_1 we have delta<q^*(G)<1-delta whp. For this critical region, we show that whp q^*(G_{n,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).