Modularity of Erdős‐Rényi random graphs

Abstract

For a given graph G, each partition of the vertices has a modularity score, 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 score. Modularity is at the heart of the most popular algorithms for community detection, so it is an important graph parameter to understand mathematically.In particular, we may want to understand the behaviour of modularity for the Erdős-Rényi random graph G n,p with n vertices and edge-probability p. Two key features which we find 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 (and disproving another conjecture from the physics literature).

Modularity and edge sampling

“…The assumption in Theorem 1.2 that the expected average degree is large is of course much stronger than the assumption in Theorem 1.1, but still e(G p ) may be much smaller than e(G). If we go much further, and assume that at most an ε-proportion of edges are missed, that is |E \ E | ≤ ε|E|, then deterministically we have |q * (G ) − q * (G)| ≤ 2ε by Lemma 5.1 from [12].…”

“…Hence by Lemma 5.1 of [12] we have |q * (G(b)) − q * (G(ka))| < 4 k ; and so q * (G(b)) > q * (G(ka)) − 4 k .…”

“…Given a constant c > 0, for each n ≥ c we let q(n, c) = E[q * (G n,c/n )], where G n,c/n is the binomial (or Erdős-Rényi) random graph with edge-probability c/n. It was conjectured in [12] that for each c > 0, q(n, c) tends to a limit q(c) as n → ∞, and it was noted there that in this case the function q(c) would be continuous in c. From Theorem 1.1 we may deduce that also q(c) would be non-increasing in c -see Section 8.…”

“…These two corollaries have very similar proofs using the corresponding theorems. The idea of the proofs is to couple the random edge sets F and E(G p ) so that they are very similar, and then use the next lemma which follows from the proofs in Section 5 of [12]. Lemma 5.1.…”

“…The unique form of an optimal partition of C 5 is into one path P 2 and one P 3 , with modularity 3 5 − 4 2 +6 2 10 2 = 3 5 − 52 100 = 2 25 . For C 5 (2) we can balance the partition, into two copies of K 2,3 , with modularity score 12 20 − 1 2 = 1 10 > 2 25 . Let G be a (fixed) graph.…”