Modularity of Erdos-Rényi random graphs

Conference item

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).

Authors: McDiarmid, C; Skerman, F

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik
• Host title: Leibniz International Proceedings in Informatics, LIPIcs
• Journal: Leibniz International Proceedings in Informatics
• Volume: 110
• Article number: 31
• Series: 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018)
• Publication date: 2018-06-01
• Acceptance date: 2018-04-09
• DOI: 10.4230/LIPIcs.AofA.2018.31
• ISSN: 1868-8969
• ISBN: 9783959770781
• Keywords: community detection; Newman-Girvan Modularity; random graphs