On the modularity of 3‐regular random graphs and random graphs with given degree sequences

• Published in: Random structures & algorithms Vol. 61; no. 4; pp. 754 - 802
• Main Authors: Lichev, Lyuben, Mitsche, Dieter
• Format: Journal Article
• Language: English
• Published: New York John Wiley & Sons, Inc 01.12.2022; Wiley Subscription Services, Inc; Wiley
• Subjects: Asymptotic properties; Graphs; Mathematics; Modularity; random graphs with given degree sequence; random regular graphs; Upper bounds
• ISSN: 1042-9832; 1098-2418
• DOI: 10.1002/rsa.21080

The modularity of a graph is a parameter that measures its community structure; the higher its value (between 0 and 1), the more clustered the graph is. In this paper we show that the modularity of a random 3‐regular graph is at least 0.667026 asymptotically almost surely (a.a.s.), thereby proving a conjecture of McDiarmid and Skerman. We also improve the a.a.s. upper bound given therein to 0.789998. For a uniformly chosen graph Gn over a given bounded degree sequence with average degree d(Gn) and with |CC(Gn)| many connected components, we distinguish two regimes with respect to the existence of a giant component. In the subcritical regime, we compute the second term of the modularity. In the supercritical regime, we prove that there is ε>0, for which the modularity is a.a.s. at least 21−μd(Gn)+ε,where μ is the asymptotically almost sure limit of |CC(Gn)|n.