New bounds on the modularity of $G(n,p)$
Katarzyna Rybarczyk, Małgorzata Sulkowska
TL;DR
The paper sharpens the understanding of modularity in the binomial random graph $G(n,p)$ by proving an improved upper bound of the form ${\rm mod}(G(n,p)) \le \frac{3+2\sqrt{2}}{2}\cdot\frac{1}{\sqrt{d}}$ for a wide range of $d=np$, using elementary probabilistic tools and uniform edge-count fluctuations. It also provides a matching-order lower bound ${\rm mod}(G(n,p)) \ge \frac{P_*+o_{d}(1)}{\sqrt{d}}$ with $P_*=0.76321\ldots$, based on minimum-bisection results of Dembo, Montanari, and Sen. The approach connects edge-count fluctuations in induced subgraphs to modularity through a uniform bound and leverages known bisection estimates to tighten the lower bound. Together, these results establish that modularity decays on the scale of $1/\sqrt{d}$ for a broad regime of $d=np$, enhancing theoretical understanding of community structure in $G(n,p)$ and complementing spectral-gap perspectives.
Abstract
Modularity is a parameter indicating the presence of community structure in the graph. Nowadays it lies at the core of widely used clustering algorithms. We study the modularity of the most classical random graph, binomial $G(n,p)$. In 2020 McDiarmid and Skerman proved, taking advantage of the spectral graph theory and a specific subgraph construction by Coja-Oghlan from 2007, that there exists a constant $b$ such that with high probability the modularity of $G(n,p)$ is at most $b/\sqrt{np}$. The obtained constant $b$ is very big and not easily computable. We improve upon this result showing that a constant under $3$ may be derived here. Interesting is the fact that it might be obtained by basic probabilistic tools. We also address the lower bound on the modularity of $G(n,p)$ and improve the results of McDiarmid and Skerman from 2020 using estimates of bisections of random graphs derived by Dembo, Montanari, and Sen in 2017.