Modularity of random intersection graphs
Katarzyna Rybarczyk
TL;DR
This work analyzes the modularity of the classical random intersection graph ${\mathcal{G}(n,m,p)}$, identifying parameter regimes where modularity reliably detects community structure and regimes where communities exist but remain hidden from modularity. Using edge-count analyses, volume estimates, Freedman and Chernoff-type bounds, and a two-block partition framework, the authors establish both lower and upper bounds on modularity across sparse regimes, including ${mp^2=o(1)}$, ${n^2mp^2\to\infty}$, and cases with ${np\to\infty}$ or ${np=o(1)}$. A key result is that modularity remains large (approaching 1) when ${mp\to 0}$ and ${np\to\infty}$ in certain ranges, while in other regimes modularity is tightly bounded by ${O}\left(\sqrt{\frac{\ln(m/n)}{np}}+\frac{n}{m}+\omega mp^2\right)$ and can vanish as ${n\to\infty}$. The paper also demonstrates a coupling to an Erdős–Rényi model ${G(n,\bar p)}$, showing that modularity of ${\mathcal{G}(n,m,p)}$ closely tracks modularity of ${G(n,\bar p)}$ in relevant settings, thus linking community-structure measures across bipartite-affiliation network models. These results illuminate when modularity is a meaningful detector of communities in affiliation-based random graphs and suggest directions for studying more general random intersection models.
Abstract
Modularity was introduced by Newman and Girvan in 2004 and is used as a measure of community structure of networks represented by graphs. In our work we study modularity of the random intersection graph model first considered by Karoński, Scheinerman, and Singer--Cohen in 1999. Since their introduction, random intersection graphs has attracted much attention, mostly due to their application as networks models. In our work we determine the range of parameters in which modularity detects well the community structure of the random intersection graphs, as well as give a range of parameters for which there is a community structure present but not revealed by modularity. We also relate modularity of the random intersection graph to the modularity of other known random graph models.