Consistency of modularity clustering on random geometric graphs

Abstract

Given a graph, the popular “modularity” clustering method specifies a partition of the vertex set as the solution of a certain optimization problem. In this paper, we discuss scaling limits of this method with respect to random geometric graphs constructed from i.i.d. points $\mathcal{X}_{n}=\{X_{1},X_{2},\ldots,X_{n}\}$, distributed according to a probability measure $\nu$ supported on a bounded domain $D\subset\mathbb{R}^{d}$. Among other results, we show, via a Gamma convergence framework, a geometric form of consistency: When the number of clusters, or partitioning sets of $\mathcal{X}_{n}$ is a priori bounded above, the discrete optimal modularity clusterings converge in a specific sense to a continuum partition of the underlying domain $D$, characterized as the solution to a “soap bubble” or “Kelvin”-type shape optimization problem.