Consistency of modularity clustering on random geometric graphs

TL;DR

This paper studies the behavior of modularity clustering on large random geometric graphs, showing that as the sample size grows, the discrete clusters converge to a continuum optimal partition of the domain.

Contribution

It establishes the geometric consistency of modularity clustering on random geometric graphs, linking discrete solutions to continuum shape optimization problems.

Findings

Discrete optimal partitions converge to continuum partitions

Consistency holds when the number of clusters is bounded

Partitions relate to Kelvin's shape optimization problem

Abstract

We consider a large class of random geometric graphs constructed from samples $\mathcal{X}_n = \{X_1,X_2,\ldots,X_n\}$ of independent, identically distributed observations of an underlying probability measure $\nu$ on a bounded domain $D\subset \mathbb{R}^d$. The popular `modularity' clustering method specifies a partition $\mathcal{U}_n$ of the set $\mathcal{X}_n$ as the solution of an optimization problem. In this paper, under conditions on $\nu$ and $D$, we derive scaling limits of the modularity clustering on random geometric graphs. Among other results, we show a geometric form of consistency: When the number of clusters is a priori bounded above, the discrete optimal partitions $\mathcal{U}_n$ converge in a certain sense to a continuum partition $\mathcal{U}$ of the underlying domain $D$, characterized as the solution of a type of Kelvin's shape optimization problem.