Consistency of modularity clustering on random geometric graphs

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 $ u$ on a bounded domain $Dsubset mathbb{R}^d$. 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 $ u$ 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 Kelvins shape optimization problem.