Consistency of Modularity Clustering on Random Geometric Graphs

Abstract

We consider a large class of random geometric graphs constructed from independent, identically distributed observations of an underlying probability measure on a bounded domain. The popular `modularity' clustering method specifies a partition of a graph as the solution of an optimization problem. In this dissertation 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 converge in a certain sense to a continuum partition of the underlying domain, characterized as the solution of a type of Kelvin's shape optimization problem.