Consistency of modularity clustering on random geometric graphs

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 Xn={X1,X2,.,Xn}, distributed according to a probability measure . supported on a bounded domain D.Rd. Among other results, we show, via a Gamma convergence framework, a geometric form of consistency: When the number of clusters, or partitioning sets of Xn 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.