Consistency of modularity clustering on random geometric graphs

We consider a large class of random geometric graphs constructed from samples Xn = X1,X2,ldots,Xn of independent, identically distributed observations of an underlying probability measure nu on a bounded domain Dsubset ℝd. The popular `modularity' clustering method specifies a partition Un of the set Xn 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 Un converge in a certain sense to a continuum partition U of the underlying domain D, characterized as the solution of a type of Kelvin's shape optimization problem.