Seminar at SMU Delhi

Given a graph, the `modularity' clustering method specifies a partition of the vertex set as the solution of a certain optimization problem. In this talk, we discuss consistency of this method with respect to random geometric graphs constructed from iid points $Y_n = (x_1, x_2, ... , x_n)$, distributed according to a probability measure v supported on a bounded domain D in $R^d$. Specifically, when the number of clusters, or partitioning sets of $Y_n$ is a priori bounded above, we derive a scaling limit: The discrete optimal modularity clusterings converge in a particular sense to a continuum partition of the underlying domain D, characterized as the solution to a `soap bubble' or `Kelvin'-type shape optimization problem.