Nonparametric identification and estimation of stochastic block models from many small networks

This paper concerns the analysis of network data when unobserved node-specific heterogeneity is present. We postulate a weighted version of the classic stochastic block model, where nodes belong to one of a finite number of latent communities and the placement of edges between them and any weight assigned to these depend on the communities to which the nodes belong. A simple rank condition is presented under which we establish that the number of latent communities, their distribution, and the conditional distribution of edges and weights given community membership are all nonparametrically identified from knowledge of the joint (marginal) distribution of edges and weights in graphs of a fixed size. The identification argument is constructive and we present a computationally-attractive nonparametric estimator based on it. Limit theory is derived under asymptotics where we observe a growing number of independent networks of a fixed size. The results of a series of numerical experiments are reported on.