The Highest Dimensional Stochastic Blockmodel with a Regularized Estimator

Abstract

In the high dimensional Stochastic Blockmodel for a random network, the number of clusters (or blocks) K grows with the number of nodes N. Two previous studies have examined the statistical estimation performance of spectral clustering and the maximum likelihood estimator under the high dimensional model; neither of these results allow K to grow faster than N^{1/2}. We study a model where, ignoring log terms, K can grow proportionally to N. Since the number of clusters must be smaller than the number of nodes, no reasonable model allows K to grow faster; thus, our asymptotic results are the "highest" dimensional. To push the asymptotic setting to this extreme, we make additional assumptions that are motivated by empirical observations in physical anthropology (Dunbar, 1992), and an in depth study of massive empirical networks (Leskovec et al 2008). Furthermore, we develop a regularized maximum likelihood estimator that leverages these insights and we prove that, under certain conditions, the proportion of nodes that the regularized estimator misclusters converges to zero. This is the first paper to explicitly introduce and demonstrate the advantages of statistical regularization in a parametric form for network analysis.