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, Lang, Dasgupta, and Mahoney, 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.

How Many Communities Are There?

“…By means of a regularized maximum likelihood estimation approach, Rohe et al (2014) further proved that this weak convergence can be achieved for K = O(N/ log 5 N ).…”

Consistent structure estimation of exponential-family random graph models with block structure

“…To demonstrate the advantages of additional structure, we consider a natural form of additional structure known as block structure. Block structure is popular in the large and growing body of literature on stochastic block models [e.g., 45,7,1,14,12,52,8,74,3,44,39,53,21,32,73,9]. We focus here on exponential-family random graph models with block structure, which allow edges within blocks to be dependent [55].…”

“…When the block structure is unknown, the first and foremost question is whether it can be recovered with high probability. A large and growing body of consistency results for stochastic block models shows that it is possible to recover the block structure of stochastic block models with high probability [e.g., 45,7,1,14,12,52,8,74,3,44,39,53,21,32,73,9]. While it is encouraging that the block structure of stochastic block models can be recovered with high probability, these results are restricted to models with independent edges within and between blocks.…”

“…Comparison with stochastic block models. There is a large and growing body of consistency results on stochastic block models [e.g., 7,14,12,52,8,3,39,53,21]. In the language of stochastic block models, the consistency result in Theorem 1 is a weak consistency result in the sense that the discrepancy between the estimated and datagenerating block structure is small with high probability.…”

Using Maximum Entry-Wise Deviation to Test the Goodness of Fit for Stochastic Block Models

“…This condition implies that k is allowed to grow linearly with n ignoring a logarithmic factor. This kind of scenario has been referred by Rohe, Qin and Fan (2014) as the highest dimensional stochastic block model as the number of communities must be smaller than the number of nodes, and no reasonable model would allow k to grow faster than that. As a result, the proposed test significantly relaxes the condition in Lei (2016).…”