Stochastic blockmodels with growing number of classes

TL;DR

This paper analyzes stochastic blockmodels for network data, showing that with growing classes and degrees, node classification errors vanish and providing finite-sample confidence bounds, validated through simulations and real data application.

Contribution

It establishes asymptotic and finite-sample results for stochastic blockmodels with a growing number of classes, including error convergence and confidence bounds.

Findings

Misclassification fraction converges to zero with growing classes and degrees.

Finite-sample confidence bounds on parameter estimates are derived.

Application to Facebook data reveals residual network structure.

Abstract

We present asymptotic and finite-sample results on the use of stochastic blockmodels for the analysis of network data. We show that the fraction of misclassified network nodes converges in probability to zero under maximum likelihood fitting when the number of classes is allowed to grow as the root of the network size and the average network degree grows at least poly-logarithmically in this size. We also establish finite-sample confidence bounds on maximum-likelihood blockmodel parameter estimates from data comprising independent Bernoulli random variates; these results hold uniformly over class assignment. We provide simulations verifying the conditions sufficient for our results, and conclude by fitting a logit parameterization of a stochastic blockmodel with covariates to a network data example comprising a collection of Facebook profiles, resulting in block estimates that reveal residual structure.