Asymptotic normality of maximum likelihood and its variational approximation for stochastic blockmodels

Variational methods for parameter estimation are an active research area, potentially offering computationally tractable heuristics with theoretical performance bounds. We build on recent work that applies such methods to network data, and establish asymptotic normality rates for parameter estimates of stochastic blockmodel data, by either maximum likelihood or variational estimation. The result also applies to various sub-models of the stochastic blockmodel found in the literature.

💡 Research Summary

This paper investigates the asymptotic distribution of two widely used estimators for the stochastic blockmodel (SBM): the classical maximum‑likelihood estimator (MLE) and a variational approximation (VA) obtained via a variational EM algorithm. The authors first formalize the SBM with a fixed number of blocks K, block proportion vector π, and inter‑block connection probability matrix Θ. Observations consist of an undirected adjacency matrix A where each entry A_{ij} is an independent Bernoulli random variable with success probability θ_{z_i z_j}, the latent block assignments z_i being i.i.d. categorical with probabilities π.

The main contribution is a pair of theorems establishing that, under a set of regularity conditions, both estimators are √n‑consistent and asymptotically normal: \