Normalized Likelihood Criteria for Model Selection in the Stochastic Block Model

TL;DR

This paper introduces normalized likelihood criteria-based estimators for determining the number of communities in stochastic block models, demonstrating their strong consistency and practical effectiveness.

Contribution

It proposes and analyzes the DNML and NMCL estimators, providing theoretical guarantees and empirical evidence of their performance in network community detection.

Findings

DNML estimator has explicit form with cubic complexity

Both NMCL and DNML estimators are strongly consistent for sparse networks

DNML performs well in unbalanced networks

Abstract

Estimating the number of communities is a fundamental problem in network analysis under the stochastic block model (SBM). In this paper, we study penalized estimators for this task based on normalized likelihood criteria. We show that a penalized estimator derived from the Normalized Maximum Likelihood (NML) is strongly consistent with a logarithmic penalty term, although its computation is intractable. To overcome this limitation, we consider the Normalized Maximum Complete Likelihood (NMCL) and the Decomposed Normalized Maximum Likelihood (DNML). The DNML admits an explicit formulation with cubic computational complexity in the number of nodes. We prove that the NMCL- and DNML- based estimators are strongly consistent for sparse networks in which the average node degree diverges with the network size. Empirical results show that the DNML estimator performs competitively with existing methods, particularly in unbalanced networks.