Likelihood-based model selection for stochastic block models

TL;DR

This paper develops a likelihood-based criterion for selecting the number of communities in stochastic block models, analyzing its asymptotic properties and extending it to degree-corrected models, suitable for large networks.

Contribution

It introduces a new model selection criterion based on likelihood ratios with proven asymptotic consistency for SBMs and DCSBMs.

Findings

Likelihood ratio statistic has normal limit under underfitting

Convergence rate established for overfitting case

Criterion remains valid with polylogarithmic degree growth

Abstract

The stochastic block model (SBM) provides a popular framework for modeling community structures in networks. However, more attention has been devoted to problems concerning estimating the latent node labels and the model parameters than the issue of choosing the number of blocks. We consider an approach based on the log likelihood ratio statistic and analyze its asymptotic properties under model misspecification. We show the limiting distribution of the statistic in the case of underfitting is normal and obtain its convergence rate in the case of overfitting. These conclusions remain valid when the average degree grows at a polylog rate. The results enable us to derive the correct order of the penalty term for model complexity and arrive at a likelihood-based model selection criterion that is asymptotically consistent. Our analysis can also be extended to a degree-corrected block model (DCSBM). In practice, the likelihood function can be estimated using more computationally efficient variational methods or consistent label estimation algorithms, allowing the criterion to be applied to large networks.