Likelihood-based model selection for stochastic block models

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.

Evaluating Overfit and Underfit in Models of Network Community Structure

“…This first group of methods includes a variety of regu- [20] Minimum description length MDL (DC-SBM) [20] Minimum description length S-NB [9] Spectral with non-backtracking matrix S-cBHm [11] Spectral with Bethe Hessian, version m S-cBHa [11] Spectral with Bethe Hessian, version a AMOS [32] Statistical test using spectral clustering LRT-WB (DC-SBM) [13] Likelihood ratio test larization approaches for choosing the number of communities, e.g., those based on penalized likelihood scores [8], [28], various Bayesian techniques including marginalization [7], [29], cross-validation methods with probabilistic models [15], [17], compression approaches like MDL [20], and explicit model comparison such as likelihood ratio tests (LRT) [13].…”

“…• Bayesian marginalization and regularized likelihood 3 approaches [6], [7], [8], • information theoretic approaches [12], • modularity based methods [5], • spectral and other embedding techniques [9], [10], [11], [60], [63], • cross-validation methods [14], [15], and • statistical hypothesis tests [13]. We note, however, that the boundaries among these classes are not rigid and one method can belong to more than one group.…”

A review of stochastic block models and extensions for graph clustering

“…The second is the increasing adopting of the DC-SBM (Karrer and Newman, 2011), not only in modifying the model, but also in aiding model selection in different ways (Yan et al, 2014, Yan, 2016, Wang and Bickel, 2017, Hu et al, 2019. While it is particularly useful for dealing with the degree heterogeneity, the results are more mixed when it is applied to real-world networks, as seen in Section 8.…”

“…Yan (2016) presented a mixed picture in which the DC-SBM always dominated the original SBM, even more so at smaller K, but the ICL criterion actually increased with K, thus potentially requiring some penalty. While Wang and Bickel (2017) selected K = 4 using their penalised log-likelihood, closer investigation revealed that one ground-truth group matched well with one inferred group, while the other ground-truth group is split into three smaller inferred groups. However, Hu et al (2019) found that, at a certain value of the tuning parameter λ (Section 7.1) the penalised loglikelihood selected K = 1, therefore arguing for their corrected BIC that has a heavier penalty.…”

“…This maximised lower bound E Q [log π(Y, Z) − log Q(Z)] is then used by Latouche et al (2012) as the approximation of the marginal log-likelihood, under the assumption that the K-L divergence is close to zero and does not depend on K.This is also being adopted byAicher et al (2015), who calculated the Bayes factor of one model to another, according to the difference of their respective approximate marginal log-likelihoods. Similarly,Hayashi et al (2016) provided an approximation to the marginal log-likelihood, this time an asymptotic one, called the fully factorised information criterion (F 2 IC) Wang and Bickel (2017). also investigated log-likelihood ratio similar toYan et al (2014), but this time for an underfitting/overfitting K against the true K.…”