Open Access
April 2017 Likelihood-based model selection for stochastic block models
Y. X. Rachel Wang, Peter J. Bickel
Ann. Statist. 45(2): 500-528 (April 2017). DOI: 10.1214/16-AOS1457


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.


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Y. X. Rachel Wang. Peter J. Bickel. "Likelihood-based model selection for stochastic block models." Ann. Statist. 45 (2) 500 - 528, April 2017.


Received: 1 October 2015; Revised: 1 February 2016; Published: April 2017
First available in Project Euclid: 16 May 2017

zbMATH: 1371.62017
MathSciNet: MR3650391
Digital Object Identifier: 10.1214/16-AOS1457

Primary: 62F05

Keywords: likelihood ratio statistic , model misspecification , Network communities , stochastic block models

Rights: Copyright © 2017 Institute of Mathematical Statistics

Vol.45 • No. 2 • April 2017
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