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February 2018 On semidefinite relaxations for the block model
Arash A. Amini, Elizaveta Levina
Ann. Statist. 46(1): 149-179 (February 2018). DOI: 10.1214/17-AOS1545

Abstract

The stochastic block model (SBM) is a popular tool for community detection in networks, but fitting it by maximum likelihood (MLE) involves a computationally infeasible optimization problem. We propose a new semidefinite programming (SDP) solution to the problem of fitting the SBM, derived as a relaxation of the MLE. We put ours and previously proposed SDPs in a unified framework, as relaxations of the MLE over various subclasses of the SBM, which also reveals a connection to the well-known problem of sparse PCA. Our main relaxation, which we call SDP-1, is tighter than other recently proposed SDP relaxations, and thus previously established theoretical guarantees carry over. However, we show that SDP-1 exactly recovers true communities over a wider class of SBMs than those covered by current results. In particular, the assumption of strong assortativity of the SBM, implicit in consistency conditions for previously proposed SDPs, can be relaxed to weak assortativity for our approach, thus significantly broadening the class of SBMs covered by the consistency results. We also show that strong assortativity is indeed a necessary condition for exact recovery for previously proposed SDP approaches and not an artifact of the proofs. Our analysis of SDPs is based on primal-dual witness constructions, which provides some insight into the nature of the solutions of various SDPs. In particular, we show how to combine features from SDP-1 and already available SDPs to achieve the most flexibility in terms of both assortativity and block-size constraints, as our relaxation has the tendency to produce communities of similar sizes. This tendency makes it the ideal tool for fitting network histograms, a method gaining popularity in the graphon estimation literature, as we illustrate on an example of a social networks of dolphins. We also provide empirical evidence that SDPs outperform spectral methods for fitting SBMs with a large number of blocks.

Citation

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Arash A. Amini. Elizaveta Levina. "On semidefinite relaxations for the block model." Ann. Statist. 46 (1) 149 - 179, February 2018. https://doi.org/10.1214/17-AOS1545

Information

Received: 1 January 2016; Revised: 1 November 2016; Published: February 2018
First available in Project Euclid: 22 February 2018

zbMATH: 06865108
MathSciNet: MR3766949
Digital Object Identifier: 10.1214/17-AOS1545

Subjects:
Primary: 62G20 , 62H99 , 90C22

Keywords: Community detection , network , semidefinite programming , Stochastic block model

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.46 • No. 1 • February 2018
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