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October 2016 Minimax rates of community detection in stochastic block models
Anderson Y. Zhang, Harrison H. Zhou
Ann. Statist. 44(5): 2252-2280 (October 2016). DOI: 10.1214/15-AOS1428

Abstract

Recently, network analysis has gained more and more attention in statistics, as well as in computer science, probability and applied mathematics. Community detection for the stochastic block model (SBM) is probably the most studied topic in network analysis. Many methodologies have been proposed. Some beautiful and significant phase transition results are obtained in various settings. In this paper, we provide a general minimax theory for community detection. It gives minimax rates of the mis-match ratio for a wide rage of settings including homogeneous and inhomogeneous SBMs, dense and sparse networks, finite and growing number of communities. The minimax rates are exponential, different from polynomial rates we often see in statistical literature. An immediate consequence of the result is to establish threshold phenomenon for strong consistency (exact recovery) as well as weak consistency (partial recovery). We obtain the upper bound by a range of penalized likelihood-type approaches. The lower bound is achieved by a novel reduction from a global mis-match ratio to a local clustering problem for one node through an exchangeability property.

Citation

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Anderson Y. Zhang. Harrison H. Zhou. "Minimax rates of community detection in stochastic block models." Ann. Statist. 44 (5) 2252 - 2280, October 2016. https://doi.org/10.1214/15-AOS1428

Information

Received: 1 July 2015; Revised: 1 December 2015; Published: October 2016
First available in Project Euclid: 12 September 2016

zbMATH: 1355.60125
MathSciNet: MR3546450
Digital Object Identifier: 10.1214/15-AOS1428

Subjects:
Primary: 60G05

Rights: Copyright © 2016 Institute of Mathematical Statistics

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Vol.44 • No. 5 • October 2016
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