The Annals of Statistics

Minimax rates of community detection in stochastic block models

Anderson Y. Zhang and Harrison H. Zhou

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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.

Article information

Ann. Statist. Volume 44, Number 5 (2016), 2252-2280.

Received: July 2015
Revised: December 2015
First available in Project Euclid: 12 September 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G05: Foundations of stochastic processes

Network community detection stochastic block model minimax rate


Zhang, Anderson Y.; Zhou, Harrison H. Minimax rates of community detection in stochastic block models. Ann. Statist. 44 (2016), no. 5, 2252--2280. doi:10.1214/15-AOS1428.

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Supplemental materials

  • Supplement to “Mimimax rates of community detection in stochastic block models”. In the supplement [31], we provide proofs of Lemma 5.2, Propositions 5.1 and 5.2. We also provide proofs for Theorems 2.1 and 3.1, which extend the minimax results of Theorems 2.2 and 3.2 to a larger parameter space $\Theta$. In addition, we state and prove the asymptotic equivalence of $I$.