Minimax Rates of Community Detection in Stochastic Block Models

TL;DR

This paper develops a comprehensive minimax theory for community detection in stochastic block models, establishing exponential rates and phase transition thresholds for various network settings.

Contribution

It introduces the first general minimax rates for community detection across diverse SBM settings, including sparse and dense networks with multiple communities.

Findings

Minimax rates are exponential, differing from typical polynomial rates.

Threshold phenomena are established for exact and partial recovery.

Upper bounds are achieved via penalized likelihood approaches.

Abstract

Recently network analysis has gained more and more attentions 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.