Optimal Noise Reduction in Dense Mixed-Membership Stochastic Block Models under Diverging Spiked Eigenvalues Condition

TL;DR

This paper addresses optimal noise reduction in dense mixed-membership stochastic block models, establishing theoretical bounds and proposing an estimator that achieves minimax optimality under general conditions.

Contribution

It introduces a new estimator for MMSB that matches the minimax lower bound, advancing the theoretical understanding of community detection in overlapping networks.

Findings

Proved minimax lower bounds on estimation error.

Developed an estimator matching the lower bounds.

Validated theoretical results through experiments.

Abstract

Community detection is one of the most critical problems in modern network science. Its applications can be found in various fields, from protein modeling to social network analysis. Recently, many papers appeared studying the problem of overlapping community detection, where each node of a network may belong to several communities. In this work, we consider Mixed-Membership Stochastic Block Model (MMSB) first proposed by Airoldi et al. MMSB provides quite a general setting for modeling overlapping community structure in graphs. The central question of this paper is to reconstruct relations between communities given an observed network. We compare different approaches and establish the minimax lower bound on the estimation error. Then, we propose a new estimator that matches this lower bound. Theoretical results are proved under fairly general conditions on the considered model. Finally, we illustrate the theory in a series of experiments.