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. (2008). 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.

翻译：社区检测是现代网络科学中最关键的问题之一，其应用涵盖从蛋白质建模到社交网络分析等多个领域。近年来，涌现出大量研究重叠社区检测的论文——在这类问题中，网络中的每个节点可能同时属于多个社区。本文考虑由Airoldi等人(2008)首次提出的混合隶属度随机块模型(MMSB)。MMSB为图结构中重叠社区结构的建模提供了相当通用的框架。本文的核心问题是在给定观测网络的情况下重建社区间的关联关系。我们比较了不同方法，确立了估计误差的极小化最优下界，并提出了一种能达到该下界的新估计量。在模型相当宽松的一般性条件下，我们证明了理论结果。最后，通过一系列实验对理论进行了验证。