A Tensor Approach to Learning Mixed Membership Community Models

TL;DR

This paper introduces a tensor spectral decomposition method for guaranteed community detection in overlapping community models, extending beyond traditional non-overlapping models like the stochastic block model.

Contribution

It provides a unified, fast tensor-based approach for learning mixed membership community models with provable guarantees, including finite sample analysis.

Findings

Guaranteed recovery of community memberships and parameters

Method matches best known scaling for stochastic block models

Efficient linear algebraic learning procedure

Abstract

Community detection is the task of detecting hidden communities from observed interactions. Guaranteed community detection has so far been mostly limited to models with non-overlapping communities such as the stochastic block model. In this paper, we remove this restriction, and provide guaranteed community detection for a family of probabilistic network models with overlapping communities, termed as the mixed membership Dirichlet model, first introduced by Airoldi et al. This model allows for nodes to have fractional memberships in multiple communities and assumes that the community memberships are drawn from a Dirichlet distribution. Moreover, it contains the stochastic block model as a special case. We propose a unified approach to learning these models via a tensor spectral decomposition method. Our estimator is based on low-order moment tensor of the observed network, consisting of 3-star counts. Our learning method is fast and is based on simple linear algebraic operations, e.g. singular value decomposition and tensor power iterations. We provide guaranteed recovery of community memberships and model parameters and present a careful finite sample analysis of our learning method. As an important special case, our results match the best known scaling requirements for the (homogeneous) stochastic block model.