Exact Recovery of Community Detection in k-partite Graph Models

TL;DR

This paper establishes sharp thresholds for exact community detection in k-partite graphs with Gaussian noise, introducing a new SDP approach that works efficiently for multiple communities.

Contribution

It provides the first sharp thresholds for exact recovery in k-community models with Gaussian perturbations and develops a novel complex SDP method for efficient detection.

Findings

Sharp thresholds for Gaussian perturbation intensity $\sigma$ for exact recovery.

A new complex SDP method enables efficient community detection for k>2.

Thresholds are independent of community size distribution.

Abstract

We study the vertex classification problem on a graph whose vertices are in $k\ (k\geq 2)$ different communities, edges are only allowed between distinct communities, and the number of vertices in different communities are not necessarily equal. The observation is a weighted adjacency matrix, perturbed by a scalar multiple of the Gaussian Orthogonal Ensemble (GOE), or Gaussian Unitary Ensemble (GUE) matrix. For the exact recovery of the maximum likelihood estimation (MLE) with various weighted adjacency matrices, we prove sharp thresholds of the intensity $\sigma$ of the Gaussian perturbation. These weighted adjacency matrices may be considered as natural models for the electric network. Surprisingly, these thresholds of $\sigma$ do not depend on whether the sample space for MLE is restricted to such classifications that the number of vertices in each group is equal to the true value. In contrast to the $\ZZ_2$-synchronization, a new complex version of the semi-definite programming (SDP) is designed to efficiently implement the community detection problem when the number of communities $k$ is greater than 2, and a common region (independent of $k$) for $\sigma$ such that SDP exactly recovers the true classification is obtained.