A Sparse Completely Positive Relaxation of the Modularity Maximization for Community Detection

TL;DR

This paper introduces a novel sparse, low-rank completely positive relaxation for modularity maximization in community detection, with an efficient algorithm and strong theoretical guarantees, outperforming existing methods on large sparse networks.

Contribution

It proposes a new relaxation approach and an efficient algorithm for community detection, with theoretical convergence and misclassification bounds, applicable to very large networks.

Findings

Outperforms existing methods in accuracy and efficiency

Effective on networks with over 50 million nodes

Provides theoretical guarantees for convergence and misclassification

Abstract

In this paper, we consider the community detection problem under either the stochastic block model (SBM) assumption or the degree-correlated stochastic block model (DCSBM) assumption. The modularity maximization formulation for the community detection problem is NP-hard in general. In this paper, we propose a sparse and low-rank completely positive relaxation for the modularity maximization problem, we then develop an efficient row-by-row (RBR) type block coordinate descent (BCD) algorithm to solve the relaxation and prove an $\mathcal{O}(1/\sqrt{N})$ convergence rate to a stationary point where $N$ is the number of iterations. A fast rounding scheme is constructed to retrieve the community structure from the solution. Non-asymptotic high probability bounds on the misclassification rate are established to justify our approach. We further develop an asynchronous parallel RBR algorithm to speed up the convergence. Extensive numerical experiments on both synthetic and real world networks show that the proposed approach enjoys advantages in both clustering accuracy and numerical efficiency. Our numerical results indicate that the newly proposed method is a quite competitive alternative for community detection on sparse networks with over 50 million nodes.