Modularity functions maximization with nonnegative relaxation facilitates community detection in networks

TL;DR

This paper introduces a nonnegative relaxation approach to modularity maximization for community detection, leading to more accurate and interpretable community assignments, including overlapping communities, outperforming traditional spectral methods.

Contribution

The paper proposes a novel nonnegative relaxation method for modularity maximization, enabling direct community assignment and overlapping community detection, improving upon spectral relaxation techniques.

Findings

Outperforms traditional spectral relaxation methods.

Effectively detects overlapping communities.

Provides solutions close to ideal community indicators.

Abstract

We show here that the problem of maximizing a family of quantitative functions, encompassing both the modularity (Q-measure) and modularity density (D-measure), for community detection can be uniformly understood as a combinatoric optimization involving the trace of a matrix called modularity Laplacian. Instead of using traditional spectral relaxation, we apply additional nonnegative constraint into this graph clustering problem and design efficient algorithms to optimize the new objective. With the explicit nonnegative constraint, our solutions are very close to the ideal community indicator matrix and can directly assign nodes into communities. The near-orthogonal columns of the solution can be reformulated as the posterior probability of corresponding node belonging to each community. Therefore, the proposed method can be exploited to identify the fuzzy or overlapping communities and thus facilitates the understanding of the intrinsic structure of networks. Experimental results show that our new algorithm consistently, sometimes significantly, outperforms the traditional spectral relaxation approaches.