Gradient flow-based modularity maximization for community detection in multiplex networks

TL;DR

This paper introduces two gradient flow-based methods for community detection in multiplex networks, offering a computationally efficient alternative to existing techniques with comparable accuracy.

Contribution

The paper presents novel gradient flow formulations for multiplex community detection, including a total variation approach and a direct modularity maximization, with efficient numerical solutions.

Findings

Methods are competitive with state-of-the-art techniques.

Significant reduction in computational complexity.

Runtimes are orders of magnitude faster for large networks.

Abstract

We propose two methods for the unsupervised detection of communities in undirected multiplex networks. These networks consist of multiple layers that record different relationships between the same entities or incorporate data from different sources. Both methods are formulated as gradient flows of suitable energy functionals: the first (MPBTV) builds on the minimization of a balanced total variation functional, which we show to be equivalent to multiplex modularity maximization, while the second (DGFM3) directly maximizes multiplex modularity. The resulting non-linear matrix-valued ordinary differential equations (ODEs) are solved efficiently by a graph Merriman--Bence--Osher (MBO) scheme. Key to the efficiency is the approximate integration of the discrete linear differential operators by truncated eigendecompositions in the matrix exponential function. Numerical experiments on several real-world multiplex networks show that our methods are competitive with the state of the art with respect to various metrics. Their major benefit is a significant reduction of computational complexity leading to runtimes that are orders of magnitude faster for large multiplex networks.