Modularity-Maximizing Network Communities via Mathematical Programming

TL;DR

This paper introduces two novel algorithms based on mathematical programming rounding techniques for maximizing network modularity, providing approximation guarantees and demonstrating competitive performance on standard benchmarks.

Contribution

The paper presents new algorithms for modularity maximization using linear and vector program rounding, with approximation guarantees and improved performance.

Findings

Algorithms perform comparably or better than existing methods.

Linear programming approach offers an a posteriori approximation guarantee.

Vector programming provides bounds for bipartitioning quality.

Abstract

In many networks, it is of great interest to identify "communities", unusually densely knit groups of individuals. Such communities often shed light on the function of the networks or underlying properties of the individuals. Recently, Newman suggested "modularity" as a natural measure of the quality of a network partitioning into communities. Since then, various algorithms have been proposed for (approximately) maximizing the modularity of the partitioning determined. In this paper, we introduce the technique of rounding mathematical programs to the problem of modularity maximization, presenting two novel algorithms. More specifically, the algorithms round solutions to linear and vector programs. Importantly, the linear programing algorithm comes with an a posteriori approximation guarantee: by comparing the solution quality to the fractional solution of the linear program, a bound on the available "room for improvement" can be obtained. The vector programming algorithm provides a similar bound for the best partition into two communities. We evaluate both algorithms using experiments on several standard test cases for network partitioning algorithms, and find that they perform comparably or better than past algorithms.