Community detection using fast low-cardinality semidefinite programming

TL;DR

This paper introduces a scalable low-cardinality semidefinite programming algorithm for community detection that outperforms existing heuristics like Louvain and Leiden, achieving near-global optimality on real-world networks.

Contribution

It generalizes local update heuristics to a semidefinite relaxation framework, enabling scalable and more accurate community detection.

Findings

Achieves global semidefinite optimality in small cases

Outperforms state-of-the-art algorithms on real datasets

Scales efficiently with network size

Abstract

Modularity maximization has been a fundamental tool for understanding the community structure of a network, but the underlying optimization problem is nonconvex and NP-hard to solve. State-of-the-art algorithms like the Louvain or Leiden methods focus on different heuristics to help escape local optima, but they still depend on a greedy step that moves node assignment locally and is prone to getting trapped. In this paper, we propose a new class of low-cardinality algorithm that generalizes the local update to maximize a semidefinite relaxation derived from max-k-cut. This proposed algorithm is scalable, empirically achieves the global semidefinite optimality for small cases, and outperforms the state-of-the-art algorithms in real-world datasets with little additional time cost. From the algorithmic perspective, it also opens a new avenue for scaling-up semidefinite programming when the solutions are sparse instead of low-rank.