Clustering via Hypergraph Modularity

TL;DR

This paper introduces a hypergraph modularity measure that extends graph modularity, providing theoretical foundations and heuristic algorithms for clustering hypergraphs more effectively than traditional graph-based methods.

Contribution

It proposes a novel hypergraph modularity function and generalizes the Chung-Lu model for hypergraphs, along with heuristic algorithms for clustering.

Findings

Hypergraph modularity often results in fewer hyperedges cut.

The proposed method generalizes existing graph modularity.

Heuristic algorithms work on small illustrative examples.

Abstract

Despite the fact that many important problems (including clustering) can be described using hypergraphs, theoretical foundations as well as practical algorithms using hypergraphs are not well developed yet. In this paper, we propose a hypergraph modularity function that generalizes its well established and widely used graph counterpart measure of how clustered a network is. In order to define it properly, we generalize the Chung-Lu model for graphs to hypergraphs. We then provide the theoretical foundations to search for an optimal solution with respect to our hypergraph modularity function. Two simple heuristic algorithms are described and applied to a few small illustrative examples. We show that using a strict version of our proposed modularity function often leads to a solution where a smaller number of hyperedges get cut as compared to optimizing modularity of 2-section graph of a hypergraph.