Hypergraph clustering by iteratively reweighted modularity maximization

Learning on graphs is a subject of great interest due to the abundance of relational data from real-world systems. Many of these systems involve higher-order interactions (super-dyadic) rather than mere pairwise (dyadic) relationships; examples of these are co-authorship, co-citation, and metabolic reaction networks. Such super-dyadic relations are more adequately modeled using hypergraphs rather than graphs. Learning on hypergraphs has thus been garnering increased attention with potential applications in network analysis, VLSI design, and computer vision, among others. Especially, hypergraph clustering is gaining attention because of its enormous applications such as component placement in VLSI, group discovery in bibliographic systems, image segmentation in CV, etc. For the problem of clustering on graphs, modularity maximization has been known to work well in the pairwise setting. Our primary contribution in this article is to provide a generalization of the modularity maximization framework for clustering on hypergraphs. In doing so, we introduce a null model for graphs generated by hypergraph reduction and prove its equivalence to the configuration model for undirected graphs. The proposed graph reduction technique preserves the node degree sequence from the original hypergraph. The modularity function can be defined on a thus reduced graph, which can be maximized using any standard modularity maximization method, such as the Louvain method. We additionally propose an iterative technique that provides refinement over the obtained clusters. We demonstrate both the efficacy and efficiency of our methods on several real-world datasets.