Hypergraph Community Detection With Fuzzy Memberships

Hypergraph community detection reveals both mesoscale structures and functional characteristics of real-life hypergraphs. Although many methods have been developed from diverse perspectives, to our knowledge, none can provide fine-grained hypergraph fuzzy community information, and the quality of identified crisp partitions is often limited. This study defines a set of novel concepts of multi-scale hypergraph fuzzy memberships that systematically quantify the partial belongingness (i.e., membership grades) among nodes, hyperedges, and hypergraph communities within the interval [0,1], thereby revealing the multi-scale partial topological affiliations. Furthermore, the multiscale hypergraph fuzzy memberships are employed to develop a general framework, named fuzzy membership-assisted hypergraph modularity optimization (FMHMO), aiming to approximate the modularity-optimal crisp hypergraph partition. The FMHMO framework comprises two key strategies: Hyperedge similarity-based hypergraph reduction (HS-HR) and fuzzy membership-based hypergraph partition recovery (FM-HPR). In particular, HS-HR reduces a hypergraph to a simple weighted graph, mapping hyperedges as nodes and their interactions as edges using novel hyperedge similarity as weights. Thereby, preserving intrahyperedge and interhyperedge topologies of the original hypergraph. FM-HPR recovers a hypergraph partition from the approximately modularity-optimal weighted graph partition by quantifying the fuzzy memberships of bridge nodes connecting multiple incident hyperedges, thereby enabling precise assignment of their crisp hypergraph community affiliations. Experimental results in both synthetic and real-world datasets indicate the superiority of FMHMO over state-of-the-art hypergraph community detection algorithms in accuracy and modularity quality.