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 finegrained 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 multiscale partial topological affiliations. Furthermore, the multi-scale 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-similaritybased hypergraph reduction (HS-HR) and fuzzy-membershipbased 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 intraand inter-hyperedge topologies of the original hypergraph. FMHPR 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.