Higher Order Fuzzy Membership in Motif Modularity Optimization

Higher order community detection (HCD) reveals both mesoscale structures and functional characteristics of real-world networks. Although many methods have been developed from diverse perspectives, to our knowledge, none can provide fine-grained higher order fuzzy community information. This study introduces a novel concept of higher order fuzzy memberships that quantify the membership grades of motifs to crisp higher order communities, thereby revealing partial community affiliations. Furthermore, we utilize higher order fuzzy memberships to enhance HCD via a general framework called fuzzy memberships-assisted motif-based evolutionary modularity. On the one hand, a fuzzy membership-based neighbor community modification strategy is designed to correct misassigned bridge nodes, thereby improving partition quality. On the other hand, a fuzzy membership-based local community merging strategy is proposed to combine excessively fragmented communities, enhancing local search ability. Experimental results indicate that the proposed framework outperforms state-of-the-art methods in both synthetic and real-world datasets, particularly in networks with ambiguous and complex structures.