Higher-order Fuzzy Membership in Motif Modularity Optimization

Higher-order community detection (HCD) reveals both mesoscale structures and functional characteristics of real-life 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 presents a novel concept of higher-order fuzzy memberships that quantify the membership grades of motifs to crisp higher-order communities, thereby revealing the partial community affiliations. Furthermore, we employ higher-order fuzzy memberships to enhance HCD via a general framework called fuzzy memberships assisted motif-based evolutionary modularity (FMMEM). In FFMEM, on the one hand, a fuzzy membership-based neighbor community modification (FM-NCM) strategy is designed to correct misassigned bridge nodes, thereby improving partition quality. On the other hand, a fuzzy membership-based local community merging (FM-LCM) strategy is also proposed to combine excessively fragmented communities for enhancing local search ability. Experimental results indicate that the FMMEM framework outperforms state-of-the-art methods in both synthetic and real-world datasets, particularly in the networks with ambiguous and complex structures.