  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.
