Clustering Coefficient Reflecting Pairwise Relationships within Hyperedges

Hypergraphs are generalizations of simple graphs that allow for the representation of complex group interactions beyond pairwise relationships. Clustering coefficients quantify local link density in networks and have been widely studied for both simple graphs and hypergraphs. However, existing clustering coefficients for hypergraphs treat each hyperedge as a distinct unit rather than a collection of potentially related node pairs, failing to capture intra-hyperedge pairwise relationships and incorrectly assigning zero values to nodes with meaningful clustering patterns. We propose a novel clustering coefficient that addresses this fundamental limitation by transforming hypergraphs into weighted graphs, where edge weights reflect relationship strength between nodes based on hyperedge connections. Our definition satisfies three key conditions: values in the range $[0,1]$, consistency with simple graph clustering coefficients, and effective capture of intra-hyperedge pairwise relationships -- a capability absent from existing approaches. Theoretical evaluation on higher-order motifs demonstrates that our definition correctly assigns values to motifs where existing definitions fail (motifs III, IV-a, IV-b of order 3), while empirical evaluation on three real-world datasets shows similar overall clustering tendencies with more detailed measurements, especially for hypergraphs with larger hyperedges. The proposed clustering coefficient enables accurate quantification of local density in complex networks, revealing structural characteristics missed by existing definitions in systems where group membership implies connections between members, such as social communities and co-authorship networks.