Developing Ollivier-Ricci Curvature for Geometric Homophily Analysis in Hypergraphs
摘要
Ollivier-Ricci curvature of graphs, a generalization of Ricci curvature in Riemannian geometry, is becoming a valuable tool for geometrically analyzing complex networks. While its generalization to hypergraphs has been explored, existing approaches do not take into account homophily structures (structural patterns determined by node class labels). Recent studies on homophily have focused on uniform hypergraphs. We develop a new notion of Ollivier-Ricci curvature for uniform hypergraphs, defined for edges (node pairs within a hyperedge) and explicitly incorporating node class labels to relate Ollivier-Ricci curvature to homophily structures. Using real-world hypergraphs, we explore its properties at both the edge and node levels by comparing it with conventional Ollivier-Ricci curvature. We show that the proposed discrete curvature captures distinctive structural features from the homophily-structure perspective, revealing notable node pairs and nodes that are difficult to detect with conventional Ollivier-Ricci curvature analysis.