Lower Ricci Curvature for Hypergraphs

TL;DR

This paper introduces hypergraph lower Ricci curvature (HLRC), a new, computationally efficient curvature measure that captures higher-order structural features in hypergraphs, enabling advanced analysis of complex systems.

Contribution

The paper presents HLRC, a novel closed-form curvature metric for hypergraphs that balances interpretability and efficiency, addressing limitations of existing methods.

Findings

HLRC effectively distinguishes community structures in hypergraphs.

HLRC uncovers latent semantic labels and tracks temporal dynamics.

HLRC supports robust hypergraph clustering based on global structure.

Abstract

Networks with higher-order interactions, prevalent in biological, social, and information systems, are naturally represented as hypergraphs, yet their structural complexity poses fundamental challenges for geometric characterization. While curvature-based methods offer powerful insights in graph analysis, existing extensions to hypergraphs suffer from critical trade-offs: combinatorial approaches such as Forman-Ricci curvature capture only coarse features, whereas geometric methods like Ollivier-Ricci curvature offer richer expressivity but demand costly optimal transport computations. To address these challenges, we introduce hypergraph lower Ricci curvature (HLRC), a novel curvature metric defined in closed form that achieves a principled balance between interpretability and efficiency. Evaluated across diverse synthetic and real-world hypergraph datasets, HLRC consistently reveals meaningful higher-order organization, distinguishing intra- from inter-community hyperedges, uncovering latent semantic labels, tracking temporal dynamics, and supporting robust clustering of hypergraphs based on global structure. By unifying geometric sensitivity with algorithmic simplicity, HLRC provides a versatile foundation for hypergraph analytics, with broad implications for tasks including node classification, anomaly detection, and generative modeling in complex systems.