提出可高效计算的高阶网络曲率度量，揭示超图深层结构。
Lower Ricci Curvature for Hypergraphs
- 基于闭式公式定义超图下界里奇曲率，兼顾可解释性与效率。
- 在真实与合成数据上准确区分社区内外的超边，捕捉动态演化。
- 适用于节点分类、异常检测等复杂系统分析，适合图神经网络研究者。
具有高阶交互作用的网络广泛存在于生物、社会和信息系统中，自然地以超图形式建模，但其结构复杂性给几何表征带来根本挑战。尽管曲率方法在图分析中表现优异，现有超图扩展存在关键权衡：组合方法如Forman-Ricci曲率仅能捕捉粗粒度特征，而几何方法如Ollivier-Ricci曲率虽表达能力强，却需代价高昂的最优传输计算。为解决此问题，我们提出超图下界里奇曲率（HLRC），一种以闭式公式定义的新曲率度量，在可解释性与效率之间实现原则性平衡。在多种合成与真实世界超图数据集上评估表明，HLRC持续揭示有意义的高阶组织结构，能够区分社区内与社区间的超边，发现潜在语义标签，追踪时间动态，并支持基于全局结构的鲁棒聚类。通过统一几何敏感性与算法简洁性，HLRC为超图分析提供了通用基础，对节点分类、异常检测及复杂系统中的生成建模具有广泛意义。
原文摘要 · Abstract (English)
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.
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