Curvature serves as a potent and descriptive invariant, with its efficacy validated both theoretically and practically within graph theory. We employ a definition of generalized Ricci curvature proposed by Ollivier, which Lin and Yau later adapted to graph theory, known as Ollivier-Ricci curvature (ORC). ORC measures curvature using the Wasserstein distance, thereby integrating geometric concepts with probability theory and optimal transport. Jost and Liu previously discussed the lower bound of ORC by showing the upper bound of the Wasserstein distance. We extend the applicability of these bounds to discrete spaces with metrics on integers, specifically hypergraphs. Compared to prior work on ORC in hypergraphs by Coupette, Dalleiger, and Rieck, which faced computational challenges, our method introduces a simplified approach with linear computational complexity, making it particularly suitable for analyzing large-scale networks. Through extensive simulations and application to synthetic and real-world datasets, we demonstrate the significant improvements our method offers in evaluating ORC.

翻译：曲率作为一种强大且描述性强的几何不变量，其有效性在图论领域已得到理论与实践的验证。我们采用由Ollivier提出、并经Lin和Yau引入图论的广义Ricci曲率定义，即Ollivier-Ricci曲率（ORC）。ORC通过Wasserstein距离来度量曲率，从而将几何概念与概率论及最优传输理论相结合。Jost和Liu先前通过证明Wasserstein距离的上界，讨论了ORC的下界。我们将这些界的适用范围扩展到具有整数度量的离散空间，特别是超图。相较于Coupette、Dalleiger和Rieck在超图ORC研究中面临的计算挑战，我们的方法引入了一种具有线性计算复杂度的简化方案，使其特别适用于分析大规模网络。通过大量模拟实验以及在合成与真实数据集上的应用，我们证明了该方法在评估ORC方面带来的显著改进。