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arXiv 2009.04306math.PRmath.COmath.MG
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随机几何图的Ollivier曲率收敛于其黎曼流形的Ricci曲率

Ollivier curvature of random geometric graphs converges to Ricci curvature of their Riemannian manifolds

Pim van der Hoorn, Gabor Lippner, Carlo Trugenberger, Dmitri Krioukov

发表信息：
Discrete & Computational Geometry, v.70(3), p.671-712 , 2023
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AI总结：

该研究严格证明了黎曼流形中随机几何图的Ollivier-Ricci曲率在适当设定下收敛于流形的Ricci曲率，适用于多种图距离与密度，是首个关联随机图与光滑空间曲率的严格结论。

摘要：

曲率是光滑空间的基本几何特征。近年来，研究者针对图等组合离散对象提出了多种曲率概念。然而，这类离散曲率概念与其光滑对应概念之间的关联仍不明确且存在争议。具体而言，目前尚无严格结论表明任何图曲率概念能收敛于光滑空间的任何传统曲率概念。本文证明，在适当设定下，黎曼流形中随机几何图的Ollivier-Ricci曲率收敛于该流形的Ricci曲率。这是首个将随机图曲率与光滑空间曲率关联起来的严格结论。本文结果适用于多种图距离概念（包括重标度最短路径距离）及多种图密度情形，其中平均度作为图规模的函数，其标度范围从近对数到近线性。

英文摘要：

Curvature is a fundamental geometric characteristic of smooth spaces. In recent years different notions of curvature have been developed for combinatorial discrete objects such as graphs. However, the connections between such discrete notions of curvature and their smooth counterparts remain lurking and moot. In particular, it is not rigorously known if any notion of graph curvature converges to any traditional notion of curvature of smooth space. Here we prove that in proper settings the Ollivier-Ricci curvature of random geometric graphs in Riemannian manifolds converges to the Ricci curvature of the manifold. This is the first rigorous result linking curvature of random graphs to curvature of smooth spaces. Our results hold for different notions of graph distances, including the rescaled shortest path distance, and for different graph densities. With the scaling of the average degree, as a function of the graph size, ranging from nearly logarithmic to nearly linear.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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