Ollivier-Ricci curvature convergence in random geometric graphs
Ollivier-Ricci curvature convergence in random geometric graphs
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DOI：
10.1103/physrevresearch.3.013211
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发表时间：
2020-08
期刊：
影响因子：
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通讯作者：
P. Hoorn;W. Cunningham;Gábor Lippner;C. Trugenberger;D. Krioukov
中科院分区：
文献类型：
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作者：
P. Hoorn;W. Cunningham;Gábor Lippner;C. Trugenberger;D. Krioukov
连续世界和离散世界之间的联系往往是难以捉摸的。一个例子是曲率。尽管图曲率存在许多不等价的定义，但严格地说，没有一个定义在任何限制下收敛于黎曼流形曲率的任何传统定义。在这里，我们表明，奥利维尔曲率的随机几何图在任何黎曼流形收敛于连续极限的Ricci曲率的基础流形。这一结果建立了第一个严格的联系之间的定义曲率适用于随机图和一个传统的定义曲率的光滑空间。
Connections between continuous and discrete worlds tend to be elusive. One example is curvature. Even though there exist numerous nonequivalent definitions of graph curvature, none is rigorously known to converge in any limit to any traditional definition of curvature of a Riemannian manifold. Here we show that Ollivier curvature of random geometric graphs in any Riemannian manifold converges in the continuum limit to Ricci curvature of the underlying manifold. This result establishes the first rigorous link between a definition of curvature applicable to random graphs and a traditional definition of curvature of smooth space.