论文标题
随机几何图的圆润曲率收敛到其riemannian歧管的Ricci曲率
Ollivier curvature of random geometric graphs converges to Ricci curvature of their Riemannian manifolds
论文作者
论文摘要
曲率是光滑空间的基本几何特征。近年来，已经为组合离散对象（例如图形）开发了不同的曲率概念。但是，这种曲率的离散概念与它们平滑的对应物之间的连接仍然潜伏在潜伏。特别是，是否有任何图形曲率概念会收敛于任何传统的光滑空间曲率概念，尚不清楚。在这里，我们证明，在适当的设置中，Riemannian歧管中随机几何图的ollivier-Ricci曲率收敛到歧管的Ricci曲率。这是将随机图的曲率连接到光滑空间的曲率的第一个严格结果。我们的结果适用于不同的图形距离概念，包括重新缩放的最短路径距离和不同的图形密度。随着平均程度的缩放，作为图形大小的函数，范围从几乎对数到几乎线性。
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.
