论文标题
随机几何图中的ollivier-Ricci曲率收敛
Ollivier-Ricci curvature convergence in random geometric graphs
论文作者
论文摘要
连续世界和离散世界之间的联系往往难以捉摸。一个例子是曲率。即使存在许多对图曲率的非定义，但众所周知，没有任何限制将任何传统的Riemannian歧管曲率定义融合。在这里，我们表明，在任何riemannian歧管中随机几何图的曲率曲率在连续限制到基础歧管的RICCI曲率中收敛，但前提是，只有将Ollivier图曲率的定义适当地推广到适用于中学图形邻域。该结果建立了适用于网络的曲率定义与光滑空间曲率的传统定义之间的第一个严格联系。
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 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, but only if the definition of Ollivier graph curvature is properly generalized to apply to mesoscopic graph neighborhoods. This result establishes the first rigorous link between a definition of curvature applicable to networks and a traditional definition of curvature of smooth spaces.
