论文标题
离散的Ollivier-RICCI曲率
Discrete Ollivier-Ricci curvature
论文作者
论文摘要
我们分析了配备了给定距离的“ $ \ dist $”（W.R.T.哪个$ \ g $的连续和离散时间ollivier-ricci曲率$ \ g $）和一般随机步行。我们表明，对于一大批马尔可夫和非马克维亚随机步行，连续的ollivier-ricci曲率已明确定义，并为存在连续的ollivier-ricci曲率提供了标准；上述结果概括了文献中先前相当有限的结构。 In addition, important properties of both discrete-time and continuous-time Ollivier-Ricci curvatures are obtained including -- to name a few -- Lipschitz continuity, concavity properties, piece-wise regularity (piece-wise linearity in the case of linear walks) for the discrete-time Ollivier-Ricci as well as Lipschitz continuity and limit-free formulation for the continuous-time Ollivier-Ricci.这些特性以前仅以非常特定的距离和非常特定的随机步行而闻名。作为Lipschitz连续性的应用，我们获得了广义连续时ollivier-Ricci曲率流的存在和唯一性。 一路上，我们通过优化McMullen的上限 - 在凸形多层的顶点数量和环境尺寸方面，对凸多角形的顶点的数量进行了鲜明的上限估计，这在凸几何形状中可能具有独立的兴趣。所述上限使我们能够绑定离散时间ollivier-ricci曲率的多项式零件的数量，这是时间多项式随机步行中时间的函数。我们建立的无极限公式使我们能够定义操作员理论的Ollivier-Ricci曲率，该曲率是在合适的操作员空间上起作用的非线性凹形。
We analyze both continuous and discrete-time Ollivier-Ricci curvatures of locally-finite weighted graphs $\G$ equipped with a given distance "$\dist$" (w.r.t. which $\G$ is metrically complete) and for general random walks. We show the continuous-time Ollivier-Ricci curvature is well-defined for a large class of Markovian and non-Markovian random walks and provide a criterion for existence of continuous-time Ollivier-Ricci curvature; the said results generalize the previous rather limited constructions in the literature. In addition, important properties of both discrete-time and continuous-time Ollivier-Ricci curvatures are obtained including -- to name a few -- Lipschitz continuity, concavity properties, piece-wise regularity (piece-wise linearity in the case of linear walks) for the discrete-time Ollivier-Ricci as well as Lipschitz continuity and limit-free formulation for the continuous-time Ollivier-Ricci. these properties were previously known only for very specific distances and very specific random walks. As an application of Lipschitz continuity, we obtain existence and uniqueness of generalized continuous-time Ollivier-Ricci curvature flows. Along the way, we obtain -- by optimizing McMullen's upper bounds -- a sharp upper bound estimate on the number of vertices of a convex polytope in terms of number of its facets and the ambient dimension, which might be of independent interest in convex geometry. The said upper bound allows us to bound the number of polynomial pieces of the discrete-time Ollivier-Ricci curvature as a function of time in the time-polynomial random walk. The limit-free formulation we establish allows us to define an operator theoretic Ollivier-Ricci curvature which is a non-linear concave functional on suitable operator spaces.
