关于赋权图的里奇曲率和
On the Sum of Ricci-Curvatures for Weighted Graphs
摘要：
本文将Lin-Lu-Yau的Ricci曲率推广到赋权图，并给出了一个简单的无极限定义。证明了赋权图Ricci曲率和的两个极值结果。加权图$G=(V,E,d)$是与距离函数$d\colon E\to [0,\infty)$相关联的无向图$G=(V,E)$。如果可能的话，通过重新定义权重，不失一般性，我们假设对于任何边缘$uv$，$u$和$v$之间的最短加权距离正好是$d(u,v)$。现在考虑一个随机游动，对于给定的函数$F(\bullet)$，它从顶点$u$到它的邻居$v$的传递概率（沿着边$uv$的跳跃移动）与$w_{uv}:=F(d(u,v))/d(u,v)$成比例。我们首先将Lin-Lu-Yau的Ricci曲率定义推广到这个加权图上，并利用所谓的$\ast$ -耦合函数给出了$κ(x, y)$的一个简单的无极限表示.总曲率$K(G)$被定义为$G$的所有边缘上的Ricci曲率之和。证明了如下定理：若$F(\bullet)$是减函数，则有$K(G)\geq 2|V| -2|E|$ ;若$F(\bullet)$是增函数，则有$K(G)\leq 2|V| -2|E|$ .当且仅当$d$是一个常数函数加上围长至少是$6$时，两个等式都成立。特别是，这些意味着一个高斯-博内定理（未加权）图围长至少$6$，其中的图形里奇曲率的几何定义方面的最佳运输。
英文摘要：
In this paper, we generalize Lin-Lu-Yau's Ricci curvature to weighted graphs and give a simple limit-free definition. We prove two extremal results on the sum of Ricci curvatures for weighted graph. A weighted graph $G=(V,E,d)$ is an undirected graph $G=(V,E)$ associated with a distance function $d\colon E\to [0,\infty)$. By redefining the weights if possible, without loss of generality, we assume that the shortest weighted distance between $u$ and $v$ is exactly $d(u,v)$ for any edge $uv$. Now consider a random walk whose transitive probability from an vertex $u$ to its neighbor $v$ (a jump move along the edge $uv$) is proportional to $w_{uv}:=F(d(u,v))/d(u,v)$ for some given function $F(\bullet)$. We first generalize Lin-Lu-Yau's Ricci curvature definition to this weighted graph and give a simple limit-free representation of $κ(x, y)$ using a so called $\ast$-coupling functions. The total curvature $K(G)$ is defined to be the sum of Ricci curvatures over all edges of $G$. We proved the following theorems: if $F(\bullet)$ is a decreasing function, then $K(G)\geq 2|V| -2|E|$; if $F(\bullet)$ is an increasing function, then $K(G)\leq 2|V| -2|E|$. Both equalities hold if and only if $d$ is a constant function plus the girth is at least $6$. In particular, these imply a Gauss-Bonnet theorem for (unweighted) graphs with girth at least $6$, where the graph Ricci curvature is defined geometrically in terms of optimal transport.
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