## Ricci Curvature and Ricci flow on Weighted Graphs

- Jan. 29, 2021
- 2:30 p.m.

## Abstract

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 $w _ {uv}$ where $w _ {uv}$ depends on $d(u,v)$. In this talk, we generalize Lin-Lu-Yauâ€™s Ricci curvature to weighted graphs and prove some results, including an extremal result on the total curvature and the existence of Ricci-flow. This is based on joint works with Shuliang Bai, An Huang, Shing-Tung Yau, Yong Lin, and Zhiyu Wang.