The weighted Forman and Lin-Lu-Yau Ricci flow on graphs

In this paper, we propose a type of Ricci flow on graphs where the probability distribution for the Lin-Lu-Yau curvature remains constant over time, and also study the related Forman curvature flow. These two curvature flows coincide on trees. We first prove the existence and uniqueness of solutions for both curvature flows in general graphs. Then, we obtain that the normalized curvature flow on trees converges to a constant curvature metric, and under the uniform measure, a complete classification of trees can be obtained based on the convergence results.