Ollivier Ricci-flow on weighted graphs (Q6652943)

Ricci flow was introduced by Richard Hamilton in the 1980s as a process that deforms the metric of a Riemannian manifold in a way formally analogous to the diffusion of heat. The paper under review is aiming at setting up similarly powerful methods in discrete geometry.\N\NIn Riemannian manifolds, after the establishment of Ricci flow equations, one important work is to verify whether this equation always has a unique smooth solution, at least for a short time, on any compact manifold of any dimension for any initial value of the metric. \N\NThe goal of this paper is a mathematical framework so that these essential questions can be answered rigorously. The authors begin with a brief explanation of the idea of discrete Ricci flow which contains two parts: the geometric meaning of Ollivier-Ricci curvature and the role of Ricci flow.\N\NThe paper is organized as follows: In Section 2, the authors introduced the notion of Ollivier-Lin-Lu-Yau Ricci curvature defined on weighted graphs and prove some related lemmas; Section 3 contains introduction on the Ricci flow equation and on the main theorem; Section 4, displays different types of solutions for continuous Ricci flows on the path graph; Section 5 proves the convergence results of normalized Ricci flow on path and star.\N\NIn conclusion, the paper proposes a normalized continuous Ricci flow for weighted graphs, based on Ollivier-Lin-Lu-Yau Ricci curvature and proves that the Ricci flow metric \(X(t)\) with initial data \(X(0)\) exists and is unique for all \(t \geq 0\) by fixing the violation of distance condition. The authors also show some explicit, rigorous examples of Ricci flows on tree graphs. Future work already underway, they are expecting results of more general Ricci flows evolving on various graphs\N\NIt is a beautifully designed paper.