The Ricci flow with prescribed curvature on graphs

TL;DR

This paper studies a Ricci flow on finite graphs with prescribed curvature, proving existence, uniqueness, and exponential convergence under certain conditions, and relates it to combinatorial Ricci flow on surface tilings.

Contribution

It introduces a Ricci flow with prescribed curvature on graphs, establishes its mathematical properties, and connects it to 2D combinatorial Ricci flow analogs.

Findings

Existence and uniqueness of solutions on general graphs.

Exponential convergence to prescribed weights under certain conditions.

Characterization of constant curvature weights related to graph substructure ratios.

Abstract

In this paper, we consider the Ricci flow with prescribed curvature on the finite graph $G=(V,E)$. For any $e$ in $E$, $$\frac{d\omega(t,e)}{dt} = -(\kappa(t,e)-\kappa^*(e))\omega(t,e), t > 0,$$ where $\omega$ is the weight function, $\kappa$ is Lin-Lu-Yau Ricci curvature, and $\kappa^*$ is the prescribed curvature. By imposing invariance of the graph distance with respect to time $t$, the Ricci flow introduced above characterizes the weight evolution governed by the Lin-Lu-Yau curvature. We first establish the existence and uniqueness of the solution to this equation on general graphs. Furthermore, for graphs with girth of at least 6, we prove that the Ricci flow converges exponentially to weights of $\kappa^*$ if and only if $\kappa^*$ is attainable (namely, there exist weights realizing $\kappa^*$). In particular, we prove that the weights for constant curvature exist if and only if $$\max_{\emptyset \neq \Omega \subsetneq V} \frac{|E(\Omega)|}{|\Omega|} < \frac{|E|}{|V|},$$ where $E(\Omega)$ denotes the set of edges within the induced subgraph of $\Omega$, and $|A|$ is the cardinality of the set $A$. Viewing edge weights as metrics on surface tilings with girth of at least 5 or the duals of triangulations with vertex degrees exceeding 5, we demonstrate that our constant Lin-Lu-Yau curvature flow serves as an analog to the 2D combinatorial Ricci flow for piecewise constant curvature metrics, thereby providing an affirmative answer to Question 2 posed by Chow and Luo (J Differ Geom, 63(1) 2002).