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

Shuliang Bai, Shuang Liu, Xin Lai

TL;DR

The paper develops a coherent framework for weighted Ricci flows on graphs using Lin-Lu-Yau and Forman curvatures, proving existence and uniqueness of solutions and uncovering fundamental convergence behavior. On trees, the two curvature notions align, yielding convergence of the Lin-Lu-Yau flow to a constant curvature and a constant-curvature metric under the uniform measure, with a complete tree classification when $m_1=m_2\equiv1$. The Forman-flow is treated via linear ODE techniques, showing weight dynamics are governed by the spectrum of a transformed matrix $\tilde{\mathbf{F}}$ and that curvature limits are tied to $-\lambda_{\max}(\tilde{\mathbf{F}})$. Together, these results connect discrete curvature notions, provide global convergence criteria, and illustrate how graph topology and edge/vertex measures determine long-time behavior, supported by simulations on representative trees. The framework offers practical tools for shaping graph geometry through curvature-driven evolution and for understanding structural features in network data.

Abstract

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.

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

TL;DR

The paper develops a coherent framework for weighted Ricci flows on graphs using Lin-Lu-Yau and Forman curvatures, proving existence and uniqueness of solutions and uncovering fundamental convergence behavior. On trees, the two curvature notions align, yielding convergence of the Lin-Lu-Yau flow to a constant curvature and a constant-curvature metric under the uniform measure, with a complete tree classification when . The Forman-flow is treated via linear ODE techniques, showing weight dynamics are governed by the spectrum of a transformed matrix and that curvature limits are tied to . Together, these results connect discrete curvature notions, provide global convergence criteria, and illustrate how graph topology and edge/vertex measures determine long-time behavior, supported by simulations on representative trees. The framework offers practical tools for shaping graph geometry through curvature-driven evolution and for understanding structural features in network data.

Abstract

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.
Paper Structure (12 sections, 14 theorems, 128 equations, 2 figures)

This paper contains 12 sections, 14 theorems, 128 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The weighted Ricci curvature on graphs
  3. The weighted Lin-Lu-Yau curvature on graphs
  4. The weighted Forman curvature for cell complex
  5. The relationship between Lin-Lu-Yau curvature and Forman curvature
  6. The weighted Ricci flow on graphs
  7. The weighted Lin-Lu-Yau Ricci flow
  8. The weighted Forman Ricci flow on graphs
  9. Examples and simulations
  10. The case $m_1=m_2\equiv1$
  11. The case $\operatorname{Deg}(x)\equiv1$
  12. Simulations

Key Result

Theorem 1.1

There exists a unique positive solution to main-equation for all $t\in(0,\infty)$ and any edge $e\in E$ with a positive initial value $\omega_0$.

Figures (2)

  • Figure 1: Comparison of Ricci flow convergence under two different measures on $K_{1,3}$ and $K_{1,6}$ graphs.
  • Figure 2: The evolution for a tree with maximum degree of $4$.

Theorems & Definitions (30)

  • Theorem 1.1: see Theorem \ref{['main1']}
  • Remark 1
  • Theorem 1.2: see Theorem \ref{['th:main']}
  • Remark 2
  • Theorem 1.3: see Theorem \ref{['the:4.1']}
  • Remark 3
  • Definition 1
  • Theorem 2.1: see JM21
  • Remark 4
  • Theorem 3.1
  • ...and 20 more