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A Ricci flow on graphs from effective resistance

Aleyah Dawkins, Vishal Gupta, Mark Kempton, William Linz, Jeremy Quail, Harry Richman, Zachary Stier

Abstract

In this paper, we introduce a new notion of curvature on the edges of a graph that is defined in terms of effective resistances. We call this the Ricci--Foster curvature. We study the Ricci flow resulting from this curvature. We prove the existence of solutions to Ricci flow on short time intervals, and prove that Ricci flow preserves graphs with nonnegative (resp. positive) curvature.

A Ricci flow on graphs from effective resistance

Abstract

In this paper, we introduce a new notion of curvature on the edges of a graph that is defined in terms of effective resistances. We call this the Ricci--Foster curvature. We study the Ricci flow resulting from this curvature. We prove the existence of solutions to Ricci flow on short time intervals, and prove that Ricci flow preserves graphs with nonnegative (resp. positive) curvature.
Paper Structure (14 sections, 14 theorems, 37 equations, 8 figures)

This paper contains 14 sections, 14 theorems, 37 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Effective resistance
  3. Edge curvature
  4. Examples
  5. Properties
  6. Edge refinement
  7. Ricci flow
  8. Existence of Ricci flow
  9. Examples
  10. Ricci flow with surgery
  11. Nonnegatively curved graphs
  12. Some lemmas
  13. Main proof
  14. Further questions

Key Result

Theorem 1.2

Suppose $(G, \ell^{(0)})$ is an edge-weighted graph. Consider the Ricci--Foster flow on $\{\ell_e(t) : e \in E(G)\}$ defined by the system of differential equations for each edge $e \in E(G)$. Then for some $T > 0$ there exists a unique solution to this system of differential equations on the interval $t \in [0, T)$.

Figures (8)

  • Figure 1: A tree with its Ricci--Foster edge curvatures.
  • Figure 2: Barbell graph and its edge curvatures.
  • Figure 3: House graph and its edge curvatures.
  • Figure 4: Edge after subdivision.
  • Figure 5: Graph with edge resistances (left) and curvatures (right).
  • ...and 3 more figures

Theorems & Definitions (48)

  • Definition 1.1: Ricci--Foster curvature
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1: Effective resistance via spanning trees grimmett
  • Proposition 2.2: Rayleigh's monotonicity law
  • Remark 2.3
  • Definition : Ricci--Foster curvature
  • Remark 3.1
  • Remark 3.2
  • Example 3.3: Trees
  • ...and 38 more