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A brief note about p-curvature on graphs

Chunyang Hu

TL;DR

This work extends Bakry-Émery curvature to graphs via the nonlinear $p$-Laplacian, defining and analyzing the $CD_p(m,K)$ condition. It provides explicit curvature calculations for paths, cycles, and star graphs, showing non-negativity at many vertices for $p\ge 2$ but potential divergence to $-\infty$ for $1<p<2$, and reveals that $\Gamma_{2,p}$ does not preserve the classical product property for $p>2$. The results include precise curvature formulas on leaves of paths and stars, and demonstrate limitations of Cartesian-product arguments in the $p$-curvature setting, while recovering standard results when $p=2$. Overall, the paper maps the curvature landscape of $p$-curvature on common graph families and highlights the delicate behavior under graph operations like Cartesian products.

Abstract

In this paper, we consider Wang's $CD_p(m,K)$ condition on graphs, which depends on the $p$-Laplacian $Δ_p$ for $p>1$ and is an extension of the classical Bakry-Émery $CD(m,K)$ curvature dimension condition. We calculate several examples including paths, cycles and star graphs, and we show that the $p$-curvature is non-negative at some vertices in the case $p\geq 2$, while it approaches to $-\infty$ in the case of $1<p<2$. In addition, we observe that a crucial property of $Γ_2$ on Cartesian products does no longer hold for $Γ_2^p$ in the case of $p > 2$. As a consequence, an analogous proof that non-negative curvature is preserved under taking Cartesian products is not possible for $p > 2$.

A brief note about p-curvature on graphs

TL;DR

This work extends Bakry-Émery curvature to graphs via the nonlinear -Laplacian, defining and analyzing the condition. It provides explicit curvature calculations for paths, cycles, and star graphs, showing non-negativity at many vertices for but potential divergence to for , and reveals that does not preserve the classical product property for . The results include precise curvature formulas on leaves of paths and stars, and demonstrate limitations of Cartesian-product arguments in the -curvature setting, while recovering standard results when . Overall, the paper maps the curvature landscape of -curvature on common graph families and highlights the delicate behavior under graph operations like Cartesian products.

Abstract

In this paper, we consider Wang's condition on graphs, which depends on the -Laplacian for and is an extension of the classical Bakry-Émery curvature dimension condition. We calculate several examples including paths, cycles and star graphs, and we show that the -curvature is non-negative at some vertices in the case , while it approaches to in the case of . In addition, we observe that a crucial property of on Cartesian products does no longer hold for in the case of . As a consequence, an analogous proof that non-negative curvature is preserved under taking Cartesian products is not possible for .
Paper Structure (9 sections, 11 theorems, 88 equations, 8 figures)

This paper contains 9 sections, 11 theorems, 88 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $CD_p(m,{\mathcal{K}})$ conditions on Paths
  4. $CD_p(m,{\mathcal{K}})$ condition at the middle vertex on $P_3$ and $P_4$
  5. $CD_p(m,{\mathcal{K}})$ conditions at a leaf on $P_N$
  6. $CD_{p}(m,{\mathcal{K}})$ condition on a general path $P_{N}$ or $P_{\infty}$
  7. $CD_p(m,{\mathcal{K}})$ condition on cycles
  8. $CD_{p}(m, {\mathcal{K}})$ condition on a star graph
  9. $p$-curvature on Cartesian product

Key Result

Proposition 3.1

In the case of $p>2$, for a path $G=P_3$ with length $2$, we assume that $\mu(v)=1$ for each vertex $v\in G$ and $w_{uv}=1$ for each edge $u\sim v$, then it satisfies the $CD_{p}(\infty,0)$ condition at the middle vertex $u\in P_{3}$.

Figures (8)

  • Figure 1: The graph $G=P_{N}$.
  • Figure 2: The graph $G=P_{3}$.
  • Figure 3: The graph $G=P_{4}$.
  • Figure 4: The graph $G=P_{N}$.
  • Figure 5: The local structure at $x$ in cycle graph $G=C_4$.
  • ...and 3 more figures

Theorems & Definitions (31)

  • Definition 2.1: $p$-Laplacian
  • Definition 2.2: $\Gamma_p$-operator
  • Definition 2.3: $\Gamma_{2,p}$-operator
  • Definition 2.4: $CD_p(m,{\mathcal{K}})$-condition on graphs
  • Definition 2.5: Bakry-Émery curvature/$CD(m,{\mathcal{K}})$-condition
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • ...and 21 more