A Comparative Study of Curvature on Trees

TL;DR

This work provides closed-form formulas for three discrete curvature notions on trees—Ollivier-Ricci, Lin-Lu-Yau, and Steinerberger curvature—bridging transportation- and distance-based approaches. It proves key relationships among these curvatures on trees, including $\\kappa^{\mathsf{or},α}_{ij} = (1-α)\\kappa^{\mathsf{lly}}_{ij}$ and explicit leaf/non-leaf bounds, and establishes a degree-diameter bound via a reverse Bonnet–Myers-type theorem. A central combinatorial distance identity enables a direct derivation of Steinerberger curvature on trees as $\\kappa^{\mathsf{s}}_i = \frac{n}{n-1}(2-d(i))$, with extensions to weighted trees discussed. The results unify curvature notions in the tree setting, offer exact formulas, and provide a practical bound framework, aided by a downloadable implementation. The work thus advances understanding of curvature on graphs and its dependence on local structure, with potential implications for graph analysis and geometric inference on trees.

Abstract

There are several interrelated notions of discrete curvature on graphs. Many approaches utilize the optimal transportation metric on its probability simplex or the distance matrix of the graph. In this survey article, we compute formulas for three different types of curvature on graphs. Along the way, we obtain a comparison result for the curvatures under consideration, a degree-diameter theorem for trees, and a combinatorial identity for certain sums of distances on trees.

Figures (11)

• Figure 1: This table contains formulas for three different notions of discrete curvature assuming the underlying graph is a tree $T=(V, E)$. Here, we take $i, j\in V$ to be any nodes that are not necessarily adjacent. We use $\mathrm{d}({i})$ to denote the combinatorial degree of $i$, and we denote by $\Delta({i, j})$ the shortest path distance between $i, j\in V$. We omit the formula for Ollivier-Ricci curvature when the laziness parameter $\alpha$ lies in $[0, 1/2)$ because of its length when typeset, see \ref{['eq:orc-on-trees']}.
• Figure 2: Ollivier-Ricci curvature, with $\alpha=1/2$.
• Figure 3: Lin-Lu-Yau curvature.
• Figure 4: Steinerberger curvature.
• Figure 6: A visualization of the Ollivier-Ricci curvature $\kappa^{\mathsf{or},\alpha}_{ij}$ at each edge of a tree on 25 nodes, with various choices of $\alpha$. Edge width is proportional to $1-\kappa^{\mathsf{or},\alpha}_{ij}$.

Theorems & Definitions (25)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4: Reverse Bonnet-Myers
• Definition 2.1: Ollivier-Ricci curvature ollivier2009ricci
• Definition 2.2: Lin-Lu-Yau curvature lin2011ricci
• Definition 2.3: Steinerberger curvature steinerberger2023curvature
• Remark 2.4
• Proposition 3.1
• proof