[Paper Review] Curvature dimension inequalities on directed graphs

Taiki Yamada|arXiv (Cornell University)|Jan 6, 2017

Mathematics and Applications1 references3 citations

TL;DR

This paper introduces curvature dimension inequalities $CD(m,K)$ for finite directed graphs by adapting the $Γ$-calculus framework to asymmetric Laplacians via a symmetrized transition probability matrix. It establishes that strongly connected directed graphs satisfy $CD(2, C - (1 - \alpha))$, where $C(v_i)$ is a local minimum of normalized weights, providing a discrete analog of Bakry-Emery curvature bounds for directed networks.

ABSTRACT

In this paper, we define the curvature dimension inequalities CD(m, K) on finite directed graphs modifying the case of undirected graphs. As a main result, we evaluate m and K on finite directed graphs.

Motivation & Objective

• To extend curvature dimension inequalities $CD(m,K)$ from undirected to directed graphs, where standard symmetric Laplacians do not apply due to asymmetric adjacency.
• To define a meaningful Ricci curvature analog for directed graphs using a symmetrized Laplacian based on transition probabilities and a stationary measure.
• To derive explicit lower bounds for $m$ and $K$ in terms of local graph structure and parameter $\alpha$.
• To establish a discrete version of Bakry-Emery curvature bounds in the context of directed graphs.

Proposed method

• Define a transition probability matrix $M_\alpha$ with self-loop weight $\alpha$ and out-edge weights $\frac{1-\alpha}{d_x}$, ensuring irreducibility for strongly connected graphs.
• Construct a symmetrized Laplacian $\Delta$ using a stationary measure $\phi$ and edge weights $w_{ij} = \phi(v_i)(M_\alpha)_{ij}$, enabling symmetric $\Gamma$-calculus.
• Apply the $\Gamma$-calculus framework: $\Gamma(f,g) = \frac{1}{2}(\Delta(fg) - f\Delta g - g\Delta f)$ and $\Gamma_2(f,g) = \frac{1}{2}(\Delta\Gamma(f,g) - \Gamma(f,\Delta g) - \Gamma(\Delta f,g))$.
• Derive an expression for $\Gamma_2(f,f)(v_i)$ in terms of second-order differences and normalized weights, isolating a non-negative term $H(f)(v_i)$.
• Bound $H(f)(v_i)$ from below using Cauchy-Schwarz and weight normalization, yielding $\Gamma_2(f,f)(v_i) \geq \frac{1}{2}(\Delta f)^2(v_i) + (C(v_i) - (1-\alpha))\Gamma(f,f)(v_i)$.
• Define $C(v_i) = \min_{v_j \in S^{\text{out}}(v_i), v_k \in S^{\text{in}}(v_i)} \{ w_{ij}/\phi(v_j), w_{ki}/\phi(v_k) \} > 0$ as a local curvature lower bound.

Experimental results

Research questions

• RQ1 Can curvature dimension inequalities $CD(m,K)$ be meaningfully defined on directed graphs, given the asymmetry of their Laplacians?
• RQ2 What is the optimal value of $m$ and the best possible $K$ in terms of local graph structure for directed graphs?
• RQ3 How can the $\Gamma$-calculus and Bakry-Emery curvature framework be adapted to asymmetric graphs?
• RQ4 What role does the parameter $\alpha \in [0,1)$ play in balancing self-loops and edge transitions in curvature estimation?
• RQ5 Is there a discrete analog of Ricci curvature lower bounds for directed graphs, and how can it be quantified?

Key findings

• The paper proves that any finite, strongly connected directed graph satisfies $CD(2, C - (1 - \alpha))$, where $m=2$ is optimal for the curvature-dimension condition.
• The curvature bound $K(v_i) = C(v_i) - (1 - \alpha)$ is explicitly quantified via $C(v_i) = \min_{v_j \in S^{\text{out}}(v_i), v_k \in S^{\text{in}}(v_i)} \{ w_{ij}/\phi(v_j), w_{ki}/\phi(v_k) \}$, with $C(v_i) > 0$.
• The derivation shows that the non-negative term $H(f)(v_i)$ in $\Gamma_2(f,f)$ is bounded below by $16\phi(v_i)(C(v_i) + \alpha)\Gamma(f,f)(v_i)$, which contributes to the curvature term.
• The method successfully symmetrizes the Laplacian using a stationary measure $\phi$ and transition matrix $M_\alpha$, enabling application of Riemannian-like curvature analysis.
• The curvature bound depends on the minimum of normalized weights $w_{ij}/\phi(v_j)$, reflecting local edge strength and balance in the graph.
• The result generalizes Bakry-Emery curvature bounds to directed graphs, providing a foundation for studying diffusion, mixing, and geometric properties in asymmetric networks.