Curvature–Dimension Inequalities on Graphs

• Curvature–dimension inequalities on graphs are synthetic conditions extending classical Ricci curvature bounds to discrete settings with detailed control on analytic and geometric properties.
• The exponential CDE'(n,K) formulation overcomes the chain rule barrier on graphs, enabling discrete Li–Yau gradient estimates, Harnack inequalities, and Gaussian heat kernel bounds.
• These inequalities lead to robust implications including volume doubling, finite-dimensional harmonic function spaces, Bonnet–Myers type bounds, and spectral gap estimates.

A curvature–dimension inequality (CD) on a graph is a synthetic analytic condition that generalizes the classical Bakry–Émery Ricci curvature lower bounds and dimension upper bounds from Riemannian geometry to discrete structures. On weighted graphs, several versions of CD inequalities, especially the exponential variant $CDE'(n,K)$, have been shown to enforce profound analytic and geometric properties, including Li–Yau type gradient bounds, Harnack inequalities, heat kernel estimates, volume growth restrictions, and topological consequences analogous to Bonnet–Myers type theorems. This framework establishes a strong analytic toolkit for the study of Markov semigroups, spectral estimates, and geometric analysis on graphs, with close connections to classical results on manifolds and significant implications for the combinatorial and probabilistic structure of both finite and infinite graphs.

1. Formalism and Definition of Curvature–Dimension Inequality on Graphs

Let $G = (V, E, w, \mu)$ be a connected, locally finite, weighted graph with symmetric positive edge weights $w_{xy}$ and positive vertex measure $\mu: V \to (0,\infty)$. The associated (normalized) Laplacian is

$$A f(x) = \frac{1}{\mu(x)} \sum_{y \sim x} w_{xy}[f(y)-f(x)].$$

The associated carré du champ and its iteration are defined as

$$\Gamma(f,g)(x) = \frac{1}{2\mu(x)} \sum_{y \sim x} w_{xy} [f(y) - f(x)] [g(y) - g(x)], \quad \Gamma(f) = \Gamma(f,f),$$

$$\Gamma_2(f)(x) = \frac{1}{2}[A \Gamma(f)(x) - 2\Gamma(f,Af)(x)].$$

The classical Bakry–Émery curvature–dimension inequality on graphs (CD(n,K)) is

$$\Gamma_2(f)(x) \geq \frac{1}{n} (Af(x))^2 + K \Gamma(f)(x)$$

for all $f$ and $x \in V$, with parameters $n > 0$, $K \in \mathbb{R}$.

However, due to the failure of the chain rule on graphs, one works instead with the exponential curvature–dimension inequality, denoted $CDE'(n,K)$. For positive $f: V \to (0,\infty)$: $$\Gamma_2(f)(x) \geq f(x)^2 [A \log f(x)]^2 / n + K \Gamma(f)(x),$$ where $A \log f(x) = (A f(x))/f(x) - \Gamma(f)(x)/f(x)^2$.

This exponential formulation is strictly stronger than $CD(n,K)$ in the graph setting and agrees with Bakry–Émery's form on manifolds, i.e., $CDE'(n,K) \iff CD(n,K)$ in the smooth diffusion case, but only $CDE'(n,K) \implies CD(n,K)$ for graphs (Horn et al., 2014, Münch, 2015).

2. Fundamental Analytic Consequences

Under $CDE'(n,0)$, the associated heat semigroup $P_t = e^{tA}$ admits a discrete Li–Yau gradient estimate: $\frac{\Gamma(u)}{u^2} \leq \frac{n}{2t}$ for any positive solution $u(t,x) = P_t f(x)$ of $\partial_t u = A u$. This is the discrete analogue of the classical Li–Yau estimate $|\nabla u|^2/u^2 \leq n/(2t)$ for manifolds with nonnegative Ricci curvature (Horn et al., 2014, Gong et al., 2018, Bauer et al., 2013). This gradient estimate yields:

• Harnack inequalities for positive solutions to the heat equation via path integration;
• Parabolic Harnack inequalities for example, for space–time cylinders and positive solutions $u$, one has $\sup_{Q_-} u \leq C \inf_{Q_+} u$ for appropriate subcylinders $Q_-, Q_+ \subset Q$;
• Two-sided Gaussian heat kernel estimates: for the continuous-time heat kernel $p(t,x,y)$,

$$\frac{c}{\mathrm{Vol}(B(x,\sqrt{t}))} e^{-C d(x,y)^2/t} \leq p(t,x,y) \leq \frac{C}{\mathrm{Vol}(B(x,\sqrt{t}))}$$

for constants $c, C$ depending only on $n$ (and possibly curvature lower bounds) (Horn et al., 2014, Gong et al., 2018).

3. Volume Doubling and Poincaré Inequality

Under $CDE'(n,0)$, graphs satisfy the volume doubling property: there exists $C = C(n)$ such that for every $x$ and $r > 0$,

$$\mathrm{Vol}(B(x,2r)) \leq C\, \mathrm{Vol}(B(x,r)),$$

with $$\mathrm{Vol}(B(x,r)) = \sum_{y: d(x,y)\le r} \mu(y)$$ (Horn et al., 2014).

The combination of volume doubling and a Poincaré inequality,

$$\sum_{y\in B(x_0,r)} \mu(y) |f(y) - f_B|^2 \leq C r^2 \sum_{y \sim z \in B(x_0,2r)} w_{yz}[f(z) - f(y)]^2$$

(with $f_B$ the mean of $f$ over the ball), is guaranteed under $CDE'(n,0)$ and a mild "A(a)" loop-parity assumption. These two, via Delmotte's result, are equivalent to two-sided Gaussian heat kernel bounds and parabolic Harnack inequalities (Horn et al., 2014, Lin et al., 2015).

4. Topological and Functional Consequences

Several discrete analogues of classical geometric results follow:

• Finite-dimensionality of polynomial-growth harmonic functions: On graphs with $CDE'(n,0)$ and the A(a) parity assumption, the space of harmonic functions with polynomial growth has finite dimension, analogously to the Yau conjecture resolved by Colding–Minicozzi for manifolds (Horn et al., 2014).
• Bonnet–Myers type theorems: For $CDE'(n,K>0)$, the diameter of the graph (for the canonical distance $d_c$) is finite and bounded by explicit functions of $n$ and $K$: $\text{diam}(G) \lesssim C(n) \sqrt{D_p / K}$ where $D_p$ measures the uniform degree ratio in the graph (see (Horn et al., 2014) for details).
• Spectral gap and eigenvalue estimates: Under $CDE'(n,K)$ with $K > 0$, Poincaré and various log-Sobolev inequalities hold, yielding a positive lower bound for the first nontrivial eigenvalue of the Laplacian (Horn et al., 2014, Gong et al., 2018).

5. Comparison of CD, CDE, and Nonlinear Variants

The CD, CDE (entropic), and nonlinear (e.g., $CD_\psi$) curvature–dimension inequalities are related but not equivalent on graphs. The $CDE'(n,K)$ inequality is stronger than the classical CD(n,K) and is adapted to address the non-diffusive chain rule obstruction present in the discrete setting (Münch, 2015, Horn et al., 2014, Münch, 2014).

• On Ricci-flat graphs in the sense of Chung–Yau and Lin–Yau, $CDE'(0,N)$ holds for explicit $N$ (Münch, 2015, Münch, 2014).
• The nonlinear $CD_\psi(d,0)$, where $\psi$ is a function such as $\log$, interpolates between logarithmic and root-type Li–Yau inequalities, and always implies the classical CD in the limit (Münch, 2014, Münch, 2015).
• On Riemannian manifolds, all these inequalities coincide.

6. Extensions: Directed, Nonlocal, and Metric Graphs

• Directed graphs: The curvature-dimension framework extends to finite, strongly-connected directed graphs by defining an appropriately symmetrized Laplacian and bilinear forms. The inequality $CD(2, C(x)-(1-a))$ holds, where $C(x)$ reflects directional minimal weights (Yamada, 2017).
• Nonlocal graphs: For symmetric Markov generators with non-local kernels (e.g., fractional Laplacians), CD inequalities may fail for heavy-tailed or long-range kernels but hold for finite-range or fast-decay cases, with dimension parameter linked to the moment condition of the kernel (Spener et al., 2019).
• Metric graphs: Weak curvature-dimension conditions such as weak Bakry–Émery ($BE_w$), weak EVI, and weak geodesic convexity are equivalent in the metric setting if one admits possibly non-sharp constants, recovering contraction, regularity, and entropy convexity properties (Krautz, 17 Dec 2025).

7. Examples, Computability, and Open Problems

Examples:

• Lattices $\mathbb{Z}^d$: $CDE(2d,0)$ (Bauer et al., 2013).
• Trees: Negative curvature, explicit bounds dependent on degree (Bauer et al., 2013, Gao, 2016).
• Complete and regular graphs: Curvature lower bounds computed directly (Ralli, 2017, Münch, 2014).
• Cartesian products: The Cartesian product of graphs preserves the CD property and the curvature function combines as a star-product (Cushing et al., 2016).
• Ricci-flat graphs: Admits $CDE'(0,N)$ with computable $N$, and various nonlinear CDψ inequalities (Münch, 2014, Münch, 2015).

Open structural problems include classifying expanders in $CD(0,\infty)$, improving dimension constants, and refining the discrete–continuum analogy for nonlinear and nonlocal operators (Cushing et al., 2016).

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References:

• "Volume doubling, Poincaré inequality and Guassian heat kernel estimate for nonnegative curvature graphs" (Horn et al., 2014).
• "Li-Yau inequality for unbounded Laplacian on graphs" (Gong et al., 2018).
• "Li-Yau inequality on graphs" (Bauer et al., 2013).
• "Curvature estimate on the finite graph with large girth" (Gao, 2016).
• "Weak curvature conditions on metric graphs" (Krautz, 17 Dec 2025).
• "Curvature-dimension inequalities for non-local operators in the discrete setting" (Spener et al., 2019).
• "Ollivier's Ricci curvature, local clustering and curvature dimension inequalities on graphs" (Jost et al., 2011).
• "Bounds on curvature in regular graphs" (Ralli, 2017).
• "Ricci curvature on birth-death processes" (Hua et al., 2017).
• "Li-Yau inequality on finite graphs via non-linear curvature dimension conditions" (Münch, 2014).
• "Global Poincaré inequality on Graphs via Conical Curvature-Dimension Conditions" (Lakzian et al., 2016).
• "Bakry-Émery curvature functions of graphs" (Cushing et al., 2016).
• "Remarks on curvature dimension conditions on graphs" (Münch, 2015).