Remarks on curvature dimension conditions on graphs

Abstract

We show a connection between the $CDE'$ inequality introduced in Horn et al. (Volume doubling, Poincaré inequality and Gaussian heat kernel estimate for nonnegative curvature graphs. arXiv:1411.5087v2, 2014) and the $CD\psi$ inequality established in Münch (Li–Yau inequality on finite graphs via non-linear curvature dimension conditions. arXiv:1412.3340v1, 2014). In particular, we introduce a $CD_\psi ^\varphi$ inequality as a slight generalization of $CD\psi$ which turns out to be equivalent to $CDE'$ with appropriate choices of $\varphi$ and $\psi$. We use this to prove that the $CDE'$ inequality implies the classical CD inequality on graphs, and that the $CDE'$ inequality with curvature bound zero holds on Ricci-flat graphs.

Appendix A

Definition A.1

(Graph) A pair $G=(V,E)$ with a finite set V and a relation $E \subset V\times V$ is called a finite graph if $(v,v) \notin E$ for all $v \in V$ and if $(v,w) \in E$ implies $(w,v) \in E$ for $v,w \in V$. For $v,w \in V$, we write $v \sim w$ if $(v,w) \in E$.

Definition A.2

(Laplacian $\Delta$) Let $G=(V,E)$ be a finite graph. The Laplacian $\Delta : C(V) := {\mathbb {R}}^V \rightarrow C(V)$ is defined for $f \in C(V)$ and $v \in V$ as $\Delta f (v) := \sum _{w \sim v} (f(w) - f(v)).$

Definition A.3

We write $\mathbb {R}^{+}:= (0,\infty )$ and $\mathbb {R}^{+}_{0}:= [0,\infty )$. Let $G=(V,E)$ be a finite graph. Then, we write $C^+(V) := \{f:V \rightarrow \mathbb {R}^{+}\}.$

Definition A.4

(Heat operator $\mathcal L$) Let $G=(V,E)$ be a graph. The heat operator $\mathcal L: C^1(V \times \mathbb {R}^{+}) \rightarrow C(V \times \mathbb {R}^{+})$ is defined by $\mathcal L(u) := \Delta u - \partial _t u$ for all $u \in C^1(V \times \mathbb {R}^{+})$. We call a function $u \in C^{1}( V \times \mathbb {R}^{+}_{0})$ a solution to the heat equation on G if $\mathcal L(u) = 0$.

Definition A.5

(Ricci-flat graphs) Let $D \in {\mathbb {N}}$. A finite graph $G = (V,E)$ is called D-Ricci-flat in $v \in V$ if all $w \in N(v):=\{v\} \cup \{w \in V: w \sim v\}$ have the degree D, and if there are maps $\eta _1,\ldots ,\eta _D : N(v) \rightarrow V$, such that for all $w \in N(v)$ and all $i, j \in \{1,\ldots ,D\}$ with $i \ne j$, one has $\eta _i(w) \sim w$, $\eta _i(w) \ne \eta _j(w)$, $\bigcup _k \eta _k(\eta _i(v)) = \bigcup _k \eta _i(\eta _k(v))$. The graph G is called D-Ricci-flat if it is D-Ricci-flat in all $v \in V$.

1.1 A.1 The CD condition via $\Gamma$ calculus

We give the definition of the $\Gamma$-calculus and the CD condition following [1].

Definition A.6

($\Gamma$-calculus) Let $G=(V,E)$ be a finite graph. Then, the gradient form or carré du champ operator $\Gamma : C(V) \times C(V) \rightarrow C(V)$ is defined by

$$\begin{aligned} 2 \Gamma (f,g) := \Delta (fg) - f\Delta g - g\Delta f. \end{aligned}$$

Similarly, the second gradient form $\Gamma _2 : C(V) \times C(V) \rightarrow C(V)$ is defined by

$$\begin{aligned} 2 \Gamma _2 (f,g) := \Delta \Gamma (f, g) - \Gamma (f, \Delta g) - \Gamma (g, \Delta f). \end{aligned}$$

We write $\Gamma (f):= \Gamma (f,f)$ and $\Gamma _2 (f):= \Gamma _2 (f,f)$.

Definition A.7

(CD inequality) Let $G=(V,E)$ be a finite graph and $d \in \mathbb {R}^{+}$. We say G satisfies the curvature-dimension inequality CD(d, K) if for all $f \in C(V)$,

$$\begin{aligned} \Gamma _2(f) \ge \frac{1}{d} (\Delta f)^2 + K \Gamma (f). \end{aligned}$$

We can interpret this as meaning that the graph G has a dimension (at most) d and a Ricci curvature larger than K.

1.2 A.2 The CDE and $CDE'$ conditions via $\widetilde{\Gamma _2}$

We give the definitions of CDE and $CDE'$ following [2, 4]

Definition A.8

(The CDE inequality) We say that a graph $G=(V,E)$ satisfies the CDE(x, d, K) inequality if for any $f \in C^+(V)$ such that $\Delta f (x) <0$, we have

$$\begin{aligned} \widetilde{\Gamma _2}(f) (x) := \Gamma _2(f)(x) - \Gamma \left( f, \frac{\Gamma (f)}{f} \right) (x) \ge \frac{1}{d} \left( \Delta f \right) ^2(x) + K \Gamma (f)(x). \end{aligned}$$

We say that CDE(d, k) is satisfied if CDE(x, d, K) is satisfied for all $x \in V$.

Definition A.9

(The $CDE'$ inequality) We say that a graph $G=(V,E)$ satisfies the $CDE'(d,K)$ inequality if for any $f \in C^+(V)$, we have

$$\begin{aligned} \widetilde{\Gamma _2}(f) \ge \frac{1}{d} f^2 \left( \Delta \log f \right) ^2 + K \Gamma (f). \end{aligned}$$

1.3 A.3 The $CD\psi$ conditions via $\Gamma ^\psi$ calculus

We give the definition of the $\Gamma ^\psi$-calculus and the $CD\psi$ condition following [5].

Definition A.10

($\psi$-Laplacian $\Delta ^\psi$) Let $\psi \in C^1(\mathbb {R}^{+})$ and let $G=(V,E)$ be a finite graph. Then, we call $\Delta ^\psi : C^+(V) \rightarrow C(V)$, defined as

$$\begin{aligned} (\Delta ^\psi f ) (v) := \left( \Delta \left[ \psi \left( \frac{f}{f(v)} \right) \right] \right) (v), \end{aligned}$$

the $\psi$ -Laplacian.

Definition A.11

($\psi$-gradient $\Gamma ^\psi$) Let $\psi \in C^1(\mathbb {R}^{+})$ be a concave function and let $G=(V,E)$ be a finite graph. We define

$$\begin{aligned} \overline{\psi }(x):= \psi '(1)\cdot (x-1) - (\psi (x) - \psi (1)). \end{aligned}$$

Moreover, we define the $\psi$-gradient as $\Gamma ^\psi : C^+(V) \rightarrow C(V)$,

$$\begin{aligned} \Gamma ^\psi := \Delta ^{\overline{\psi }}. \end{aligned}$$

Definition A.12

(Second $\psi$-gradient $\Gamma _2^\psi$) Let $\psi \in C^1(\mathbb {R}^{+})$, and let $G=(V,E)$ be a finite graph. Then, we define $\Omega ^\psi : C^+(V) \rightarrow C(V)$ by

$$\begin{aligned} (\Omega ^\psi f ) (v) := \left( \Delta \left[ \psi ' \left( \frac{f}{f(v)} \right) \cdot \frac{f}{f(v)} \left[ \frac{\Delta f}{f} - \frac{(\Delta f)(v)}{f(v)} \right] \right] \right) (v). \end{aligned}$$

Furthermore, we define the second $\psi$-gradient $\Gamma _2^\psi : C^+(V) \rightarrow C(V)$ by

$$\begin{aligned} 2 \Gamma _2^\psi (f) := \Omega ^\psi f + \frac{\Delta f \Delta ^\psi f}{f} - \frac{\Delta \left( f \Delta ^\psi f\right) }{f}. \end{aligned}$$

Definition A.13

($CD\psi$ condition) Let $G=(V,E)$ be a finite graph, $K \in {\mathbb {R}}$ and $d \in \mathbb {R}^{+}$. We say G satisfies the $CD\psi (d,K)$ inequality if for all $f \in C^+(V)$, one has

$$\begin{aligned} \Gamma _2^\psi ( f) \ge \frac{1}{d} \left( \Delta ^\psi f \right) ^2 + K\Gamma ^\psi ( f). \end{aligned}$$