Wang’s CD_p(m,K) Curvature Condition

• Wang’s CD_p(m,K) condition is a nonlinear generalization of the Bakry–Émery curvature-dimension criterion using the p-Laplacian, distinguishing its behavior from the classical p=2 case.
• It establishes a local curvature criterion via higher-order difference operators on graphs, indicating that nonnegative p-curvature holds in specific regimes such as path leaves and certain cycles, while failing in others.
• The failure of tensorization for p>2 prevents extending nonnegative curvature to Cartesian graph products, impacting applications in spectral theory and geometric analysis on discrete structures.

Wang’s $CD_p(m,K)$ condition is a nonlinear generalization of the Bakry–Émery $CD(m,K)$ curvature dimension condition to graph settings, defined via the $p$-Laplacian operator for $p>1$. For a weighted undirected graph $G=(V, w, \mu)$, it yields a local curvature criterion involving higher-order difference operators. This framework extends the Bakry–Émery theory to nonlinear settings by replacing the standard Laplacian and carré du champ operators with their $p$-analogs, revealing new geometric and analytic phenomena on discrete spaces, especially concerning the behavior of "nonnegative $p$-curvature" and its stability under taking Cartesian products of graphs (Hu, 22 Jan 2026).

1. Definitions and Operator Structure

Given a graph with finite positive vertex measure $\mu: V\to(0,\infty)$ and symmetric edge weights $w:V\times V\to[0,\infty)$, consider $p>1$ and $f:V\to\mathbb{R}$:

• $p$-Laplacian:

$$\Delta_p f(x) = \frac{1}{\mu(x)} \sum_{y\sim x} w_{xy}\, |f(y)-f(x)|^{p-2} (f(y)-f(x)).$$

• $p$-gradient (first-order carré du champ):

$$\Gamma_p(f,g)(x) = \frac{p-1}{2\mu(x)} \sum_{y\sim x} w_{xy}\, |f(y)-f(x)|^{p-2}(g(y)-g(x))(h(y)-h(x)),$$

with $$\Gamma_p(f):=\Gamma_p(f,f)(x) = \frac{p-1}{2\mu(x)} \sum_{y\sim x} w_{xy} |f(y)-f(x)|^p$$.

• Iterated carré du champ:

$$\Gamma_{2,p}(f)(x) = \frac{1}{p(p-1)\mu(x)} \sum_{y\sim x} w_{xy}|f(y)-f(x)|^{p-2} (\Gamma_p(f)(y)-\Gamma_p(f)(x)) \ - \frac{1}{(p-1)^2} \Gamma_{p,f}(f, \Delta_p f)(x),$$

where $\Gamma_{p,f}(f, \Delta_p f)$ uses the same direction $f$.

These recover the classical Bakry–Émery operators when $p=2$ (i.e., $\Delta_2 = \Delta$, $\Gamma_2$, and standard $\Gamma_{2,2}$ expressions).

2. The $CD_p(m,K)$ Condition

The $CD_p(m,K)$ condition at $x\in V$, for all positive $f:V\rightarrow (0,\infty)$, is defined by: $$\Gamma_{2,p}(f)(x) \geq \frac{p-1}{m} (\Delta_p f(x))^2 + K (\Gamma_p(f)(x))^{(2p-2)/p}.$$ The largest $K$ for which this holds is called the $p$-Bakry–Émery curvature, denoted $\,_{p,x}G(m)$. The $CD_p(\infty, 0)$ (“nonnegative $p$-curvature”) condition simplifies to $\Gamma_{2,p}(f)(x) \geq 0$ for all admissible $f$.

For $p=2$, this coincides with the Bakry–Émery curvature-dimension criterion. For $p \neq 2$, it yields a genuinely nonlinear curvature notion, creating new regimes in discrete curvature theory (Hu, 22 Jan 2026).

3. Explicit Evaluation on Graph Examples

Analytic expressions for $\Gamma_{2,p}$ have been worked out on paths, cycles, and star graphs. All computations use unweighted, unit-measure graphs:

• Path Graphs ($P_N$):
  • For the middle vertex in $P_3$ or $P_4$, explicit calculations show that for $p>2$, $\Gamma_{2,p}(f)(u)\geq 0$, so $CD_p(\infty,0)$ holds. For $1$B/|A|\to 0$), $\Gamma_{2,p}(f)(u)\to-\infty$: nonnegative $p$-curvature fails.
  • At leaves of any path $P_N$, it is shown for all $p>1$ that $CD_p(\infty,0)$ holds, due to nonnegativity of the associated function $g(x)$ at its minimum.
  • On general vertices in $P_N$ ($N \geq 5$), for $p>2$, all analysis terms in the decomposition of $\Gamma_{2,p}(f)(u)$ are nonnegative.
• Cycles ($C_d$, $d\ge3$):
  • For most $d$, the combinatorial structure local to a vertex is as in $P_N$; thus, $CD_p(\infty,0)$ holds for $p>2$. For small cycles $C_3$ and $C_4$, separate analysis confirms the result for $p>2$ and failure for $1
• Star Graphs ($S_{\ell+1}$):
  • At a leaf $u$, the curvature constant is given explicitly and becomes negative as soon as the number of leaves $\ell>2p+1$. For $\ell \leq 2p+1$, nonnegative $p$-curvature holds.

The following table summarizes the regimes for $CD_p(\infty,0)$:

Graph/Vertex	$p\ge2$	$1
Path (middle)	Curvature $\geq0$	Curvature $\to -\infty$
Path (leaf)	Curvature $\geq0$	Curvature $\geq0$
Star (leaf, $\ell \le 2p+1$)	Curvature $\geq0$	Curvature may fail
Star (leaf, $\ell>2p+1$)	Curvature $<0$	Curvature $<0$
Cycle ($d\ge4$)	Curvature $\geq0$	Curvature $\to -\infty$

4. Failure of Tensorization for $p>2$

For the classical Laplacian ($p=2$), the product operator satisfies

$$\Gamma_2 f(x,y) \geq \Gamma_2(f^x)(y) + \Gamma_2(f_y)(x),$$

enabling nonnegative curvature to "tensorize" under Cartesian products of graphs, which is essential for extending curvature lower bounds to graph products.

By contrast, for the $p$-versions:

• $\Delta_p$ and $\Gamma_p$ are additive under graph products: $\Delta_p f(x,y)=\Delta_p(f_y)(x)+\Delta_p(f^x)(y)$ and similarly for $\Gamma_p$.
• However, for $\Gamma_{2,p}$ with $p>2$, explicit calculations (involving mixed-difference terms) yield counterexamples where

$$\Gamma_{2,p}(f)(x,y) - \Gamma_{2,p}(f^x)(y) - \Gamma_{2,p}(f_y)(x) < 0.$$

Consequently, the natural proof strategy establishing preservation of nonnegative curvature under product operations fails for $p>2$. This marks a significant divergence from the $p=2$ Bakry–Émery theory (Hu, 22 Jan 2026).

5. Main Theoretical Consequences

• For $p\ge2$, nonnegative $p$-curvature, i.e., $CD_p(\infty,0)$, is abundant in paths (all vertices), cycles (all $d\ge3$), and star graphs at leaves up to $\ell\le2p+1$ neighbors.
• In contrast, for $1$P_3$, $P_4$, and $C_4$ display $\Gamma_{2,p}\to -\infty$, and so $CD_p(\infty,0)$ fails dramatically.
• On star graphs, the explicit formula for the leaf-curvature constant exhibits sharp dependence on the number of leaves, becoming negative for large stars ($\ell>2p+1$).
• The failure of the tensorization property for $p>2$ prevents extension of nonnegative $p$-curvature to Cartesian products by Bakry–Émery style arguments, even though $\Delta_p$ and $\Gamma_p$ retain their additivity properties.

This structure highlights a dichotomy: the $p$-Bakry–Émery framework recovers the classical $p=2$ situation, but for $p\neq2$ affords new geometric features, including strong dependence on graph structure, lack of tensorization, and a break-down of local nonnegativity in certain regimes.

6. Connections and Limitations

Wang’s $CD_p(m,K)$ condition recovers Bakry–Émery for $p=2$ and generalizes it to a truly nonlinear curvature notion for $p\ne2$ in discrete settings. The dichotomy in curvature behavior (between $p\ge 2$ and $1$p>2$ demonstrate that simple extension of classical results to these nonlinear contexts is not possible. This complexity underscores both the potential and the limitations of applying nonlinear curvature-dimension inequalities in network analysis, spectral theory, and metric geometry on graphs (Hu, 22 Jan 2026).