Consider a graph $G(V,E)$, $d$-regular on $n$ vertices with normalized laplacian $L$ given by $Lf(x)=\frac{1}{d}\sum\limits_{y\in V}f(y)-f(x)$ for $x\in V$ and $f\in\ell^2(V)$. For $f,g\in\ell^2(V)$, the carré du champ operator $\Gamma(f,g)\in\ell^2(V)$ is defined as $\frac{1}{2}(L(fg)-f(Lg)-(Lf)g)$ and the iterated carré du champ operator $\Gamma_2(f,g)$ is defined as $\frac{1}{2}(L\Gamma(f,g)-\Gamma(f,Lg)-\Gamma(Lf,g))$. The Bakry–Émery curvature $\kappa$ is defined as the infimum of $\frac{\Gamma_2(f,f)(x)}{\Gamma(f,f)(x)}$ over (nonconstant) $f\in\ell^2(V)$ and $x\in V$ (for any $\kappa'\le\kappa$ it is sometimes said that "$G$ satisfies $CD(\kappa',\infty)$").
Many papers use this notion of curvature to prove nontrivial results about graphs. My question is: how should one intuitively think of this notion of curvature for graphs in light of notions of manifold curvature? (e.g., Ollivier–Ricci curvature seems to have a nice interpretation as measuring volume change when doing parallel transport on a graph.)