Let $G:=(E,V,W)$ be a weighted graph and let $d_G$ be its graph metric, defined by on any two edges $e_1,e_2\in E$ by
$$
d_G(e_1,e_2)\triangleq
\inf_{\gamma}\, \sum_{v\in \gamma} W(v),\qquad\tag{0}\label{0}
$$
where the infimum is taken over all sequences of vertices $\gamma=(v_1,\dots,v_t)$ connecting $e_1$ to $e_2$; where $W:V\rightarrow \mathbb{R}$ is the *weight function* defined on unweighted graph $(E,V)$.

In Ballmann's book Lectures on spaces of nonpositive curvature conditions are given which guarantee that $d_G$ has non-positive curvature (*in the sense of Alexandrov*); namely, there is some $c>0$ satisfying:
$$
\inf_{v}\, W(v)\geq c.\tag{1}\label{1}
$$
My question is, when is $(E,d_G)$ a $\operatorname{CAT}(\kappa)$ space, for some $\kappa\in \mathbb{R}$?

My hypothesis is that there exist some partition $C_1,\dotsc,C_N$ of $E$ into open balls such that \eqref{1} holds; where the infimum in \eqref{0} is taken over all paths $\gamma$ contained in $C_i$; where $e_1,e_2\in C_i$.