  Systems which consist of many localized constituents interacting with each
other can be represented by complex networks. Consistently, network science has
become highly popular in vast fields focusing on natural, artificial and social
systems. We numerically analyze the growth of $d$-dimensional geographic
networks (characterized by the index $\alpha_G\geq0$; $d = 1, 2, 3, 4$) whose
links are weighted through a predefined random probability distribution, namely
$P(w) \propto e^{-|w - w_c|/\tau}$, $w$ being the weight $ (w_c \geq 0; \; \tau
> 0)$. In this model, each site has an evolving degree $k_i$ and a local energy
$\varepsilon_i \equiv \sum_{j=1}^{k_i} w_{ij}/2$ ($i = 1, 2, ..., N$) that
depend on the weights of the links connected to it. Each newly arriving site
links to one of the pre-existing ones through preferential attachment given by
the probability $\Pi_{ij}\propto \varepsilon_{i}/d^{\,\alpha_A}_{ij}
\;\;(\alpha_A \ge 0)$, where $d_{ij}$ is the Euclidean distance between the
sites. Short- and long-range interactions respectively correspond to
$\alpha_A/d>1$ and $0\leq \alpha_A/d \leq 1$; $\alpha_A/d \to \infty$
corresponds to interactions between close neighbors, and $\alpha_A/d \to 0$
corresponds to infinitely-ranged interactions. The site energy distribution
$p(\varepsilon)$ corresponds to the usual degree distribution $p(k)$ as the
particular instance $(w_c,\tau)=(2,0)$. We numerically verify that the
corresponding connectivity distribution $p(\varepsilon)$ converges, when
$\alpha_A/d\to\infty$, to the weight distribution $P(w)$ for infinitely narrow
distributions (i.e., $\tau \to \infty, \,\forall w_c$) as well as for $w_c\to0,
\, \forall\tau$.
