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Systems which consist of many localized constituents interacting with eachother can be represented by complex networks. Consistently, network science hasbecome highly popular in vast fields focusing on natural, artificial and socialsystems. We numerically analyze the growth of $d$-dimensional geographicnetworks (characterized by the index $\alpha_G\geq0$; $d = 1, 2, 3, 4$) whoselinks 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$\varepsilon_i \equiv \sum_{j=1}^{k_i} w_{ij}/2$ ($i = 1, 2, ..., N$) thatdepend on the weights of the links connected to it. Each newly arriving sitelinks to one of the pre-existing ones through preferential attachment given bythe probability $\Pi_{ij}\propto \varepsilon_{i}/d^{\,\alpha_A}_{ij}\;\;(\alpha_A \ge 0)$, where $d_{ij}$ is the Euclidean distance between thesites. 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 theparticular instance $(w_c,\tau)=(2,0)$. We numerically verify that thecorresponding connectivity distribution $p(\varepsilon)$ converges, when$\alpha_A/d\to\infty$, to the weight distribution $P(w)$ for infinitely narrowdistributions (i.e., $\tau \to \infty, \,\forall w_c$) as well as for $w_c\to0,\, \forall\tau$.

Statistical mechanical approach of complex networks with weighted links