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Statistical mechanical approach of complex networks with weighted links

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 $α_G\geq0$; $d = 1, 2, 3, 4$) whose links are weighted through a predefined random probability distribution, namely $P(w) \propto e^{-|w - w_c|/τ}$, $w$ being the weight $ (w_c \geq 0; \; τ> 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 $Π_{ij}\propto \varepsilon_{i}/d^{\,α_A}_{ij} \;\;(α_A \ge 0)$, where $d_{ij}$ is the Euclidean distance between the sites. Short- and long-range interactions respectively correspond to $α_A/d>1$ and $0\leq α_A/d \leq 1$; $α_A/d \to \infty$ corresponds to interactions between close neighbors, and $α_A/d \to 0$ corresponds to infinitely-ranged interactions. The site energy distribution $p(\varepsilon)$ corresponds to the usual degree distribution $p(k)$ as the particular instance $(w_c,τ)=(2,0)$. We numerically verify that the corresponding connectivity distribution $p(\varepsilon)$ converges, when $α_A/d\to\infty$, to the weight distribution $P(w)$ for infinitely narrow distributions (i.e., $τ\to \infty, \,\forall w_c$) as well as for $w_c\to0, \, \forallτ$.