Statistical mechanical approach of complex networks with weighted links

Systems that 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) >= 0; d = 1, 2, 3, 4) whose links are weighted through a predefined random probability distribution, namely P(w) proportional to e(-vertical bar w-wc vertical bar/tau), w being the weight (w(c) >= 0 tau > 0). In this model, each site has an evolving degree k(i) and a local energy epsilon(i) Sigma(ki)(j=1)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) proportional to epsilon(i)/d(ij)(alpha A)(alpha(A) >= 0), where dz 3 is the Euclidean distance between the sites. Short- and long-range interactions respectively correspond to alpha(A)/d > 1 and 0 <= alpha(A)/d <= 1; alpha(A)/d -> infinity corresponds to interactions between close neighbors, and alpha(A)/d -> 0 corresponds to infinitely-ranged interactions. The site energy distribution p(epsilon) corresponds to the usual degree distribution p(k) as the particular instance (w(c), tau) = (1,0). We numerically verify that the corresponding connectivity distribution p(epsilon) converges, when alpha(A)/d -> infinity, to the weight distribution P(w) for infinitely wide distributions (i.e. tau -> infinity, for all w(c)) as well as for w(c) -> 0, for all tau. Finally, we show that p(epsilon) is well approached by the q-exponential distribution e(q)(-beta)(q vertical bar)(epsilon-wc'vertical bar )[0 <= w(c)' (w(c), alpha(A)/d) <= w(c)], which optimizes the nonadditive entropy S-q under simple constraints; q depends only on alpha(A)/d, thus exhibiting universality.