Statistical mechanical approach of complex networks with weighted links

Abstract

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 α G ⩾ 0; d = 1, 2, 3, 4) whose links are weighted through a predefined random probability distribution, namely P ( w ) ∝ e − | w − w c | / τ , w being the weight (w c ⩾ 0; τ > 0). In this model, each site has an evolving degree k i and a local energy ε i ≡ ∑ j = 1 k i w i j / 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 i j ∝ ε i / d i j α A ( α A ⩾ 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 ⩽ α A/d ⩽ 1; α A/d → ∞ corresponds to interactions between close neighbors, and α A/d → 0 corresponds to infinitely-ranged interactions. The site energy distribution p(ɛ) corresponds to the usual degree distribution p(k) as the particular instance (w c, τ) = (1, 0). We numerically verify that the corresponding connectivity distribution p(ɛ) converges, when α A/d → ∞, to the weight distribution P(w) for infinitely wide distributions (i.e. τ → ∞, ∀w c) as well as for w c → 0, ∀τ. Finally, we show that p(ɛ) is well approached by the q-exponential distribution e q − β q | ε − w c ′ | [ 0 ⩽ w c ′ ( w c , α A / d ) ⩽ w c ] , which optimizes the nonadditive entropy S q under simple constraints; q depends only on α A/d, thus exhibiting universality.

Non-additive entropies and statistical mechanics at the edge of chaos: a bridge between natural and social sciences

“…The connections between cities as well. In fact, the growth of virtually all asymptotically scale-free networks based on preferential attachment follow a q-statistical distribution of the number of degrees or, more generally speaking, of the site energies: see [92][93][94][95][96][97][98][99] and references therein. The definition of site (or local) energy is illustrated in figure 10, for a network with randomly weighted links.…”

Nonextensive Footprints in Dissipative and Conservative Dynamical Systems

“…Rather unexpectedly a priori, some ubiquitous classes of growing networks-usually referred to as scale-free ones-are closely related [78][79][80][81][82][83][84][85][86] with various of the previous complex many-body systems. The relationship is neatly caused by the assumption of preferential attachment along the network growth.…”

“…The relationship is neatly caused by the assumption of preferential attachment along the network growth. To illustrate this, we refer here to the d-dimensional model focused on in [85]. The attachment probability for the newly arrived site i is assumed to be given by…”

“…If the attachment probability was assumed to decay exponentially with distance as Π ij ∝ e −d ij /∆ (∆ > 0), instead of as the power-law (43), the short-range region (i.e., q = 1) would emerge for any ∆ > 0. [85] where further details can be seen.…”

“…In Section 4, we have focused on a d-dimensional network growth model which, by definition of its preferential attachment rule, is essentially scale-free [85]; its interaction decays as 1/r α A (α A ≥ 0). We have obtained an energy distribution given by…”