Statistical mechanical approach of complex networks with weighted links. (1st June 2022)

• Record Type: Journal Article
• Authors: Oliveira, Rute; Brito, Samuraí; da Silva, Luciano R; Tsallis, Constantino
• Is Part Of: Journal of statistical mechanics. (2022:Jun.)
• Publication Date: 2022-06-01
• Subjects: network dynamics; Statistical mechanics -- Periodicals; Mechanics -- Statistical methods -- Periodicals; 530.1305
• DOI: 10.1088/1742-5468/ac6f51
• Languages: English
• ISSNs: 1742-5468

Abstract

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.