Boltzmann–Gibbs statistical mechanics applies satisfactorily to a plethora of systems. It fails however for complex systems generically involving nonlocal space–time entanglement. Its generalization based on nonadditive q-entropies adequately handles a wide class of such systems. We show here that scale-invariant networks belong to this class. We numerically study a d-dimensional geographically located network with weighted links and exhibit its ‘energy’ distribution per site at its quasi-stationary state. Our results strongly suggest a correspondence between the random geometric problem and a class of thermal problems within the generalised thermostatistics. The Boltzmann–Gibbs exponential factor is generically substituted by its q-generalisation, and is recovered in the $$q=1$$ q = 1 limit when the nonlocal effects fade away. The present connection should cross-fertilise experiments in both research areas.

In the realm of Boltzmann–Gibbs statistical mechanics, there are three well known isomorphic connections with random geometry, namely, (i) the Kasteleyn–Fortuin theorem, which connects the [Formula: see text] limit of the [Formula: see text]-state Potts ferromagnet with bond percolation, (ii) the isomorphism, which connects the [Formula: see text] limit of the [Formula: see text]-state Potts ferromagnet with random resistor networks, and (iii) the de Gennes isomorphism, which connects the [Formula: see text] limit of the [Formula: see text]-vector ferromagnet with self-avoiding random walk in linear polymers. We provide here strong numerical evidence that a similar isomorphism appears to emerge connecting the energy [Formula: see text]-exponential distribution [Formula: see text] (with [Formula: see text] and [Formula: see text]) optimizing, under simple constraints, the nonadditive entropy [Formula: see text] with a specific geographic growth random model based on preferential attachment through exponentially distributed weighted links, [Formula: see text] being the characteristic weight.

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.

A ciência das redes é um campo multidisciplinar que oferece um arcabouço amplo para se estudar propriedades estatísticas de uma variedade de fenômenos. No cerne do seu sucesso, está o fato de que os sistemas, por mais complexos que sejam seus constituintes ou interações, podem ser representados por um simples grafo, um conjunto de nós conectados por arestas. Nesta abordagem, processos de natureza muito diferentes, como a internet, colaborações científicas, ou redes de proteínas, se tornam semelhantes do ponto de vista da rede, o que nos permite não somente entender de maneira unificada as redes naturais mas também otimizar e projetar redes artificiais mais eficientes. Dentro deste contexto, este artigo tem dois objetivos. Primeiramente, apresentar os principais conceitos da ciência das redes, tais como grafos, propriedade de mundo pequeno, distribuição de conectividade entre outros, assim como alguns dos principais modelos de redes já propostos. O segundo objetivo é aplicar este ferramental para analisar uma rede real, mais precisamente a rede de pesquisadores do Instituto Nacional de Ciência e Tecnologia de Informação Quântica. Nossos resultados mostram que do ponto de vista estatístico a rede estudada é bem descrita por uma lei de potência truncada, com um alto grau de interconectividade entre os participantes. Um aglomerado, formado por 8 comunidades menores, contém 85% dos cientistas da rede. O número médio de colaborações da rede é próximo de 5 e a média de artigos publicados está acima de 13 durante o período de duração do projeto. A rede possui um alto grau de agregação, com valor de C = 0.4, mostrando que os colaboradores de um dado cientista também tendem a colaborar entre si. Palavras-chave: Redes Complexas, Ciência das Redes, Informação Quântica.Network science is a multidisciplinary field that offers a broad framework for studying statistical properties of a variety of phenomena. At the core of its success is the fact that systems, in spite of the complexity of their constituents or interactions, can be represented as a simple graph, a set of nodes connected by edges. In this approach, processes of a very different nature, such as the internet, scientific collaborations, or protein networks, become similar from a network point of view, which allows not only to understand natural networks in a unified way, but also to optimize and design more efficient artificial networks. Within this context, this article has two objectives. First, present the main concepts of network science, such as graphs, the small-world property, connectivity distribution, among others, as well as some of the main network models proposed in the literature. The second objective is to apply network science to analyze a real network, more precisely the network of researchers from the National Institute of Science and Technology of Quantum Information. Our results show that from a statistical point of view the studied network is well described by a truncated power law, with a high degree of interconnectivity among the participants. A...

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