Role of dimensionality of complex networks with metrics: Connection with nonextensive statistical mechanics

The study of networks is perceived in several fields of the science since many real systems can be modeled as networks. The networks are everywhere from social science to physics, biology, economics and other areas. Over the last decade a large number of empirical studies has been identifying peculiar properties in very different networks, for instance, the Internet and Online Social Networks (e.g. Facebook) to citation networks and networks of neurons. Over the past years the concept of nonextensive statistical mechanics has been extremely successful in applications of the complex systems, in particular, on the complex networks. Deep connections are known to exist between scale-free networks and non-Gibbsian statistics. For example, the typical degree distributions at the thermodynamical limit are of the form $P(k) \propto e_q^{-k/\kappa}$, where the $q$-exponential form $e_q^z \equiv [1+(1-q)z]^{\frac{1}{1-q}}$ optimizes the nonadditive entropy $S_q$, basis of nonextensive statistical mechanics (which recovers Boltzmann-Gibbs statistical mechanics at the $q \to 1$ limit). We introduce and study here $d$-dimensional growing networks with preferential attachment involving Euclidean distances through $r_{ij}^{-\alpha_A} \; (\alpha_A \ge 0)$. Reinforcing the connection with $q$-statistics we numerically verify that the $q$-exponential degree distributions exhibit, for both $q$ and $\kappa$, universal dependences on the ratio $\alpha_A/d$. Moreover, the $q=1$ limit is exponentially achieved by increasing $\alpha_A/d$ to infinity.