Role of dimensionality in complex networks: Connection with nonextensive statistics

Deep connections are known to exist between scale-free networks and non-Gibbsian statistics. For example, 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 \(S_q\) (which, for \(q\to 1\), recovers the Boltzmann-Gibbs entropy). We introduce and study here \(d\)-dimensional geographically-located networks which grow with preferential attachment involving Euclidean distances through \(r_{ij}^{-\alpha_A} \; (\alpha_A \ge 0)\). Revealing the connection with \(q\)-statistics, we numerically verify (for \(d\) =1, 2, 3 and 4) 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 rapidly achieved by increasing \(\alpha_A/d\) to infinity.