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 $P(k)\propto e_q^{-k\mathrm{/}\kappa }$, 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 ${\alpha }_A\mathrm{/}d$. Moreover, the $q=1$ limit is rapidly achieved by increasing ${\alpha }_A\mathrm{/}d$ to infinity.