Role of dimensionality in complex networks

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 [Image: see text], where the q-exponential form [Image: see text] optimizes the nonadditive entropy S(q) (which, for q → 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 [Image: see text]. 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 k, universal dependences on the ratio α(A)/d. Moreover, the q = 1 limit is rapidly achieved by increasing α(A)/d to infinity.