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 P(k) proportional to e(q)(-k/k), where the q-exponential form e(q)(z) = [1+ (1-q) z](1/1-q) 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 (-alpha)(rij) (alpha(A) >= 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 k, universal dependences on the ratio alpha(A)/d. Moreover, the q = 1 limit is rapidly achieved by increasing alpha(A)/d to infinity.