Role of dimensionality in complex networks

Abstract

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) ∝ eq-k/k, where the q-exponential form eqz= [1 + (1-q)z]z]1/1-q. optimizes the nonadditive entropy Sq (which, for q → 1, recovers the Boltzmann-Gibbs entropy). We introduce and study here d-dimensional geographicallylocated networks which grow with preferential attachment involving Euclidean distances through rij-α (αA ≥ 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 αA/d. Moreover, the q = 1 limit is rapidly achieved by increasing αA/d to infinity.