Role of dimensionality in complex networks: Connections with nonextensive statistical mechanics
complex networks, nonextensive statistical mechanics, universality
Studies in complex networks are quite current and promote the integration of several areas of knowledge. Has been verified in previous research that the nonextensive statistical mechanics is the more suitable approach to describe the complex networks when there is long-range interactions between your constituents. At the thermodynamic limit the degree distribution is of the form P(k) ∝ e^(k/κ), where e_q is the q-exponential defined by e^z ≡ [1 + (1 - q)z]^(1/(1-q) ) which optimizes the non-additive entropy S_q (when q→1, the Boltzmann-Gibbs entropy is recovered). In this thesis we have introduced a study of the d-dimensional geographic networks (Natal Model) which grow with preferential attachment involving Euclidean distance by introducing the term r^(-α_A ) (α_A ≥ 0) into the preferential attachment rule. Given the connection between complex networks and the q-statistic, we numerically verified (for d = 1,2,3 e 4) that the degree distributions exhibit, for both q and κ, universal dependencies with respect to the variable α_A/d. In addition, the limit q = 1 is quickly reached when α_A/d → ∞. We also verified that other properties of the network also have universal dependencies with respect to α_A/d, such as: shortest path length ⟨l⟩, dynamic exponent β (from connectivity time evolution of the sites) and the entropy S_q of the degree distribution.