A generalised model for asymptotically-scale-free geographical networks. (14th April 2020) - Cinardi, Nicola; Rapisarda, Andrea; Tsallis, Constantino
Abstract: We consider a generalised d -dimensional model for asymptotically-scale-free geographical networks. Central to many networks of this kind, when considering their growth in time, is the attachment rule, i.e. the probability that a new node is attached to one (or more) preexistent nodes. In order to be more realistic, a fitness parameter for each node i of the network is also taken into account to reflect the ability of the nodes to attract new ones. Our d -dimensional model takes into account the geographical distances between nodes, with different probability distribution for which sensibly modifies the growth dynamics. The preferential attachment rule is assumed to be where k i is the connectivity of the i th pre-existing site and characterizes the importance of the euclidean distance r for the network growth. For special values of the parameters, this model recovers respectively the Bianconi–Barabási and the Barabási–Albert ones. The present generalised model is asymptotically scale-free in all cases, and its degree distribution is very well fitted with q -exponential distributions, which optimizes the nonadditive entropy S q, given by, with depending uniquely only on the ratio and the fitness distribution. Hence this model constitutes a realization of asymptotically-scale-free geographical networks within nonextensive statistical mechanics, where k plays the role of energy and plays the role of temperature. General scaling laws are also found for q as a function of the parameters of the model.