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 ki is the connectivity of the ith 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 Sq, 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.