A generalised model for asymptotically-scale-free geographical networks

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 $\eta_i \in [0,1]$ 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 $\eta$ which sensibly modifies the growth dynamics. The preferential attachment rule is assumed to be $\Pi_i\propto k_i \eta_i r_{ij}^{-\alpha_A}$ where $k_i$ is the connectivity of the $i$th pre-existing site and $\alpha_A$ characterizes the importance of the euclidean distance r for the network growth. For special values of the parameters, this model recovers respectively the Bianconi-Barab\'{a}si and the Barab\'{a}si-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 optimise the nonadditive entropy $S_q$, given by $p(k) \propto e_q^{-k/\kappa} \equiv 1/[1+(q-1)k/\kappa]^{1/(q-1)}$, with $(q,\kappa)$ depending uniquely only on the ratio $\alpha_A/d$ 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 $\kappa$ plays the role of temperature. General scaling laws are also found for q as a function of the parameters of the model.