Role of dimensionality in preferential attachment growth in the Bianconi-Barabasi model

Scale-free networks are quite popular nowadays since many systems are well represented by such structures. In order to study these systems, several models were proposed. However, most of them do not take into account the nodeto-node Euclidean distance, i.e. the geographical distance. In real networks, the distance between sites can be very relevant, e.g. those cases where it is intended to minimize costs. Within this scenario we studied the role of dimensionality d in the Bianconi-Barabasi model with a preferential attachment growth involving Euclidean distances. The preferential attachment in this model follows the rule Pi(i) proportional to eta(i)k(i)/r(ij)(alpha A) (1 <= i < j; alpha(A) >= 0), where eta(i) characterizes the fitness of the ith site and is randomly chosen within the (0, 1] interval. We verified that the degree distribution P(k) for dimensions d = 1, 2, 3, 4 are well fitted by P(k) proportional to e(q)(-k/k), where e(q)(-k/k) is the q-exponential function naturally emerging within nonextensive statistical mechanics. We determine the index q and. as functions of the quantities alpha A and d, and numerically verify that both present a universal behavior with respect to the scaled variable alpha A/d. The same behavior also has been displayed by the dynamical beta exponent which characterizes the steadily growing number of links of a given site.