Relevância da dimensionalidade no modelo de Bianconi-Barabási

Abstract

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 node-to-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-Barabási model with a preferential attachment growth involving Euclidean distances. The preferential attachment in this model follows the rule Πi ∝ ηiki/rαA ij (1 ≤ i < j; αA ≥ 0), where ηi characterizes the tness of the i-th site and is randomly chosen within the (0, 1] interval. We veri ed that the degree distribution P(k) for dimensions d = 1, 2, 3, 4 are well tted by P(k) ∝ e −k/κ q , where e −k/κ q is the q-exponential function naturally emerging within nonextensive statistical mechanics. We determine the index q and κ as functions of the quantities αA and d, and numerically verify that both present a universal behavior with respect to the scaled variable αA/d. The same behavior also has been displayed by the dynamical β exponent which characterizes the steadily growing number of links of a given site.