Progress of Theoretical Physics Supplement No. 162, 2006 131 Preferential Attachment Scale-Free Growth Model with Random Fitness and Connection with Tsallis Statistics

We introduce a model which consists in a planar network which grows by adding nodes at a distance r from the pre-existing barycenter. Each new node position is randomly located through the distribution law P (r) ∝ 1/rγ with γ> 1. The new node j is linked to only one pre-existing node according to the probability law P (i ↔ j) ∝ ηiki/rαAij (1 ≤ i < j; αA ≥ 0); ki is the number of links of the ith node, ηi is its fitness (or quality factor), and rij is the distance. We consider in the present paper two models for ηi. In one of them, the single fitness model (SFM [D. J. B. Soares, C. Tsallis, A. M. Mariz and L. R. da Silva, Europhys. Lett. 70 (2005), 70.]), we consider ηi = 1 ∀ i. In the other one, the uniformly distributed fitness model (UDFM), ηi is chosen to be uniformly distributed within the interval (0, 1]. We have determined numerically the degree distribution P (k). This distribution appears to be well fitted with P (k) = P (1) k−λ e−(k−1)/κq with q ≥ 1, where exq ≡ [1 + (1 − q)x]1/(1−q) (ex1 = ex) is the q-exponential function naturally emerging within Tsallis nonextensive statistical mechanics. We determine, for both models, the entropic index q as a function of αA. Additionally, we determine the average topological (or chemical) distance within the network, and the time evolution of the average number of links 〈ki〉. W