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/r γ with γ > 1. The new node j is linked to only one pre-existing node according to the probability law P; k i is the number of links of the i th node, η i is its fitness (or quality factor), and r ij is the distance. We consider in the present paper two models for η i . 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 η i = 1 ∀ i. In the other one, the uniformly distributed fitness model (UDFM), η i 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 Pis the q-exponential function naturally emerging within Tsallis nonextensive statistical mechanics. We determine, for both models, the entropic index q as a function of α A . Additionally, we determine the average topological (or chemical) distance within the network, and the time evolution of the average number of links k i . We obtain that, asymptotically, k i ∝ (t/i) β , (i coincides with the input-time of the i th node) and β(α A ) to both cases.Scale-free networks are very popular nowadays 2)-5) due to their uncountable applications in different fields of knowledge. These models are typically associated with some physical quantities that are characterized by power-law asymptotic behavior, instead of the usual exponential laws. Most of these models do not take into account the node-to-node Euclidean distance, i.e., the geographical distance. One of them which does take into account this aspect of the problem has been introduced recently 1) and discussed. In this example, as well as in others, 6), 7) strong connection has been revealed with nonextensive statistical mechanics. 8), 9) In the present paper, we follow the lines of Ref. 1) which we extend in the sense that we include now a local variable denominated fitness or quality factor, we call this variable η i . The model studied by Ref. 1) is herein referred to as the Single Fitness Model (SFM), that is, η i = 1 ∀ i, and we also study the Uniformly Distributed Fitness Model (UDFM), in * )
