  We introduce a two-dimensional growth model where every new site is located,
at a distance $r$ from the barycenter of the pre-existing graph, according to
the probability law $1/r^{2+\alpha_G} (\alpha_G \ge 0)$, and is attached to
(only) one pre-existing site with a probability $\propto k_i/r^{\alpha_A}_i
(\alpha_A \ge 0$; $k_i$ is the number of links of the $i^{th}$ site of the
pre-existing graph, and $r_i$ its distance to the new site). Then we
numerically determine that the probability distribution for a site to have $k$
links is asymptotically given, for all values of $\alpha_G$, by $P(k) \propto
e_q^{-k/\kappa}$, where $e_q^x \equiv [1+(1-q)x]^{1/(1-q)}$ is the function
naturally emerging within nonextensive statistical mechanics. The entropic
index is numerically given (at least for $\alpha_A$ not too large) by $q =
1+(1/3) e^{-0.526 \alpha_A}$, and the characteristic number of links by $\kappa
\simeq 0.1+0.08 \alpha_A$. The $\alpha_A=0$ particular case belongs to the same
universality class to which the Barabasi-Albert model belongs. In addition to
this, we have numerically studied the rate at which the average number of links
$<k_i>$ increases with the scaled time $t/i$; asymptotically, $<k_i > \propto
(t/i)^\beta$, the exponent being close to $\beta={1/2}(1-\alpha_A)$ for $0 \le
\alpha_A \le 1$, and zero otherwise.
  The present results reinforce the conjecture that the microscopic dynamics of
nonextensive systems typically build (for instance, in Gibbs $\Gamma$-space for
Hamiltonian systems) a scale-free network.
