Preferential attachment growth model and nonextensive statistical mechanics

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 $1/r^{2+α_G} (α_G \ge 0)$, and is attached to (only) one pre-existing site with a probability $\propto k_i/r^{α_A}_i (α_A \ge 0$; $k_i$ 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 $α_G$, by $P(k) \propto e_q^{-k/κ}$, 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 $α_A$ not too large) by $q = 1+(1/3) e^{-0.526 α_A}$, and the characteristic number of links by $κ\simeq 0.1+0.08 α_A$. The $α_A=0$ 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, $<k_i > \propto (t/i)^β$, the exponent being close to $β={1/2}(1-α_A)$ for $0 \le α_A \le 1$, and zero otherwise. The present results reinforce the conjecture that the microscopic dynamics of nonextensive systems typically build (for instance, in Gibbs $Γ$-space for Hamiltonian systems) a scale-free network.

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