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/r2+αG (αG ≥ 0), and is attached to (only) one pre-existing site with a probability ∝ kᵢ/rαAᵢ (αA ≥ 0; kᵢ is the number of links of the iᵗʰ site of the pre-existing graph, and rᵢ 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 αG, by P(k) ∝ eq-k/κ, where eqˣ ≡ [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 αA not too large) by q = 1+(1/3) e-0.526 αA, and the characteristic number of links by κ 0.1+0.08 αA. The αA=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 <kᵢ> increases with the scaled time $t/i$; asymptotically, <kᵢ > ∝ (t/i)^β, the exponent being close to β=1/2(1-αA) for 0 ≤ αA ≤ 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.