Preferential attachment growth model and nonextensive statistical mechanics

Abstract

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 $\propto k_i/r_i^{{\alpha }_A}({\alpha }_A\ge 0$; $k_i$ 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.

5 pages including 5 figures (the original colored figures 1 and 5a can be asked directly to the authors)