Preferential attachment growth model and nonextensive statistical mechanics

• Published in: Europhysics letters 2005-04, Vol.70 (1), p.70-76
• Main Authors: Soares, D. J. B, Tsallis, C, Mariz, A. M, Silva, L. R. da
• Format: Article
• Language: English
• Subjects: 05.70.Ln; 89.75.-k; 89.75.Hc
• ISSN: 0295-5075; 1286-4854
• DOI: 10.1209/epl/i2004-10467-y

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 $1/r^{2+\alpha_G}$ $(\alpha_G > 0)$, and is attached to (only) one pre-existing site with a probability $\propto{k}_i/r^{\alpha_A}_i$ ($\alpha_A\ge0$; ki is the number of links of the i-th 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 $\alpha_G$, 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 $\kappa\simeq0.1+0.08\,\alpha_A$. 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 $\langle k_i\rangle$ increases with the scaled time $t/i$; asymptotically, $\langle{k_i}\rangle\propto(t/i)^{\beta}$, the exponent being close to $\beta=\frac{1}{2}(1-\alpha_A)$ for $0\le\alpha_A\le1$, 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.