Europhys. Lett.
Volume 70, Number 1, April 2005
Page(s) 70 - 76
Section General
Published online 09 March 2005
Europhys. Lett., 70 (1), pp. 70-76 (2005)
DOI: 10.1209/epl/i2004-10467-y

Preferential attachment growth model and nonextensive statistical mechanics

D. J. B. Soares1, 2, C. Tsallis3, 4, A. M. Mariz1 and L. R. da Silva1

1  Departamento de Física Teórica e Experimental Universidade Federal do Rio Grande do Norte Campus Universitario, 59072-970 Natal-RN, Brazil
2  Departamento de Física, Universidade Federal do Ceará 60451-970, Fortaleza-Ce, Brazil
3  Centro Brasileiro de Pesquisas Físicas - Rua Xavier Sigaud 150 22290-180 Rio de Janeiro-RJ, Brazil
4  Santa Fe Institute - 1399 Hyde Park Road, Santa Fe, NM 87501, USA

received 27 October 2004; accepted in final form 8 February 2005
published online 9 March 2005

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 $\Gamma$-space for Hamiltonian systems) a scale-free network.

05.70.Ln - Nonequilibrium and irreversible thermodynamics.
89.75.Hc - Networks and genealogical trees.
89.75.-k - Complex systems.

© EDP Sciences 2005