Preferential attachment growth model and nonextensive statistical mechanics

D. J. B. Soares; C. Tsallis; A. M. Mariz; L. R. da Silva

doi:10.1209/epl/i2004-10467-y

All issues

Volume 70 / No 1 (April 2005)

Europhys. Lett., 70 1 (2005) 70-76

Abstract

Issue		Europhys. Lett. Volume 70, Number 1, April 2005


Page(s)		70 - 76
Section		General
DOI		https://doi.org/10.1209/epl/i2004-10467-y
Published online		09 March 2005

Europhys. Lett., 70 (1), pp. 70-76 (2005)

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: 8 February 2005

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 $1/r^{2+\alpha_G}$ $(\alpha_G > 0)$ , and is attached to (only) one pre-existing site with a probability $\propto{k}_i/r^{\alpha_A}_i$ ( $\alpha_A\ge0$ ; 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 $\kappa\simeq0.1+0.08\,\alpha_A$ . 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.

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

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