Preferential attachment growth model and nonextensive statistical mechanicsD. 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 , and is attached to (only) one pre-existing site with a probability ( ; 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 , by , where is the function naturally emerging within nonextensive statistical mechanics. The entropic index is numerically given (at least for not too large) by , and the characteristic number of links by . The 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 increases with the scaled time t/i; asymptotically, , the exponent being close to for , and zero otherwise. The present results reinforce the conjecture that the microscopic dynamics of nonextensive systems typically build (for instance, in Gibbs -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.
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