Ising model in scale-free networks: a Monte Carlo simulation.

Abstract

The Ising model in uncorrelated scale-free networks has been studied by means of Monte Carlo simulations. These networks are characterized by a degree (or connectivity) distribution P (k) approximately k(-gamma). The ferromagnetic-paramagnetic transition temperature has been studied as a function of the parameter gamma. For gamma>3 our results agree with earlier analytical calculations, which found a phase transition at a temperature T(c) (gamma) in the thermodynamic limit. For gamma< or =3, a ferromagnetic-paramagnetic crossover occurs at a size-dependent temperature T(co), and the system is in the ordered ferromagnetic state at any temperature for a system size N--> infinity. For gamma=3 and large enough N, the crossover temperature is found to be T(co) approximately A ln N, with a prefactor A proportional to the mean degree. For 2<gamma<3, we obtain T(co) approximately <k>N(z), with an exponent z that decreases as gamma increases. This exponent is found to be lower than predicted by earlier calculations.