Ising model in scale-free networks: A Monte Carlo simulation

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)~k^-γ. The ferromagnetic-paramagnetic transition temperature has been studied as a function of the parameter γ. For γ>3 our results agree with earlier analytical calculations, which found a phase transition at a temperature Tc(γ) in the thermodynamic limit. For γ3, a ferromagnetic-paramagnetic crossover occurs at a size-dependent temperature Tco, and the system is in the ordered ferromagnetic state at any temperature for a system size N→∞. For γ=3 and large enough N, the crossover temperature is found to be Tco≈A0.2em0exln0.2em0exN, with a prefactor A proportional to the mean degree. For 2<γ<3, we obtain Tco~⟨k⟩Nᶻ, with an exponent z that decreases as γ increases. This exponent is found to be lower than predicted by earlier calculations.