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) ∼ 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 γ ≤ 3, 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 ≈ A ln N , with a prefactor A proportional to the mean degree. For 2 < γ < 3, we obtain Tco ∼ k N z , with an exponent z that decreases as γ increases. This exponent is found to be lower than predicted by earlier calculations.PACS numbers: 64.60. Cn, 05.50.+q, 89.75.Hc,84.35.+i Complex networks describe several kinds of natural and artificial systems (social, biological, technological, economic), and are currently employed as models to study various processes taking place in real-life systems [1][2][3]. In last years, new models of complex networks have been introduced, motivated by empirical data in different fields. Thus, the so-called small-world [4] and scale-free (SF) networks [5] incorporate various aspects of real systems. These complex networks provide us with the underlying topological structure to study processes such as spread of infections [6], signal propagation [1,7], and cooperative phenomena [8][9][10][11].In a SF network the degree distribution, P (k), where k is the number of links connected to a node, has a powerlaw decay P (k) ∼ k −γ . This kind of networks have been found in particular in social systems [12], in protein interaction networks [13], in the internet [14], and in the world-wide web [15]. In both natural and artificial networks, the exponent γ controlling the degree distribution is usually in the range 2 < γ < 3 [3,16].Cooperative phenomena in complex networks are expected to display unusual characteristics, associated to the peculiar topology of these systems. In this context, the Ising model on SF networks has been studied with several theoretical techniques [9,17,18], and its critical behavior was found to be dependent on the exponent γ. In particular, when k 2 is finite, there appears a ferromagnetic (FM) to paramagnetic (PM) transition at a finite temperature T c . On the contrary, when k 2 diverges (as happens for γ ≤ 3), the system remains in its ordered FM phase at any temperature, and no phase transition occurs in the thermodynamic limit.Here we investigate the FM-PM transition for the Ising model in scale-free networks with various values of the exponent γ. We employ Monte Carlo (MC) simulations to obtain the transition temperature, and compare it with that predicted in earlier calculations. Our results confirm those of analytical calculations for γ > 3, and are u...

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