Ising model in clustered scale-free networks.

The Ising model in clustered scale-free networks has been studied by Monte Carlo simulations. These networks are characterized by a degree distribution of the form P(k) ~ k-γ for large k. Clustering is introduced in the networks by inserting triangles, i.e., triads of connected nodes. The transition from a ferromagnetic (FM) to a paramagnetic (PM) phase has been studied as a function of the exponent γ and the triangle density. For γ>3 our results are in line with earlier simulations, and a phase transition appears at a temperature Tc(γ) in the thermodynamic limit (system size N → ∞). For γ ≤ 3, a FM-PM crossover appears at a size-dependent temperature Tco, so the system remains in a FM state at any finite temperature in the limit N → ∞. Thus, for γ = 3, Tco scales as lnN, whereas for γ<3, we find Tco ~ JNz, where the exponent z decreases for increasing γ. Adding motifs (triangles in our case) to the networks causes an increase in the transition (or crossover) temperature for exponent γ>3 (or ≤ 3). For γ>3, this increase is due to changes in the mean values 〈k〉 and 〈k²〉, i.e., the transition is controlled by the degree distribution (nearest-neighbor connectivities). For γ ≤ 3, however, we find that clustered and unclustered networks with the same size and distribution P(k) have different crossover temperature, i.e., clustering favors FM correlations, thus increasing the temperature Tco. The effect of a degree cutoff kcut on the asymptotic behavior of Tco is discussed. [ABSTRACT FROM AUTHOR]