Critical behavior of the Ising model in annealed scale-free networks

TL;DR

This paper analyzes the critical behavior of the Ising model on annealed scale-free networks, deriving finite-size scaling laws and confirming them with numerical simulations, revealing similarities and differences with quenched networks.

Contribution

It provides the first analytic derivation of finite-size scaling near criticality for the Ising model on annealed scale-free networks, accounting for degree cutoff effects.

Findings

Finite-size scaling characterized by four regimes.

Analytic results match numerical simulations.

Differences between annealed and quenched networks persist.

Abstract

We study the critical behavior of the Ising model in annealed scale-free (SF) networks of finite system size with forced upper cutoff in degree. By mapping the model onto the weighted fully connected Ising model, we derive analytic results for the finite-size scaling (FSS) near the phase transition, characterized by the cutoff-dependent two-parameter scaling with four distinct scaling regimes, in highly heterogeneous networks. These results are essentially the same as those found for the nonequilibrium contact process in annealed SF networks, except for an additional complication due to the trivial critical point shift in finite systems. The discrepancy of the FSS theories between annealed and quenched SF networks still remains in the equilibrium Ising model, like some other nonequilibrium models. All of our analytic results are confirmed reasonably well by numerical simulations.