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

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.

Figure 1

Schematic plot of $⟨\tilde{m}⟩$ versus $ϵ$ in the exponential networks.

Figure 2

Schematic plot of $⟨\tilde{m}⟩$ versus ${ϵ}_b$ in the annealed SF networks with $\lambda >5$ and (a) ${\omega }_{\text{nat}}\le \omega <2\left(\lambda -3\right)$ and (b) $\omega >2\left(\lambda -3\right)$. Note that the bulk critical point ${ϵ}_b=0$ is outside of the critical region in (b).

Figure 3

Schematic plots of $⟨\tilde{m}⟩$ versus (a) $ϵ$ and $⟨\tilde{m}⟩$ versus (b) ${ϵ}_b$ in the annealed SF networks with $3<\lambda <5$ and $\omega >{\omega }_{\text{nat}}$.

Figure 4

(Color online) Monte Carlo simulation data at $\lambda =6$. (a) The order parameters ${⟨\tilde{m}⟩}_c$ at $ϵ=0$ and ${⟨\tilde{m}⟩}_{b,c}$ at ${ϵ}_b=0$ with different values of $\left(d,\omega \right)=\left(2.25,5\right)$, (3,6), and (3,7) are plotted with symbols. They are compared with the analytic results of Eqs. (24, 37) which are drawn with lines. Numerical values of the coefficients are $A_e\simeq 0.815860$, ${\tilde{A}}_e\simeq 0.687983$, and $B_e\simeq 1.03920$. (b) Scaling plot of $⟨\tilde{m}⟩N^{\beta /\overline{\nu }}$ versus $\left|ϵ\right|N^{1/\overline{\nu }}$ at $\omega =6$ with $\beta =1/2$ and $\overline{\nu }=2$. Data with different values of $N$ fall onto the scaling function $\tilde{\psi }\left(\left|ϵ\right|N^{1/2}\right)$ drawn with the solid curve.

Figure 5

(Color online) Monte Carlo simulation data at $\lambda =4$. The parameter values of $d$ are 2.57 for $\omega =3$ and 1 for the other values of $\omega$. (a) Numerical data for ${⟨\tilde{m}⟩}_c$ and ${⟨\tilde{m}⟩}_{b,c}$ represented with symbols are compared with the analytic results represented with straight lines. (b) Scaling plot of $⟨\tilde{m}⟩N^{1/\left(\lambda -1\right)}$ versus $\left|ϵ\right|N^{\left(\lambda -3\right)/\left(\lambda -1\right)}$ at $\omega ={\omega }_{\text{nat}}=3$. The straight line has the slope of $\beta =1/\left(\lambda -3\right)=1$. We also show the scaling collapse of the numerical data at $\omega =4$ according to the FSS forms in (c) Eq. (51) and (d) Eq. (52). The solid line and the dashed line have a slope of 1/2 and $\beta =1/\left(\lambda -3\right)=1$, respectively. We use same symbols of Fig. 4b and the symbol ◇ for $N=32768\times {10}^3$.

Figure 6

(Color online) The histogram for ${⟨\tilde{m}⟩}_c/\left[{⟨\tilde{m}⟩}_c\right]$, the order parameter at $ϵ=0$ normalized with the ensemble average, where network parameters are $\lambda =4.0$, $k_0=3$, $d=1$, and $\omega ={\omega }_{\text{nat}}=3$. The curves from different values of $N$ collapse onto a single curve.