Nonequilibrium dynamics of the Ising model on heterogeneous networks with an arbitrary distribution of threshold noise

The Ising model on networks plays a fundamental role as a testing ground for understanding cooperative phenomena in complex systems. Here we solve the synchronous dynamics of the Ising model on random graphs with an arbitrary degree distribution in the high-connectivity limit. Depending on the distribution of the threshold noise that governs the microscopic dynamics, the model evolves to nonequilibrium stationary states. We obtain an exact dynamical equation for the distribution of local magnetizations, from which we find the critical line that separates the paramagnetic from the ferromagnetic phase. For random graphs with a negative binomial degree distribution, we demonstrate that the stationary critical behavior as well as the long-time critical dynamics of the first two moments of the local magnetizations depend on the distribution of the threshold noise. In particular, for an algebraic threshold noise, these critical properties are determined by the power-law tails of the distribution of thresholds. We further show that the relaxation time of the average magnetization inside each phase exhibits the standard mean-field critical scaling. The values of all critical exponents considered here are independent of the variance of the negative binomial degree distribution. Our work highlights the importance of certain details of the microscopic dynamics for the critical behavior of nonequilibrium spin systems.

Figure 1

(a) Stationary magnetization $m$ and (b) variance $\mathrm{Var}\left(m\right)$ of the distribution ${\mathcal{P}}_{\mathrm{h}}\left(m\right)$ of local magnetizations as a function of the rescaled temperature $T/T_c$ [see Eq. (40)] for the hyperbolic tangent distribution ${\mu }_{\mathrm{h}}$ of the threshold noise. The parameter $1/\alpha$ is the relative variance of the negative binomial degree distribution [see Eq. (32)]. The solid lines follow from the analytic equations (38), (39), and (42). The symbols are results obtained from numerical simulations of Eq. (1) for $N={10}^4$ and mean degree $c={10}^2$. The vertical bars are the standard deviation calculated from ten independent runs of the simulations. The random graph samples in the numerical simulations are generated from Eq. (29).

Figure 2

The stationary distribution ${\mathcal{P}}_{\mathrm{h}}\left(m\right)$ of local magnetizations inside the ferromagnetic phase of the Ising model on random graphs with a negative binomial degree distribution with $\alpha =2.5$ [see Eq. (32)] and the hyperbolic tangent distribution ${\mu }_{\mathrm{h}}$ of the threshold noise.

Figure 3

(a) The stationary magnetization $m$ and (b) the variance $\mathrm{Var}\left(m\right)$ of the distribution ${\mathcal{P}}_{\mathrm{a}}\left(m\right)$ of local magnetizations as a function of the reduced temperature $(T_c-T)/T_c$ [see Eq. (40)] for $\alpha =1$ and the algebraic distribution ${\mu }_{\mathrm{a}}$ of the threshold noise. The symbols are numerical results obtained from Eq. (43), while the dashed lines are the analytic expressions of Eqs. (46) and (47).

Figure 4

Dynamics of the average magnetization $m\left(t\right)$ inside the ferromagnetic phase of the Ising model on random graphs with a negative binomial degree distribution with $\alpha =0.625$. The dashed lines are derived from Eqs. (19) and (22), while the symbols are numerical simulations of Eq. (1) for $N={10}^4$ and mean degree $c=100$. The main panel shows results for the hyperbolic tangent distribution and for the algebraic distribution [see Eq. (35)] of the threshold noise with temperature $T=T_c/2$. The random graphs in the numerical simulations of the main panel are sampled from Eq. (29). The inset compares numerical simulation results obtained from the configuration model and from the effective matrix of Eq. (29) for the distribution ${\mu }_{\mathrm{h}}$. The vertical bars are the standard deviations calculated from ten independent simulations.

Figure 5

(a) Dynamics of the global magnetization $m\left(t\right)$ and of the (b) variance $\mathrm{Var}\left[m\right(t\left)\right]$ of the distribution of local magnetizations at the critical temperature $T=T_c$. The degrees follow a negative binomial degree distribution with relative variance $1/\alpha$. The main panels show results for $\alpha =1$, different initial conditions $m\left(0\right)$, and for the algebraic distribution ${\mu }_a\left(\zeta \right)$ of the threshold noise with $\kappa =2$. The inset in (a) illustrates the long-time power-law decay of $m\left(t\right)$ for $m\left(0\right)=0.4$ and different $\alpha$ (the other parameters are the same as in the main panels): $\alpha =2$ ($\diamond$), $\alpha =1$ ($\times$), and $\alpha =0.5$ ($+$).

Figure 6

Relaxation time $\tau$ of the average magnetization as a function of $\alpha$ for fixed temperature $T=2J$, approaching the critical value ${\alpha }_c=1$ [see Eq. (40)] from the (a) ferromagnetic and from the (b) paramagnetic phase. The quantity $1/\alpha$ is the relative variance of the negative binomial degree distribution. Each data point is obtained by fitting the long-time dynamics of $m\left(t\right)$, derived from the recurrence Eqs. (19) and (22), with the exponential function of Eq. (49).