First-order phase transition in a majority-vote model with inertia

We generalize the original majority-vote model by incorporating inertia into the microscopic dynamics of the spin flipping, where the spin-flip probability of any individual depends not only on the states of its neighbors, but also on its own state. Surprisingly, the order-disorder phase transition is changed from a usual continuous or second-order type to a discontinuous or first-order one when the inertia is above an appropriate level. A central feature of such an explosive transition is a strong hysteresis behavior as noise intensity goes forward and backward. Within the hysteresis region, a disordered phase and two symmetric ordered phases are coexisting and transition rates between these phases are numerically calculated by a rare-event sampling method. A mean-field theory is developed to analytically reveal the property of this phase transition.

Figure 1

Absolute magnetization $\left|m\right|$ as a function of noise intensity $f$ on ER random networks with different inertia parameter $\theta$: (a) $\theta =0$, (b) $\theta =0.2$, (c) $\theta =0.3$, and (d) $\theta =0.35$. The squares and circles correspond to forward and backward simulations, respectively. The network parameters are $N=10000$ and $\langle k\rangle =20$.

Figure 2

(a) Time series of magnetization $m$ show the transition events between ordered and disordered phases within the hysteresis region. Also shown is the PDF of $m$ for (b) $f=0.136$, (c) $f=0.138$, and (d) $f=0.14$. The other parameters are $\theta =0.3, N=500$, and $\langle k\rangle =20$.

Figure 3

(a) Logarithm of transition rates $lnR$ as a function of $f$ for different $N$. Closed symbols correspond to the transition rate from the disordered state to the ordered state and open symbols to the transition rate from the ordered state to the disordered state. (b) Logarithm of transition rates $lnR$ as a function of $N$ for $f=0.144$. The lines indicate the linear fitting $ln{R}_{1\left(2\right)}\sim -{\nu }_{1\left(2\right)}N$. The inset shows the fitting exponents ${\nu }_{1\left(2\right)}$ as a function of $f$.

Figure 4

Theoretical result of $\left|m\right|$ as a function of $f$ for three typical values of $\theta$: (a) $\theta =0.15$, (b) $\theta =0.23$, and (c) $\theta =0.3$. In (c) circles within the hysteresis region indicate the unstable solution.

Figure 5

Phase diagram in the $\theta -f$ plane for three types of networks with two different average degrees, $\langle k\rangle =20$ (top panels) and $\langle k\rangle =40$ (bottom panels) (from left to right): RD RNs, ER RNs, and BA SFNs. Lines and symbols correspond to the theoretical and simulation results, respectively. The $f_{c_F}$ are indicated by solid lines and squares and $f_{c_B}$ by dashed lines and circles. The sizes of all the networks are the same: $N=10000$.

Figure 6

The $q=3$ state IMV model. The order parameter $m$ is plotted as a function of noise intensity $f$ on ER random networks with different inertia factor $\theta$: (a) $\theta =0$, (b) $\theta =0.1$, (c) $\theta =0.2$, and (d) $\theta =0.3$. The squares and circles correspond to forward and backward simulations, respectively. The network parameters are $N=10000$ and $\langle k\rangle =20$.

Figure 7

The $q=5$ state IMV model. The order parameter $m$ is plotted as a function of noise intensity $f$ on ER random networks with different inertia factor $\theta$: (a) $\theta =0$, (b) $\theta =0.1$, (c) $\theta =0.2$, and (d) $\theta =0.3$. The squares and circles correspond to forward and backward simulations, respectively. The network parameters are $N=10000$ and $\langle k\rangle =20$.