Critical noise of majority-vote model on complex networks

The majority-vote model with noise is one of the simplest nonequilibrium statistical model that has been extensively studied in the context of complex networks. However, the relationship between the critical noise where the order-disorder phase transition takes place and the topology of the underlying networks is still lacking. In this paper, we use the heterogeneous mean-field theory to derive the rate equation for governing the model's dynamics that can analytically determine the critical noise $f_c$ in the limit of infinite network size $N\to \infty$. The result shows that $f_c$ depends on the ratio of $\langle k\rangle$ to $\langle k^{3/2}\rangle$, where $\langle k\rangle$ and $\langle k^{3/2}\rangle$ are the average degree and the $3/2$ order moment of degree distribution, respectively. Furthermore, we consider the finite-size effect where the stochastic fluctuation should be involved. To the end, we derive the Langevin equation and obtain the potential of the corresponding Fokker-Planck equation. This allows us to calculate the effective critical noise $f_c\left(N\right)$ at which the susceptibility is maximal in finite-size networks. We find that the $f_c-f_c\left(N\right)$ decays with $N$ in a power-law way and vanishes for $N\to \infty$. All the theoretical results are confirmed by performing the extensive Monte Carlo simulations in random $k$-regular networks, Erdös-Rényi random networks, and scale-free networks.

Figure 1

Graphic demonstration of steady-state solutions of $Q$. When $f$ is less than a critical value $f_c$, there are three solutions. One is at $Q=\frac{1}{2}$ corresponding to a disordered phase and the other two correspond to two symmetric ordered phases. For $f>f_c$, there is only one solution at $Q=\frac{1}{2}$.

Figure 2

The critical noise $f_c$ obtained from the theory (lines) and MC simulation (symbols) on (a) RkRN, (b) ERRN, and (c, d) SFN.

Figure 3

The susceptibility ${\chi }^{\prime }$ as a function of noise $f$ for some different network sizes: $N=1000,2000,5000,10000$. (a) RkRN: $k=20$, (b) ERRN: $\langle k\rangle =20$, (c) SFN: $\gamma =2.4$ and $k_0=20$, and (d) $\gamma =3$ and $k_0=20$. Lines and symbols indicate the results from theory and MC simulation, respectively.

Figure 4

Log-log plot of the differences of the effective critical noise $f_c$ for finite size networks with $f_c$, $\Delta f\left(N\right)=f_c-f_c\left(N\right)$, with $N$ on (a) RkRN: $k=20$, ERRN: $\langle k\rangle =20$, and (b) SFN: $k_0=20$. $\Delta f\left(N\right)$ power law decays with $N$: $\Delta f\left(N\right)\sim N^{-\nu }$. Lines and symbols indicate the results from theory and MC simulation, respectively. The inset of Fig. 4 shows the exponent $\nu$ as a function of $\gamma$.