Phase ordering in coupled noisy bistable systems on scale-free networks

We study a system consisting of diffusively coupled noisy bistable elements on a scale-free random network. This system exhibits an order-disorder phase transition as the noise intensity is varied. The phase ordering process takes place consecutively and in order of the degrees, reflecting strong degree heterogeneity of the scale-free network. A nonlinear Fokker-Planck equation describing the network dynamics is derived under mean-field approximation of the network, and is used to explain the phase ordering dynamics of the system.

Figure 1

Snapshots of the distribution of the elements during numerical evolution of Eq. (1) at (a) $t=0$, (b) $t=2$, (c) $t=10$, (d) $t=100$, (e) $t=500$, and (f) $t=10000$, where the state $x_i$ of the element $i$ is plotted as a function of the degree $k_i$ for $i=1,\cdots ,N$. The black stars and the blue squares represent stable and unstable fixed points of Eq. (7) under mean-field approximation, respectively, where $\overline{x}\left(t\right)$ is set to the mean field estimated using Eq. (6) from the states of the elements for each $t$. The network parameters are $N=1800$, $k_{\min }=6$, and $\gamma =2.5$, yielding $k_{\max }=87$. The system parameters are $\epsilon =0.02$ and $D=0.02$.

Figure 2

Snapshots of the probability density function $P(x,t;k)$ for $k=k_{\min },\cdots ,k_{\max }$ during the numerical evolution of the nonlinear Fokker-Planck equation (12) at (a) $t=0$, (b) $t=2$, (c) $t=10$, (d) $t=100$, (e) $t=500$, and (f) $t=10000$, where $P(x,t;k)$ is shown in the density plot (the bars indicate values of the probability densities). The network parameters are $k_{\min }=6$, $k_{\max }=87$, and $\gamma =2.5$, which are chosen to be the same as those used in Fig. 1, and the system parameters are $\epsilon =0.02$ and $D=0.02$.

Figure 3

Dependence of the resulting mean field $x^{\left(1\right)}$ on the presumed mean field $x^{\left(0\right)}$ calculated from Eqs. (13) and (14) for $D=0.02<D_{\mathrm{c}}$ and $D=0.4>D_{\mathrm{c}}$. The parameter values are $k_{\min }=6$, $k_{\max }=87$, $\gamma =2.5$, and $\epsilon =0.02$.

Figure 4

[(a)–(c)] Dependence of the mean field on the bifurcation parameter $D$ near the critical point for (a) $\epsilon =0.005$, (b) $\epsilon =0.02$, and (c) $\epsilon =0.2$. The distribution of the residence time $\tau \left(x\right)$ during which $|\overline{x}|$ stayed in the range $x\le |\overline{x}|<x+\Delta x$ is depicted in the density plot (the bars indicate values of the probability densities), where $\Delta x=0.02$ and $\tau \left(x\right)$ is normalized as ${\sum }_x\tau \left(x\right)=1$. The solid lines show self-consistent solutions $x^{*}$ of the NLFPE, and the broken line indicates $D_{\mathrm{c}}$, the critical point of the NLFPE. (d) Dependence of the critical value $D_{\mathrm{c}}$ on the coupling intensity $\epsilon$. The parameters of the network are $k_{\min }=6$, $k_{\max }=87$, and $\gamma =2.5$.

Figure 5

[(a)–(c)] Typical snapshots of the original network system with $N=1800$ elements sufficiently after initial relaxation obtained by direct numerical simulation of Eq. (1) at $\epsilon =0.02$ (which satisfies the condition $\epsilon k_{\min }<1<\epsilon k_{\max }$). The black stars and blue squares depict stable and unstable fixed points of Eq. (7), respectively, where the mean field $\overline{x}$ is estimated using Eq. (6). [(d)–(f)] Stationary probability density functions $P_{\mathrm{st}}(x;k)$ of the nonlinear Fokker-Planck equation at $\epsilon =0.02$, shown in the density plot (the bars indicate values of the probability densities). The noise intensity is specified as $D=0.02$ in (a) and (d), $D=0.14$ in (b) and (e), and $D=0.2$ in (c) and (f). The parameters of the network are $k_{\min }=6$, $k_{\max }=87$, and $\gamma =2.5$.

Figure 6

Typical snapshots of the original network system with $N=1800$ elements sufficiently after initial relaxation (left column) and the stationary probability density functions of the nonlinear Fokker-Planck equation (right column, shown in the density plot where the bars indicate values of the probability densities). The black stars and blue squares in left column depict stable and unstable fixed points of Eq. (7), respectively, where the mean field $\overline{x}$ is estimated using Eq. (6). [(a)–(c)] $\epsilon =0.005$ (satisfying $\epsilon k_{\max }<1$) with (a) $D=0.01$, (b) $D=0.07$, and (c) $D=0.1$. [(d)–(f)] $\epsilon =0.2$ (satisfying $\epsilon k_{\min }>1$) with (d) $D=0.02$, (e) $D=0.7$, and (f) $D=1$. The parameters of the network are $k_{\min }=6$, $k_{\max }=87$, and $\gamma =2.5$.

Figure 7

(a) Evolution of ${\overline{x}}_k$ of the elements with degree $k$ of the original network system ($N=1800$) and ${\langle x\rangle }_k$ of the approximate NLFPE for several values of $k$. (b) Evolution of ${\overline{x}}_k$ and the stable fixed point $x_{\mathrm{g}}\left(k\right)$ minimizing Eq. (8) for several values of $k$, where $x_{\mathrm{g}}\left(k\right)$ is calculated by using the mean field $\overline{x}=\langle x\rangle$ obtained from the NLFPE. (c) Snapshots of ${\overline{x}}_k$ and ${\langle x\rangle }_k$ plotted as functions of $k$ at several values of $t$. (d) Snapshots of ${\overline{x}}_k$ and $x_{\mathrm{g}}\left(k\right)$ plotted as functions of $k$ for several values of $t$. The system parameters are $N=1800$, $k_{\min }=6$, $k_{\max }=87$, $\gamma =2.5$, $D=0.02$, and $\epsilon =0.02$. The mean field of the original network system is averaged over 50 realizations.

Figure 8

Dependence of the potential barrier $V_{\mathrm{u}}$ on the degree $k$ for $\langle x\rangle =0$ (a), $\langle x\rangle =0.3$ (b), $\langle x\rangle =0.6$ (c), and $\langle x\rangle =0.9$ (d).

Figure 9

(a) Evolution of the mean field $\overline{x}\left(t\right)$ during the relaxation of the original network system ($N=1800$, averaged over 50 realizations) and the mean field $\langle x\rangle \left(t\right)$ of the approximate NLFPE at several values of $\epsilon$. (b) Relaxation time $T$ of the original network system ($N=1800$, averaged over 50 realizations) and of the NLFPE, which is defined as the time taken for the overall mean field $\overline{x}\left(t\right)$ or $\langle x\rangle \left(t\right)$ to exceed $0.95\times x^{*}$, plotted with respect to $\epsilon$. The dashed line is the Kramers’ escape time $T=C\exp [-V_{\mathrm{u}}(\overline{x};k_{\min })/D]$ with $\overline{x}$ of Eq. (19), where the prefactor $C$ is determined by the least-squares fit. The network parameters are $k_{\min }=6$, $k_{\max }=87$, and $\gamma =2.5$, and the noise intensity is fixed at $D=0.02$.