Critical slowing down in networks generating temporal complexity

We study a nonlinear Langevin equation describing the dynamic variable $X\left(t\right)$, the mean field (order parameter) of a finite size complex network at criticality. The conditions under which the autocorrelation function of $X$ shows any direct connection with criticality are discussed. We find that if the network is prepared in a state far from equilibrium, $X\left(0\right)=1$, the autocorrelation function is characterized by evident signs of critical slowing down as well as by significant aging effects, while the preparation $X\left(0\right)=0$ does not generate evident signs of criticality on $X\left(t\right)$, in spite of the fact that the same initial state makes the fluctuating variable $\eta \left(t\right)\equiv sgn\left(X\right(t\left)\right)$ yield significant aging effects. These latter effects arise because the dynamics of $\eta \left(t\right)$ are directly dependent on crucial events, namely the re-crossings of the origin, which undergo a significant aging process with the preparation $X\left(0\right)=0$. The time scale dominated by temporal complexity, aging, and ergodicity breakdown of $\eta \left(t\right)$ is properly evaluated by adopting the method of stochastic linearization which is used to explain the exponential-like behavior of the equilibrium autocorrelation function of $X\left(t\right)$.

Figure 1

Age-dependent correlation function with preparation $X\left(0\right)=1$. The bottom doublet, solid black line (theoretical), dashed line (numerical), refers to $t_a=0$. The top doublet, solid line (theoretical), dashed line (numerical) refers to $t_a=100$. $\gamma =0.01,\sigma =0.001$.

Figure 2

The correlation function with preparation $X\left(0\right)=1$ at different ages. The solid line denotes $t_a=0$, the dotted line $t_a=100$, and the bottom dashed line $t_a=1000$. These three curves are numerical. The top dashed line is the theoretical equilibrium ($t_a=\infty$) correlation function, the exponential function of Eq. (31) with the parameter $\Gamma$ given by Eq. (32). $\gamma =0.01,\sigma =0.001$.

Figure 3

The (solid) green curve is the numerical correlation function with preparation $X\left(0\right)=1$ with $t_a={10}^4$ and the (dashed) black curve is the exponential function of Eq. (31) with the parameter $\Gamma$ given by Eq. (32). $\gamma =0.01,\sigma =0.001$. Numerical work shows that preparation $X\left(0\right)=0$ with $t_a={10}^4$ generates indistinguishable results.