Noise-induced desynchronization and stochastic escape from equilibrium in complex networks

Complex physical systems are unavoidably subjected to external environments not accounted for in the set of differential equations that models them. The resulting perturbations are standardly represented by noise terms. If these terms are large enough, they can push the system from an initial stable equilibrium point, over a nearby saddle point, outside of the basin of attraction of the stable point. Except in some specific cases, the distance between these two points is not known analytically. Focusing on Kuramoto-like models and under simple assumptions on this distance, we derive conditions under which such noise terms perturb the dynamics strongly enough that they lead to stochastic escape from the initial basin of attraction. We numerically confirm the validity of that criterion for coupled oscillators on four very different complex networks. We find in particular that, quite counterintuitively, systems with inertia leave their initial basin faster than or at the same time as systems without inertia, except for strong white-noise perturbations.

Figure 1

Time evolution of the winding number $q$ for Eq. (1) on a single-cycle network with $n=83$ nodes, $m=0$ (red lines), and $\frac{m}{d}/\frac{d}{{\lambda }_2}=10/175$ (blue dashed lines). (a) Noise with short correlation time ${\lambda }_2{\tau }_0/d=5.7\times {10}^{-4}$. (b) Noise with longer correlation time ${\lambda }_2{\tau }_0/d=0.03$.

Figure 2

Color-coded survival probability $\mathbb{P}$ for Eq. (1) with $m=0$. (a) Single-cycle network with $n=83$ and nearest-neighbor coupling; (b) single-cycle network with $n=83$, nearest- and third-neighbor coupling; (c) U.K. transmission network with $n=120$; (d) small-world network with $n=200$ nodes. Yellow dashed lines give the boundary of the region of validity of the inequality in Eq. (8) with $m=0$ and $\mathrm{\Delta }$ obtained analytically for panel (a) and numerically for panels (b)–(d). Observation times $T_{\mathrm{obs}}$ correspond to comparable dimensionless parameters ${\lambda }_2{T}_{\mathrm{obs}}/d=143$ (a), 143 (b), 130 (c), and 115 (d).

Figure 3

Color-coded survival probability $\mathbb{P}$ for Eq. (1) with $m=0$ for a single-cycle network with $n=83$ and nearest-neighbor coupling; ${\lambda }_2{T}_{\mathrm{obs}}/d=14.3$ (a), 143 (b), 569 (c). The yellow dashed line gives the boundary of the region of validity of the inequality in Eq. (8) with $m=0$ and $\mathrm{\Delta }$ obtained analytically.

Figure 4

Escape time $T_{\mathrm{esc}}$ from the initial basin of attraction vs noise amplitude, $\delta P_0$, for cycle networks with $n=83$ (a), $n=249$ (b), and the U.K. transmission network (c). The noise correlation time corresponds to ${\lambda }_2{\tau }_0/d=8.6\times {10}^{-3}$ (a), ${\lambda }_2{\tau }_0/d=9.6\times {10}^{-4}$ (b), and ${\lambda }_2{\tau }_0/d=0.02$ (c). Blue circles are averages over 40 realizations of noise. Red crosses correspond to Eq. (9), with $\beta \cong 5/8$ (a, b) and $\beta \cong 2/5$ (c).

Figure 5

Color-coded difference in survival probability $\mathbb{P}$ with and without inertia for a single-cycle network with $n=83$ with nearest- and third-neighbor coupling obtained from 20 realizations of noise; (a) $\frac{m}{d}/\frac{d}{{\lambda }_2}=0.25/0.35$, (b) $2.5/0.35$, and (c) $25/0.35$. The yellow dashed lines give the boundary of the region of validity of the inequality in Eq. (8), as discussed in the main text.

Figure 6

Maximum value $\delta P_0^{*}$ of the noise amplitude obtained from Eqs. (C2) and (C3) for large (blue) and short (green, red) time correlation, ${\tau }_0$, as a function of the size of the cyclic network $n$. For the red curve, we consider a constant ratio ${\tau }_0/d=0.001$. For the green curve we consider a constant ratio ${\lambda }_2{\tau }_0/d=0.001$ where ${\lambda }_2=2-2cos(2\pi /n)$ depends on the size of the network.

Figure 7

(a) Illustration of the connections of a vertex to its first and second neighbors on a cycle. (b) Illustration of the connections of a vertex to its nearest and third neighbors on a cycle. (c) Illustration of the U.K. network with $n=120$ vertices and $m=165$ edges. (d) Illustration of our small-world network with $n=200$ vertices. Its relative clustering coefficient is $C\left({\mathcal{G}}_p\right)/C\left({\mathcal{G}}_0\right)\approx 0.89$, and its relative characteristic path length is $L\left({\mathcal{G}}_p\right)/L\left({\mathcal{G}}_0\right)\approx 0.32$.

Figure 8

Example of the time evolution of the 120 angles of the U.K. network [Fig. 7]. We clearly see two angles jumping from a value close to 0 to a value close to $2\pi$. The state of the system at the time given by the vertical dashed line is our candidate for a 1-saddle $\mathit{\phi }$.

Figure 9

Histograms of the 2-norm distance from the fixed point of the set of 1-saddles found numerically for the cycle with third-neighbor (a), the U.K. network (b), and the small-world network (c). We found (a) 284 1-saddles for the cycle with third neighbor, with smallest 2-norm $n_{min}\approx 3.12$, and quartiles of the 2-norms $(Q_1,Q_2,Q_3)\approx (3.12,8.61,8.61)$; (b) 788 1-saddles for the U.K. network, with smallest 2-norm $n_{min}\approx 3.13$, and quartiles of the 2-norms $(Q_1,Q_2,Q_3)\approx (7.24,10.02,12.17)$; and (c) 4956 1-saddles for the small-world network, with smallest 2-norm $n_{min}\approx 3.13$, and quartiles of the 2-norms $(Q_1,Q_2,Q_3)\approx (10.74,12.13,13.95)$. The yellow dashed lines indicate the three quartiles $Q_1$, $Q_2$, and $Q_3$, and the red dashed lines indicate the norm of the closest 1-saddle.

Figure 10

The two 1-saddles, ${\mathit{\phi }}^{\left(1\right)}$ and ${\mathit{\phi }}^{\left(2\right)}$, with smallest 2-norm, for the cycle network, with nearest and third neighbors. (a) ${\mathit{\phi }}^{\left(1\right)}$: all angles are equal, except one which is $\pi$ apart from all others. The 2-norm of this 1-saddle is $\sim 3.12$. (b) ${\mathit{\phi }}^{\left(2\right)}$: all angles are slightly displaced compared to their neighbors. The 2-norm of this 1-saddle is $\sim 8.61$. This configuration is, in our opinion, more likely to occur under noisy perturbations applied to all nodes.

Figure 11

Color plot of the difference of fraction of trajectories that stay in the initial basin of attraction with finite inertia compared to $m=0$ for a cycle network of $n=83$ nodes. Timescales are $\frac{m}{d}/\frac{d}{{\lambda }_2}=10/175$.