Time-varying multiplex network: Intralayer and interlayer synchronization

A large class of engineered and natural systems, ranging from transportation networks to neuronal networks, are best represented by multiplex network architectures, namely a network composed of two or more different layers where the mutual interaction in each layer may differ from other layers. Here we consider a multiplex network where the intralayer coupling interactions are switched stochastically with a characteristic frequency. We explore the intralayer and interlayer synchronization of such a time-varying multiplex network. We find that the analytically derived necessary condition for intralayer and interlayer synchronization, obtained by the master stability function approach, is in excellent agreement with our numerical results. Interestingly, we clearly find that the higher frequency of switching links in the layers enhances both intralayer and interlayer synchrony, yielding larger windows of synchronization. Further, we quantify the resilience of synchronous states against random perturbations, using a global stability measure based on the concept of basin stability, and this reveals that intralayer coupling strength is most crucial for determining both intralayer and interlayer synchrony. Lastly, we investigate the robustness of interlayer synchronization against a progressive demultiplexing of the multiplex structure, and we find that for rapid switching of intralayer links, the interlayer synchronization persists even when a large number of interlayer nodes are disconnected.

Figure 1

Variation of (a) intralayer and (b) interlayer synchronization errors with respect to the intralayer coupling parameter $\epsilon$ (interlayer coupling strength $\lambda =\frac{\epsilon }{6}$) for different values of rewiring frequencies $f=0.001$ (blue circle), 0.01 (red square), 0.1 (green diamond), 1.0 (magenta lower triangle), and 10 (black upper triangle). The insets in the respective panels show the magnified version of errors for a better visualization.

Figure 2

Maximum Lyapunov exponent corresponding to (a) intralayer synchronization and (b) interlayer synchronization with respect to the parameter $\epsilon$ where $\lambda =\frac{\epsilon }{6}$ for the same set of rewiring frequencies as in Fig. 1.

Figure 3

Variation of basin stability for (a) intralayer and (b) interlayer synchronous state with respect to the parameter $\epsilon$ where $\lambda =\frac{\epsilon }{6}$ with different values of rewiring frequencies: $f=0.001$ (blue circle), $f=0.01$ (red square), $f=0.1$ (green diamond), $f=1.0$ (magenta lower triangle), and $f=10$ (black upper triangle).

Figure 4

Synchronization errors (first row) and basin stability measures (second row) in the $\epsilon \text{−}\lambda$ plane for (a), (c) intralayer synchronization and (b), (d) interlayer synchronization. Here $f=0.1$.

Figure 5

Basin stability in the $f\text{−}\epsilon$ plane where $\lambda =\frac{\epsilon }{6}$, for intralayer synchronization where (a) $k=2$ and (c) $k=3$; interlayer synchronization where (b) $k=2$ and (d) $k=3$.

Figure 6

Variation of (a) interlayer synchronization error and (b) corresponding basin stability with respect to the number of demultiplexed nodes $\nu$ for various intralayer coupling strength $\epsilon =0.26$ (blue circle), 0.5 (red square), 1.0 (green diamond), 1.5 (magenta star), and 2.0 (black triangle). (c) Phase diagram in the $\nu \text{−}\epsilon$ plane for interlayer synchronization for $f=0.1$ and $\lambda =0.2$. (d) Maximum number of demultiplexed nodes ${\nu }^{*}$ preserving interlayer synchrony against increasing $\epsilon$ for various rewiring frequencies $f=0.001$ (blue circle), 0.01 (red square), 0.1 (green diamond), 1.0 (magenta star), and 10 (black triangle). Here $\lambda =0.2$.