Interlayer antisynchronization in degree-biased duplex networks

With synchronization being one of nature's most ubiquitous collective behaviors, the field of network synchronization has experienced tremendous growth, leading to significant theoretical developments. However, most previous studies consider uniform connection weights and undirected networks with positive coupling. In the present article, we incorporate the asymmetry in a two-layer multiplex network by assigning the ratio of the adjacent nodes' degrees as the weights to the intralayer edges. Despite the presence of degree-biased weighting mechanism and attractive-repulsive coupling strengths, we are able to find the necessary conditions for intralayer synchronization and interlayer antisynchronization and test whether these two macroscopic states can withstand demultiplexing in a network. During the occurrence of these two states, we analytically calculate the oscillator's amplitude. In addition to deriving the local stability conditions for interlayer antisynchronization via the master stability function approach, we also construct a suitable Lyapunov function to determine a sufficient condition for global stability. We provide numerical evidence to show the necessity of negative interlayer coupling strength for the occurrence of antisynchronization, and such repulsive interlayer coupling coefficients cannot destroy intralayer synchronization.

Figure 1

A multiplex network: We here visualize a duplex network (a multiplex network with two layers) with the help of Gephi [70]. Each of these layers consists of a connected intralayer network. This network used for the numerical experiments (unless stated otherwise) contains 12 nodes and 21 links. The six interlayer edges (dotted lines) connect the replica nodes and help to connect the two connected layers.

Figure 2

The variation of $F_{\mathrm{Layer}1}, F_{\mathrm{Layer}2}$, and $F_{\mathrm{Replica}}$ as a function of interlayer coupling strength $k_R$: We choose the multiplex network shown in Fig. 1, and place an identical limit cycle oscillator (14) on top of each node with ${\omega }_i=\omega =3$. We vary the interlayer coupling strength $k_R$ from 0.01 to $-0.01$ with fixed space $-0.0001$ and fixed intralayer coupling strength $k_A=0.1$. For each of these $200k_R\mathrm{s}$, we choose the initial condition of each oscillator randomly within the interval $[-1,1]\times [-1,1]$. For $k_R>0$, the system remains in interlayer phase synchronization (i.e., $F_{\mathrm{Replica}}=2$) beyond a critical value of $k_R$. However, the system attains interlayer antisynchronization $(F_{\mathrm{Replica}}=0)$ for a suitable negative interlayer coupling strength. Each of these subfigures is drawn with different adjacency matrices ${\tilde{\mathcal{A}}}^{\left[\alpha \right]}$ using the multiplex network in Fig. 1. Panel (a) represents the results for hub-attracting intralayer matrix $(\beta =1)$, whereas panel (c) depicts the results for the hub-repelling intralayer matrix $(\beta =-1)$. The middle panel (b) shows the results for the unweighted intralayer matrix $(\beta =0)$. Irrespective of the chosen value of $\beta$, the system settles down to an interlayer antisynchronized state for negative interlayer coupling strength (see blue square markers). In spite of choosing negative $k_R$, each layer maintains intralayer synchronization as $F_{\mathrm{Layer}1}$ (red plus ($+$) markers) $=F_{\mathrm{Layer}2}$ (magenta circle markers) $=2$ throughout the panels.

Figure 3

Intralayer synchronization and interlayer antisynchronization for hub-attracting intralayer matrix $(\beta =1)$: All the trajectories of the first layer collapse to a single trajectory (red line), and similarly, the trajectories of the second layer oscillate within $[-1,1]$ maintaining the same path (magenta line) in (a). This attests to the occurrence of intralayer synchronization. The sum $(x_{1,i}+x_{2,i})$ converges to a fixed value zero (blue line) after the transient. This validates the emergence of interlayer antisynchronization. Panel (b) contemplates the appearance of two clusters. The SL oscillators of the first layer lie within a synchronized group, and the oscillators of the second layer stay in another cluster. Due to the presence of repulsive interlayer coupling strength $k_R=-0.1$, these two clusters maintain a constant phase difference of $\pi$. All the panels are drawn using random initial conditions from $[-1,1]\times [-1,1]$. We choose the oscillators on top of the node 1, 2, and 7, respectively, from the multiplex network given in Fig. 1. We plot the phase portrait of these oscillators after the transient. Clearly, we have $x_{1,1}=x_{1,2}=-x_{2,1}$. More importantly, panels (a) and (c) confirm our analytical calculation revealing each SL oscillator evolves with a unit radius after reaching the interlayer antisynchronization manifold and the intralayer synchronization manifold. For each panel, we choose $k_A=\left|k_R\right|=0.1$.

Figure 4

The intralayer synchronization and intralayer antisynchronization with identical Thomas' cyclically symmetric attractor: The figures are drawn for (a) $b=0.10$, (b) $b=0.20$, and (c) $b=0.30$. We consider the same multiplex network with 12 vertices and 21 edges, shown in Fig. 1. Initial conditions are chosen randomly from $[-4,4]\times [-4,4]\times [-4,4]$. The interlayer coupling strength $k_R$ is set at $-0.3$, and the intralayer coupling strength $k_A$ is kept fixed at 1.0. We choose $\beta =-1$; thus, we have the hub-repelling intralayer matrices. In all panels, we find $(x_{1,i}+x_{2,i})$ converges to a fixed value zero after the initial transient. Hence, the emergence of interlayer antisynchronization is confirmed. Moreover, all the trajectories of the same layer collapse into a single trajectory [shown with the red (magenta) line for the first (second) layer].

Figure 5

The interlayer antisynchronization error $E$ (blue) and the maximum transverse Lyapunov exponent ${\mathrm{\Lambda }}_{max}$ (red) as a function of the interlayer coupling strength $k_R$: Both layers contain the same ring intralayer network with four vertices. In (a) for the SL oscillator, we set the system's parameter at $k_A=0.1$ and $\omega =3.0$, whereas in (b) for the Thomas cyclically symmetric attractor, we set $k_A=1.0$ and $b=0.2$. $E$ reduces to zero suggesting the occurrence of interlayer antisynchronization. Simultaneously, ${\mathrm{\Lambda }}_{max}$ crosses zero and becomes negative, revealing the appearance of interlayer antisynchronization. This indicates that our local stability condition agrees quite well with our numerical simulation.