Robustness of multilayer oscillator networks

We consider the robustness of multilayer networks composed of active and inactive oscillators from the viewpoint of interlayer coupling effects through the aging transition [H. Daido and K. Nakanishi, Phys. Rev. Lett. 93, 104101 (2004)]. We show in detail that two-layer networks increase or decrease their robustness depending on interlayer coupling schemes compared with single-layer networks. In addition, we find that an increase of mismatches of oscillator types (active or inactive) among interlayer-connected oscillators reduces the robustness of the networks with mean-field, chain, and diffusive interlayer couplings in two-layer networks. Moreover, we discuss the robustness of networks with more than two layers.

Figure 1

(a) Example of a layered network with $L=2$ and $N=5$. The red (light gray) and blue (dark gray) circles indicate active and inactive oscillators, respectively. The proportion of inactive elements is $p=0.4$. The black solid and green dashed lines indicate global intralayer couplings and local interlayer couplings, respectively. (b) Order parameter $\left|Z\right|$ is plotted against $p$ for three cases. The red line with $°$ corresponds to a single-layer network ($L=1$). The green line with $*$ corresponds to a multilayer network ($L=2$) in case II with $D=1.5$, where its aging transition point $p_c$ is greater than that of the single-layer network. The blue line with $\times$ corresponds to a multilayer network ($L=2$) in case III with $D=8$, where its aging transition point $p_c$ is smaller than that of the single-layer network.

Figure 2

Values of the aging transition point $p_c$ are plotted against the interlayer coupling strength $D$ in cases I ($+$), II ($\times$), and III ($°$). The results obtained from the reduced and original systems are indicated by lines and symbols, respectively.

Figure 3

Color in the $(Q,R)$ plane indicates the value of $p_c$, which is obtained from the reduced systems. The lines with ${45}^{°}$ (red), ${90}^{°}$ (blue), and ${135}^{°}$ (black) from the origin account for cases I, II, and III, respectively. The thicker and thinner lines correspond to $D>0$ and $D<0$, respectively. The values of the coupling strength $K$ are set at (a) $0$, (b) $2$, and (c) $8$.

Figure 4

Color in the $(Q,R)$ plane indicates the value of $r_1^{*}$ for ${\alpha }_1=2$ and ${\alpha }_2=-1$. The value of $r_1^{*}$ equals $\sqrt{{\alpha }_1}=\sqrt{2}$ on the black curve. This black curve can be obtained analytically as Eq. (9). On the right-hand side of the black curve, $r_1^{*}>\sqrt{2}$. On the contrary, on the left-hand side, $r_1^{*}<\sqrt{2}$. The value of $r_1^{*}$ is equal to $0$ in the black region. The region can also be obtained as Eq. (8).

Figure 5

All the figures are for the diffusive interlayer coupling (case III). (a) The order parameter $\left|Z\right|$ is plotted against $p$ for ($°$),($*$), and ($\times$) pairs with $D=3$. (b) The values of $p_c$ are plotted against the mismatched parameter $s$ for $D=1$ ($+$), $3$ ($\times$), and $8$ ($°$).

Figure 6

All the figures are for the diffusive interlayer coupling (case III). (a) The values of $p_c$ are plotted against $D$ for networks with $L=1,2,3$, and $4$. (b) The values of $p_c^{(L)}$ and $s_L$ are plotted against $L$ for $D=10$.