Effects of network robustness on explosive synchronization

Current studies have shown that there is a positive correlation between the network assortativity and robustness and that the assortativity also plays an important role in explosive synchronization. In this paper, taking the network robustness as a global property, we investigate its significance as well as the influence of its interaction with the assortativity on explosive synchronization. Our numerical results demonstrate that explosive synchronization is suppressed in extreme situations of both the robustness and assortativity. In addition, through appropriate adjustments of them, a maximum hysteresis area between the forward and backward transitions can be reached. Furthermore, our results might also provide reference for those who are interested in effects of network structure on synchronization, though this problem is still challenging as we show in the discussion.

Figure 1

The magnitude of synchronization $\Re$ versus the coupling strength $\lambda$ for the forward and backward transitions in networks with $n={10}^3$ and $\langle k\rangle =6.0$. (a) The initial SF network constructed using the BA model with $\mathcal{F}=0.197$ and $r=-0.067$. The solid and dashed lines, respectively, correspond to the forward and backward transitions, and between them is the hysteresis area represented by $\mathcal{S}$. Also, the maximal jump sizes of them are denoted by ${\mathcal{J}}_e$ and ${\mathcal{J}}_b$, accordingly. (b) $r=0.000$ with $\mathcal{F}=0.278$ adjusted by ${\xi }^{\delta }\left(r\right)$ (red circle) and $\mathcal{F}=0.250$ with $r=-0.033$ through ${\xi }^{\delta }\left(\mathcal{F}\right)$ (blue square) on the initial SF network. (c) $r=0.050$ with $\mathcal{F}=0.354$ and $\mathcal{F}=0.300$ with $r=-0.006$. (d) $r=-0.050$ with $\mathcal{F}=0.213$ and $\mathcal{F}=0.150$ with $r=-0.077$.

Figure 2

The magnitude of synchronization $\Re$ versus the coupling strength $\lambda$ for forward and backward transition on networks with $n={10}^3$ and $\langle k\rangle =6.0$ under different assortativity $r$ and robustness $\mathcal{F}$. (a–d) ${\xi }^{\delta }(r|{\xi }^{\delta }\left(\mathcal{F}\right)\equiv 0.20)$. (e, g) ${\xi }^{\delta }\left(r\right|{\xi }^{\delta }\left(\mathcal{F}\right)\equiv 0.35)$. (f, h) ${\xi }^{\delta }(r|{\xi }^{\delta }\left(\mathcal{F}\right)\equiv 0.10)$.

Figure 3

The jump size $J$ and hysteresis area $\mathcal{S}$ of the network robustness $\mathcal{F}$ and assortativity $r$. (a) The forward synchronization ${\mathcal{J}}_e$. (b) The backward synchronization ${\mathcal{J}}_b$. (c) $\mathcal{S}$. The solid and dashed curves correspond to $\mathcal{F}$ of $r$ with ${\xi }^{\delta }\left(r\right)$ and $r$ of $\mathcal{F}$ with ${\xi }^{\delta }\left(\mathcal{F}\right)$, respectively. Each result is the average of 20 network realizations.

Figure 4

The magnitude of synchronization $\Re$ versus the coupling strength $\lambda$ for forward and backward transition on networks with $n={10}^4$ and $\langle k\rangle =6.0$ for different $r$ and $\mathcal{F}$. The result of the paradigm network from the BA model is reported in panel (a).

Figure 5

The magnitude of synchronization $\Re$ versus the coupling strength $\lambda$ for forward and backward transition on networks with $n={10}^3$ and $\langle k\rangle =6.0$ under different assortativity $r$ and robustness $\mathcal{F}$. (a, e, f) ${\xi }^{\delta }(\mathcal{F}|{\xi }^{\delta }\left(r\right)\equiv 0.000)$. (b, g, h) ${\xi }^{\delta }(\mathcal{F}|{\xi }^{\delta }\left(r\right)\equiv -0.050)$. (c) ${\xi }^{\delta }(\mathcal{F}|{\xi }^{\delta }\left(r\right)\equiv +0.150)$. (d) ${\xi }^{\delta }(\mathcal{F}|{\xi }^{\delta }\left(r\right)\equiv -0.200)$.

Figure 6

The hysteresis area $\mathcal{S}$ versus the perturbations of network assortativity $r$ and robustness $\mathcal{F}$ on networks with $\langle k\rangle =6.0$, $n={10}^3$ (the same initial network as Fig. 2) for (a, b, e–h) and $n={10}^4$ (the same initial network as in Fig. 4) for (c, d). (a, b) Respectively for ${\xi }^{\delta }\left(r\right|\mathcal{F})$ and ${\xi }^{\delta }\left(\mathcal{F}\right|r)$ with $r\left(0\right)=-0.067$ and $\mathcal{F}\left(0\right)=0.197$. (c, d) Respectively for ${\xi }^{\delta }\left(r\right|\mathcal{F})$ and ${\xi }^{\delta }\left(\mathcal{F}\right|r)$ with $r\left(0\right)=-0.029$ and $\mathcal{F}\left(0\right)=0.187$. (e, g) ${\xi }^{\delta }\left(r\right)$ with $r\left(0\right)=0.000$ and $r\left(0\right)=0.050$, accordingly. (f, h) ${\xi }^{\delta }\left(\mathcal{F}\right)$ with $\mathcal{F}\left(0\right)=0.197$ and $\mathcal{F}\left(0\right)=0.250$, accordingly.

Figure 7

The hysteresis area $\mathcal{S}$ and the network robustness $\mathcal{F}$ or assortativity $r$ versus $r$ or $\mathcal{F}$ for forward (blue) and backward (green) evolutions on networks with $\langle k\rangle =6.0$ and $n={10}^3$.

Figure 8

The magnitude of synchronization $\Re$ versus the coupling strength $\lambda$ for forward and backward transition on networks with $\langle k\rangle =6.0$, $r=0.000$ and $\mathcal{F}=0.200$ for (a) $n={10}^3$ and (b) $n={10}^4$. ${\xi }^{\delta }(r|{\xi }^{\delta }\left(\mathcal{F}\right|r))$ here means that $\mathcal{F}$ is first adjusted to 0.200 by keeping $r$ constant and then change $r$ to 0.000 with fixed $\mathcal{F}\equiv 0.200$.