Explosive synchronization in a general complex network

Explosive synchronization (ES) has recently attracted much attention, where its two necessary conditions are found to be a scale-free network topology and a positive correlation between the natural frequencies of the oscillators and their degrees. Here we present a framework for ES to be observed in a general complex network, where a positive correlation between coupling strengths of the oscillators and the absolute of their natural frequencies is assumed and the previous studies are included as specific cases. In the framework, the previous two necessary conditions are replaced by another one, thus fundamentally deepening the understanding of the microscopic mechanism toward synchronization. A rigorous analytical treatment by a mean field is provided to explain the mechanism of ES in this alternate framework.

Figure 1

Synchronization diagrams for different networks. (a) and (b) represent the cases of a fully connected network for the Lorentzian and Gaussian distributions, respectively, with ${\omega }_0=0$ and $\gamma =0.5$ in (a) and $\mu =0$ and ${\sigma }^2=1$ in (b). (c) and (d) represent the cases of ER and UCM networks, respectively, with the Lorentzian distribution of the natural frequencies and average degree $\langle k\rangle =6$.

Figure 2

The evolutionary dynamics along the forward continuation in the network model of Fig. 1a. (a) ${\omega }_i^{\text{eff}}$ vs $\lambda$. (b) $\Delta {\theta }_i$ vs $\lambda$.

Figure 3

Case of class II. The curves with squares and circles in (a)–(d) correspond to Figs. 1a, 1b, 1c, 1d, respectively, where $g\left(\omega \right)\sim {\omega }^{-\beta }$ with $\omega >0$ and $\beta =1.5$ in (a), (c), and (d), and $g\left(\omega \right)=\sqrt{\frac{2}{\pi }}\exp (-\frac{{\omega }^2}{2})$ with $\omega >0$ in (b). The curves with up triangles and down triangles in (c) and (d) denote the case of $\langle k\rangle =24$.

Figure 4

(a) and (b) represent the cases of ER and UCM networks in class I, respectively, where the curves with squares, circles, and triangles represent the cases of $\langle k\rangle =6$, 12, and 24, respectively, and the solid curve represents the theoretical result from Eq. (7). (c) $R\left(x\right)$ vs $x$ for fully connected networks in class II where the solid and dashed curves correspond to Figs. 3a and 3b, respectively.