Analysis of cluster explosive synchronization in complex networks

Correlations between intrinsic dynamics and local topology have become a new trend in the study of synchronization in complex networks. In this paper, we investigate the influence of topology on the dynamics of networks made up of second-order Kuramoto oscillators. In particular, based on mean-field calculations, we provide a detailed investigation of cluster explosive synchronization (CES) [Phys. Rev. Lett. 110, 218701 (2013)] in scale-free networks as a function of several topological properties. Moreover, we investigate the robustness of discontinuous transitions by including an additional quenched disorder, and we show that the phase coherence decreases with increasing strength of the quenched disorder. These results complement the previous findings regarding CES and also fundamentally deepen the understanding of the interplay between topology and dynamics under the constraint of correlating natural frequencies and local structure.

Figure 1

Parameter space of the pendulum [Eq. (19)]: (a) for increasing coupling strength and (b) decreasing coupling strength. The red (dark gray) area indicates the existence of a stable fixed point, whereas the gray area indicates the parameter combinations that give rise to a stable limit cycle. The dots in the parameter space represent oscillators with degree $k=8$ and degree $k=20$, which start with incoherence in (a) [coherence in (b)] and approach synchronous states (incoherence) for increasing (decreasing) coupling strength $\lambda$ with $\alpha =0.1,D=0.1,\langle k\rangle =10,C=-3,N=3000$, and $P\left(k\right)\backsim k^{-\gamma }$, where $\gamma =3$.

Figure 2

Analytical (in blue) and numerical (in red) analysis of the order parameter $r$ (a) and $C\left(\lambda r\right)$ with increasing coupling strength (b) for synchronization diagrams. We set the value $C\left(\lambda r\right)$ to be 0 if $\lambda <{\lambda }_c^I$. The analytical plots are calculated from Eqs. (12), (15), and (18) with the synchronized boundary Eq. (22) for increasing coupling and for decreasing coupling (24). Here the simulations are conducted with $\alpha =0.1,D=0.1$, and Barabási-Albert scale-free networks characterized by $N=3000,\langle k\rangle =10$, and $k_{min}=5$.

Figure 3

(a) Average frequency ${\langle \omega \rangle }_k$ and (b) order parameter ${\langle r\rangle }_k$ of each cluster from simulations with $\langle k\rangle =10$. Solid lines denote synchronized clusters at the onset of synchronization. Dashed lines denote clusters composed of large degree nodes. The simulation parameters are the same as in Fig. 2.

Figure 4

Synchronized degrees from analytical and simulation results with increasing coupling strength. The yellow (light gray) shading shows the range of synchronized degrees from the simulations, and the red (dark gray) shading shows the range from the analytical results.

Figure 5

Results with increasing coupling strength $\lambda$. Part (a) shows the order parameter $r$ vs $\lambda$. The red (blue) curve denotes the simulation (analytical) results. Part (b) shows $C\left(\lambda r\right)$ vs $\lambda$. The critical coupling is $0.6$. As in Fig. 2, we take the value from solid lines. The analytical results are obtained from Eqs. (12), (15), (18), and (22). Here the simulations are conducted with $\alpha =0.1,D=0.1$, and Barabási-Albert scale-free networks characterized by $N=3000,\langle k\rangle =12$, and $k_{min}=6$.

Figure 6

Results are obtained with the same parameter values as in Fig. 5. Part (a) shows the evolution of the average frequency of each cluster ${〈\omega 〉}_k$ as a function of $\lambda$, and (b) indicates the evolution of the order parameter of each cluster ${〈r〉}_k$ of $\lambda$. Solid lines indicate the clusters synchronized at the critical threshold. The simulation parameters are the same as in Fig. 5.

Figure 7

Mean values of critical coupling strength $\langle {\lambda }_c^I\rangle$ for increasing coupling with different minimal degrees $k_{min}$. The gray shading indicates the standard deviation. Simulations at each minimal degree are conducted as in Fig. 2. All networks have $N=3000$.

Figure 8

Synchronization diagrams for networks with degree distribution $P\left(k\right)\sim k^{-\gamma }$ for different exponent $\gamma$. The analytical plots are calculated from the summation of Eqs. (11) and (14). All networks have $N=3000$ and $\langle k\rangle =10$.

Figure 9

Synchronization diagrams with respect to different probabilities $q$ with $q=0$ (a), $q=1$ (b), $q=2$ (c), and $q=6$ (d). Analytical results are obtained from the summation of $r_{\mathrm{lock}}^{\mathrm{q}}$ and $r_{\mathrm{drift}}^{\mathrm{q}}$. Here, we use the same network topology as in Fig. 2 with $N=3000,\alpha =0.1,D=0.1$, and set $g\left(\varepsilon \right)=1/2q$ with $\varepsilon \in [-q,q]$.