Explosive Synchronization Transitions in Scale-Free Networks

Explosive collective phenomena have attracted much attention since the discovery of an explosive percolation transition. In this Letter, we demonstrate how an explosive transition shows up in the synchronization of scale-free networks by incorporating a microscopic correlation between the structural and the dynamical properties of the system. The characteristics of the explosive transition are analytically studied in a star graph reproducing the results obtained in synthetic networks. Our findings represent the first abrupt synchronization transition in complex networks and provide a deeper understanding of the microscopic roots of explosive critical phenomena.

Figure 1

Synchronization diagrams $r(\lambda )$ for different networks constructed using the model in 25. The $\alpha$ values in each panel are (a) $\alpha =1$ (ER), (b) $\alpha =0.6$, (c) $\alpha =0.2$ and (d) $\alpha =0$ (BA). The panels show both forward and backward continuations in $\lambda$ using $\delta \lambda =0.02$. The size of the networks is $N={10}^3$ and $⟨k⟩=6$.

Figure 2

The panels show the evolution of the effective frequencies of the nodes along the (forward) continuation in the model networks of Fig. 1. The colored dots account for single-node values (colors stand for their respective degree) while solid lines show the average value of the effective frequencies of nodes having the same degree.

Figure 3

Panel (a) shows the synchronization diagrams $r(\lambda )$ for several SF networks constructed via the configurational model. All the networks have a degree distribution $P(k)\sim k^{-\gamma }$ with $\gamma =2.4$, 2.7, 3.0, and 3.3 while $N={10}^3$. The steps of the continuation are set to $\delta \lambda =0.02$. In panel (b) we show the synchronization diagrams of the same SF networks without the local correlation between degrees and natural frequencies, i.e., ${\omega }_i\ne k_i$, while $g(\omega )\sim {\omega }^{-\gamma }$.

Figure 4

Synchronization diagrams for the star graph [see the inset in plot (a)]. In (a) we show the (forward and backward) continuation diagrams for $K=10$ while (b) shows the forward continuation diagrams for star graphs of different sizes as indicated.