Designing networks with specific synchronization transitions independent of the system’s dynamics

Synchronization transition, a phenomenon widely observed in various networks, can be classified into two types: explosive and continuous transitions. While the continuous transition is relatively well-understood, the underlying mechanisms of the explosive transition present a challenge to researchers. Most studies, however, pointed out the interaction between node dynamics and network topology as the fundamental feature of explosive synchronization in networks. In this study, we demonstrate the impact of Laplacian eigenvalue on the type of synchronization transition observed in complex systems. Through simulations, our results show that the sparsity of the Laplacian eigenvalues is the key to designing the synchronization transitions. The findings reveal that the sparse distributions lead to a jump in the evolution of the order parameter, explosive synchronization. Additionally, the nodes exhibit continuous synchronization in cases where the eigenvalues are broadly dispersed. This study examines three distinct chaotic systems – Chen, Rössler, and Lorenz – with five different Laplacian eigenvalue distributions. Surprisingly, the outcomes demonstrate that the synchronization transition is not mainly dependent on the internal dynamics of the nodes, highlighting a pivotal characteristic that can be used to design the precise control of complex networks.