Perfect synchronization in Sakaguchi–Kuramoto model on directed complex networks

In this study, we analytically derive optimal frequency sets for achieving perfect synchronization at a targeted coupling strength in directed networks of phase-frustrated oscillators. With that aim, we construct a synchrony alignment function (SAF) that helps optimize the synchronization properties of that network and acts as a quantitative indicator of the network’s synchronization level. We conduct extensive numerical simulations of the Sakaguchi–Kuramoto (SK) model on both directed scale-free and directed Erdős–Rényi networks to validate the suggested optimal frequency configuration. Numerical simulations demonstrate that the analytically determined frequency set ensures not only stable, perfect synchronization at the desired point in the network but also outperforms other frequency set options in achieving a high degree of synchronization in its vicinity. However, we observe that the synchronization level decreases after reaching perfect synchronization at the targeted point with our derived frequency set. In order to resist this synchronization loss, we give a network reconstruction approach without changing the number of edges in the network. Following the approach, significant improvement in the level of synchronization is achieved. The stability of the system’s perfect synchronization state is assessed through a low-dimensional model of the network, while its robustness is evaluated by adding Gaussian noise to the derived frequency set.