In most realistic networked systems where synchronization is relevant, strong synchronization may also be related to pathological activities such as epileptic seizures and Parkinson disease in neural networks, to extinction in ecology, and social catastrophes in epidemic spreading. Recent experiments have revealed that various levels of synchronization are needed to avoid pathological activities and enable a system to have flexibility and robustness. Despite the growing interest in network dynamics, the present state of knowledge cannot explain the various levels of synchronization observed in real-life complex systems.A fundamental goal of this project is the development of a theory for emergence spontaneous of synchronization in heterogeneous networks of interacting dynamical systems. A core intention is to determine how the network ability to complement various levels of stable synchronization depends on the network structural parameters such as degree distribution and spectral properties of the graph. These results play a vital role towards the ultimate goal of understanding the emerging spontaneous collective motion in complex systems.