Synchronization of networks of coupled oscillators has long been an important topic in relating a lot of issues. Network structure leads to nontrivial phase transitions, such as explosive synchronization and cluster synchronization. Besides the usual diversity in natural frequencies of oscillators, heterogeneity in the coupling strength (i.e., weighted networks) is also an important characteristic in many real systems and induces the emergence of various coherent states in the route to synchronization.
In this talk we consider the influence of natural-frequency heterogeneities on the coupling and develop the order parameter analysis and the self-consistency approach to get the low-dimensional behaviors of coupled phase oscillators. Rigorous stability for the phase-locked state is analyzed from different levels. We give general conditions for the stability of the phase-locked state based on the properties of eigen spectrum and the associated eigenvectors. We reveal universal routes to synchronization from the incoherent state to the oscillatory state and then to the phase-locked state. These findings are significant in revealing the universality and the diversity of synchronization transitions of networks of oscillators with heterogeneous couplings.