Effects of the frequency-degree correlation on local synchronization in complex networks

We investigate the effects of the frequency-degree correlation on local synchronization in complex networks with Kuramoto oscillators. We find that a discontinuous synchronization transition occurs in the local patterns for heterogenous networks while for homogenous networks, the local synchronization transition remains continuous. Then, we extend our study to a general frequency-degree correlation case and conclude that the positive correlation does not change the local synchronization patterns while in the case of a negative correlation, the local synchronization transition degenerates to second order. Moreover, the correlation parameter α is verified to have a strong influence on the synchronization level. In particular, smaller |α| results in higher synchronization ability and faster speed to a synchronized state. Our study provides a deeper understanding of the effects of the frequency-degree correlation on network synchronization.