Effect of mixing parts of modular networks on explosive synchronization

Synchronization of coupled oscillators on networks has been investigated in a wide range of topologies. One of the latest findings is the explosive synchronization in the scale-free network with a positive frequency-degree correlation (Gómez G. J. et al., Phys. Rev. Lett., 106 (2011) 128701). In this letter, we generalize this study and explore the effect of mixing parts on the Kuramoto model with positive correlation between frequencies and degrees. It is shown that small or weak mixing parts on module networks may accelerate the synchronization of the whole network while large and strong mixing parts may hinder synchronization. In particular, by altering the mixing part of a joint-star network, a two-step shaped transition of synchronization is observed with theoretical analysis on the critical points. Our findings indicate that mesoscopic structures should be of importance to affect network explosive synchronization.