Explosive transitions to synchronization in weighted static scale-free networks

The emergence of explosive synchronization transitions in networks of phase oscillators has become recently one of the most interesting topics. We simulate the Kuramoto model on top of a family of weighted static scale-free networks. It is found that when the strength of the network's edge is linearly correlated with frequency gap of pair of oscillators at its ends, i.e., the microscopic correlation exponent beta is equal to 1, the model with the degree distribution exponent gamma > 3 undergoes a first-order phase transition, while the transition becomes second order at 2 < gamma <= 3. We also find that in homogeneous networks (gamma -> infinity) the explosive synchronization is replaced by a continuous phase transition when the microscopic correlation exponent beta is changed from positive to negative. This is a new discovery of explosive synchronization transitions in weighted complex networks, which provides a fresh angle and tool to understand this explosive behavior.