Reduction of oscillator dynamics on complex networks to dynamics on complete graphs through virtual frequencies
J. Gao and K. Efstathiou
We consider the synchronization of oscillators in complex networks where there is an interplay between the oscillator dynamics and the network topology. Through a remarkable variable transformation and the introduction of virtual frequencies we show that Kuramoto oscillators on annealed networks, with or without frequency-degree correlation, can be transformed to Kuramoto oscillators on complete graphs with a re-arranged, virtual frequency distribution, which encodes both the natural frequency distribution (dynamics) and the degree distribution (topology). We exploit this observation to give alternative, straightforward, explanations to a variety of phenomena that have been observed in complex networks, such as explosive synchronization and vanishing synchronization onset. We further extend this method to the study of the frequency-weighted coupling model which also exhibits explosive synchronization.