Reduction of oscillator dynamics on complex networks to dynamics on complete graphs through virtual frequencies

J. Gao and K. Efstathiou
Arxiv preprint

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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.