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

## TL;DR

This paper introduces a transformation that simplifies the analysis of oscillator synchronization on complex networks by reducing it to dynamics on complete graphs with virtual frequencies, capturing both topology and dynamics.

## Contribution

It presents a novel transformation using virtual frequencies that maps complex network oscillator dynamics to simpler complete graph dynamics, enabling better understanding of synchronization phenomena.

## Key findings

- Explains explosive synchronization phenomena.
- Provides a unified framework for different network topologies.
- Simplifies analysis of oscillator synchronization.

## Abstract

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 transformation in parameter space and the introduction of virtual frequencies we show that Kuramoto oscillators on annealed networks, with or without frequency-degree correlation, and Kuramoto oscillators on complete graphs with frequency-weighted coupling can be transformed to Kuramoto oscillators on complete graphs with a re-arranged, virtual frequency distribution, and uniform coupling. The virtual frequency distribution encodes both the natural frequency distribution (dynamics) and the degree distribution (topology). We apply this transformation to give direct explanations to a variety of phenomena that have been observed in complex networks, such as explosive synchronization and vanishing synchronization onset.