# On the onset of synchronization of Kuramoto oscillators in scale-free networks

## TL;DR

This paper investigates the onset of synchronization in Kuramoto oscillators on scale-free networks, revealing that the critical coupling strength vanishes for certain degree exponents as network size grows, contrasting with finite values for others.

## Contribution

It compares mean-field approaches with numerical simulations to clarify how the critical coupling depends on the degree distribution exponent in large scale-free networks.

## Key findings

- Critical coupling vanishes for 2<γ≤3 as network size increases.
- Critical coupling remains finite for γ>3.
- Differences between synchronization and epidemic thresholds are highlighted.

## Abstract

Despite the great attention devoted to the study of phase oscillators on complex networks in the last two decades, it remains unclear whether scale-free networks exhibit a nonzero critical coupling strength for the onset of synchronization in the thermodynamic limit. Here, we systematically compare predictions from the heterogeneous degree mean-field (HMF) and the quenched mean-field (QMF) approaches to extensive numerical simulations on large networks. We provide compelling evidence that the critical coupling vanishes as the number of oscillators increases for scale-free networks characterized by a power-law degree distribution with an exponent $2 < \gamma \leq 3$, in line with what has been observed for other dynamical processes in such networks. For $\gamma > 3$, we show that the critical coupling remains finite, in agreement with HMF calculations and highlight phenomenological differences between critical properties of phase oscillators and epidemic models on scale-free networks. Finally, we also discuss at length a key choice when studying synchronization phenomena in complex networks, namely, how to normalize the coupling between oscillators.