# Critical dynamics of the Kuramoto model on sparse random networks

## TL;DR

This study investigates the critical dynamics of the Kuramoto model on sparse random networks, revealing mean-field universality class behavior and highlighting challenges in finite-size scaling analysis due to strong corrections.

## Contribution

It provides more accurate estimates of the critical coupling and explores the scaling behavior, emphasizing differences from all-to-all coupling models.

## Key findings

- Critical coupling strength estimated more accurately.
- Correlation exponents compatible with mean-field universality.
- Strong corrections affect order-parameter scaling and critical exponent estimates.

## Abstract

We consider the Kuramoto model on sparse random networks such as the Erd\H{o}s-R\'enyi graph or its combination with a regular two-dimensional lattice and study the dynamical scaling behavior of the model at the synchronization transition by large-scale, massively parallel numerical integration. By this method, we obtain an estimate of critical coupling strength more accurate than obtained earlier by finite-size scaling of the stationary order parameter. Our results confirm the compatibility of the correlation-size and the temporal correlation-length exponent with the mean-field universality class. However, the scaling of the order parameter exhibits corrections much stronger than those of the Kuramoto model with all-to-all coupling, making thereby an accurate estimate of the order-parameter exponent hard. We find furthermore that, as a qualitative difference to the model with all-to-all coupling, the effective critical exponents involving the order-parameter exponent, such as the effective decay exponent characterizing the critical desynchronization dynamics show a non-monotonic approach toward the asymptotic value. In the light of these results, the technique of finite-size scaling of limited size data for the Kuramoto model on sparse graphs has to be treated cautiously.