Critical behavior of the relaxation rate, the susceptibility, and a pair correlation function in the Kuramoto model on scale-free networks

TL;DR

This paper investigates how network heterogeneity influences the critical dynamics of the Kuramoto model on scale-free networks, revealing that certain critical behaviors are universal while others depend on degree distribution moments.

Contribution

It provides analytical and numerical analysis of the relaxation rate, susceptibility, and pair correlation function in the Kuramoto model on complex networks, highlighting the effects of degree distribution moments.

Findings

Critical slowing down occurs at the phase transition when the second moment is finite.

Susceptibility is inversely proportional to the relaxation rate.

Network heterogeneity affects the field dependence of relaxation and susceptibility.

Abstract

We study the impact of network heterogeneity on relaxation dynamics of the Kuramoto model on uncorrelated complex networks with scale-free degree distributions. Using the Ott-Antonsen method and the annealed-network approach, we find that the critical behavior of the relaxation rate near the synchronization phase transition does not depend on network heterogeneity and critical slowing down takes place at the critical point when the second moment of the degree distribution is finite. In the case of a complete graph we obtain an explicit result for the relaxation rate when the distribution of natural frequencies is Lorentzian. We also find a response of the Kuramoto model to an external field and show that the susceptibility of the model is inversely proportional to the relaxation rate. We reveal that network heterogeneity strongly impacts a field dependence of the relaxation rate and the susceptibility when the network has a divergent fourth moment of degree distribution. We introduce a pair correlation function of phase oscillators and show that it has a sharp peak at the critical point, signaling emergence of long-range correlations. Our numerical simulations of the Kuramoto model support our analytical results.