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We study the impact of network heterogeneity on relaxation dynamics of theKuramoto model on uncorrelated complex networks with scale-free degreedistributions. Using the Ott-Antonsen method and the annealed-network approach,we find that the critical behavior of the relaxation rate near thesynchronization phase transition does not depend on network heterogeneity andcritical slowing down takes place at the critical point when the second momentof the degree distribution is finite. In the case of a complete graph we obtainan explicit result for the relaxation rate when the distribution of naturalfrequencies is Lorentzian. We also find a response of the Kuramoto model to anexternal field and show that the susceptibility of the model is inverselyproportional to the relaxation rate. We reveal that network heterogeneitystrongly impacts a field dependence of the relaxation rate and thesusceptibility when the network has a divergent fourth moment of degreedistribution. We introduce a pair correlation function of phase oscillators andshow that it has a sharp peak at the critical point, signaling emergence oflong-range correlations. Our numerical simulations of the Kuramoto modelsupport our analytical results.

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