  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.
