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

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.

Figure 1

Relaxation rate $1/{\tau }_r$ versus coupling $K$ in the Kuramoto model on a complete graph in the region $K<K_c$. The solid curve represents a numerical solution of Eq. (29) in the case of a Gaussian distribution of the natural frequencies with the mean frequency $\langle \omega \rangle =0$ and the variance $\sigma =1$. The dashed line represents the linear approximation Eq. (40).

Figure 2

Susceptibility $\chi$ [Eq. (52)] versus the coupling $K$ near the critical point $K_c$ in the Kuramoto model on scale-free networks. Results of numerical simulations. (a) $\chi$ in a scale-free network with the degree exponent $\gamma =6$ in the fields $h=0.01$ (circles), $h=0.02$ (stars), and $h=0.03$ (diamonds). (b),(c) $1/\chi$ at $\gamma =6$ and 4, respectively. Solid lines are lines $A(K_c-K)/K_c$ and $B(K-K_c)/K_c$ with the slopes $A=2$ and $B=4$ in panel (b) and $A=2$ and $B=2$ in panel (c). The mean degree of the networks is $\langle q\rangle =20$. Each point on the plots is obtained by averaging over ${10}^4$ time steps in the iterative procedure and 100 network realizations.

Figure 3

Susceptibility $\chi$, Eq. (52), and correlation function $C$, Eq. (61), versus coupling $K$ in the Kuramoto model on a scale-free network with the degree exponent $\gamma =6$ in the field $h=0.01$. In simulations, the mean degree of the network is $\langle q\rangle =20$.

Figure 4

(a) Pair correlation function $C$ [Eq. (61)] versus the coupling $K$ near the critical point $K_c$ in a scale-free network with the degree exponent $\gamma =6$ in the fields $h=0.01$ (circles), $h=0.02$ (stars), and $h=0.03$ (diamonds). Panel (b) shows $1/C$. Solid lines are lines $A(K_c-K)/K_c$ and $B(K-K_c)/K_c$ with the slopes $A=2$ and $B=4$. In simulations, the mean degree of the network is $\langle q\rangle =20$.