Synchronization in the random-field Kuramoto model on complex networks

We study the impact of random pinning fields on the emergence of synchrony in the Kuramoto model on complete graphs and uncorrelated random complex networks. We consider random fields with uniformly distributed directions and homogeneous and heterogeneous (Gaussian) field magnitude distribution. In our analysis, we apply the Ott-Antonsen method and the annealed-network approximation to find the critical behavior of the order parameter. In the case of homogeneous fields, we find a tricritical point above which a second-order phase transition gives place to a first-order phase transition when the network is either fully connected or scale-free with the degree exponent $\gamma >5$. Interestingly, for scale-free networks with $2<\gamma \le 5$, the phase transition is of second-order at any field magnitude, except for degree distributions with $\gamma =3$ when the transition is of infinite order at $K_c=0$ independent of the random fields. Contrary to the Ising model, even strong Gaussian random fields do not suppress the second-order phase transition in both complete graphs and scale-free networks, although the fields increase the critical coupling for $\gamma >3$. Our simulations support these analytical results.

Figure 1

$K\text{−}h$ phase diagram of the Kuramoto model on a complete graph in the presence of homogeneous random fields of magnitude $h$. The solid line corresponds to the numerical solution of Eq. (23) and the points to our simulations. There are three regions: (I) asynchronous state, (II) region of hysteresis, and (III) partially synchronous state. $T=(K_c,h_c)=(2\sqrt{3},\sqrt{2})$ is a tricritical point below which there is a second-order phase transition, whereas above it there is a first-order phase transition. The networks were generated using the static model $(N={10}^4$ and $\langle q\rangle =10$) [24, 25]. The differential equations for each oscillator were solved using a fourth-order Runge-Kutta method with time step $\mathrm{\Delta }t=0.01$.

Figure 2

Critical behavior of the order parameter in scale-free networks in the presence of homogeneous random fields $\left(h=2\right)$. The symbols correspond to simulations of the model (averaged over 30 network realizations), whereas the lines in panels (a) and (c) are the linear fits. The line in panel (b) is $f\left(K\right)\propto \frac{1}{K}exp(-1/K)$. The networks were generated using the static model ($N=2\times {10}^4$ and $\langle q\rangle =10$) [24, 25]. The differential equations for each oscillator were solved using a fourth-order Runge-Kutta method with time step $\mathrm{\Delta }t=0.01$.