Frequency synchronization in a random oscillator network

We study the frequency synchronization of a randomly coupled oscillators. By analyzing the continuum limit, we obtain a sufficient condition for the mean-field-type synchronization. We especially find that the critical coupling constant $K$ becomes 0 in the random scale-free network, $P\left(k\right)\propto k^{-\gamma }$, if $2<\gamma ⩽3$. Numerical simulations in finite networks are consistent with this analysis.

Figure 1

$(\omega ,d\theta ∕dt)$ distribution of oscillators in a random network, for $K=0.15$ and $K=0.30$.

Figure 2

$(\omega ,\theta )$ distribution of oscillators in a random network; $K=0.15$ and $K=0.30$.

Figure 3

$(\omega ,\theta )$ distribution of oscillators with degree 3, 5, and 7 in a random network; $K=0.30$.

Figure 4

Interaction dependence of mean-field parameter $r_{\mathrm{av}}$.

Figure 5

Interaction dependence of the mean-field parameter in the random scale-free network for $N=500$, 1000, 2000, and 4000. The arrow shows $K_c$ estimated from Eq. (12). Simulations for each parameter are carried out for at least 50 realizations of the networks.

Figure 6

The relation between $N^{0.25}{r}_{\mathrm{av}}$ and $K∕K_c$, where $K_c$ is the value estimated from Eq. (12).