Reduction of oscillator dynamics on complex networks to dynamics on complete graphs through virtual frequencies

We consider the synchronization of oscillators in complex networks where there is an interplay between the oscillator dynamics and the network topology. Through a remarkable transformation in parameter space and the introduction of virtual frequencies we show that Kuramoto oscillators on annealed networks, with or without frequency-degree correlation, and Kuramoto oscillators on complete graphs with frequency-weighted coupling can be transformed to Kuramoto oscillators on complete graphs with a rearranged, virtual frequency distribution and uniform coupling. The virtual frequency distribution encodes both the natural frequency distribution (dynamics) and the degree distribution (topology). We apply this transformation to give direct explanations to a variety of phenomena that have been observed in complex networks, such as explosive synchronization and vanishing synchronization onset.

Figure 1

Transition processes ($r$ vs $\lambda$) and rearranged distributions $G\left(\nu \right)$ of oscillators with the frequency-degree correlation $\omega =Ak$, $A={\langle k\rangle }^{-1}$, in annealed complex networks: (a)–(c) scale-free networks $g\left(k\right)\sim k^{-\gamma }$ with (a) $\gamma =2.6$, (b) $\gamma =3$, and (c) $\gamma =3.4$; (d) random network with exponential degree distribution $g\left(k\right)\sim e^{-k}$. The minimum degree is assumed $k_0=50$. Theoretical predictions (solid and dashed curves) are compared to numerical results obtained for $N=10000$ oscillators in the forward process (blue $▵$ with increasing $\lambda$) and backward process (red $▿$ with decreasing $\lambda$).

Figure 2

Transition processes ($r$ vs $\lambda$) and rearranged distributions $G\left(\nu \right)$ of oscillators without frequency-degree correlation in annealed complex networks: scale-free networks $g\left(k\right)\sim k^{-\gamma }$ with (a), (c) $\gamma =2.6$ and (b), (d) $\gamma =3.4$. The minimum degree is $k_0=50$. The maximum degree is $k_{\mathrm{m}}=\infty$ (a), (b) and $k_{\mathrm{m}}=500$ (c), (d). In all cases the distribution of natural frequencies is Gaussian, $g_{\omega }\left(\omega \right)=(1/\sqrt{2\pi {\sigma }^2})exp(-{\omega }^2/2{\sigma }^2)$ with $\sigma =\langle k\rangle$. In (c), (d) the theoretically obtained curves are compared to numerical results for $N=10000$ oscillators, shown as in Fig. 1.

Figure 3

Transition processes ($r$ vs $\lambda$) and rearranged distributions $G\left(\nu \right)$ of frequency-weighted-coupling oscillators: (a) out-coupling model and (b) in-coupling model. The natural frequencies' density is Gaussian, $g_{\omega }\left(\omega \right)=(1/\sqrt{2\pi })exp(-{\omega }^2/2)$. The theoretically obtained curves are compared to numerical results for $N=10000$ oscillators, shown as in Fig. 1.