Onset of synchronization of Kuramoto oscillators in scale-free networks

Despite the great attention devoted to the study of phase oscillators on complex networks in the last two decades, it remains unclear whether scale-free networks exhibit a nonzero critical coupling strength for the onset of synchronization in the thermodynamic limit. Here, we systematically compare predictions from the heterogeneous degree mean-field (HMF) and the quenched mean-field (QMF) approaches to extensive numerical simulations on large networks. We provide compelling evidence that the critical coupling vanishes as the number of oscillators increases for scale-free networks characterized by a power-law degree distribution with an exponent $2<\gamma \le 3$, in line with what has been observed for other dynamical processes in such networks. For $\gamma >3$, we show that the critical coupling remains finite, in agreement with HMF calculations and highlight phenomenological differences between critical properties of phase oscillators and epidemic models on scale-free networks. Finally, we also discuss at length a key choice when studying synchronization phenomena in complex networks, namely, how to normalize the coupling between oscillators.

Figure 1

Comparison between the estimations of $K_c$ by susceptibilities $\chi$ [Eq. (14)] and ${\chi }_r$ [Eq. (15)]. Networks generated according to the UCM with degree distribution $P\left(k\right)\sim k^{-\gamma }$, with $\gamma =2.25$ and $k_{min}=5$. Natural frequencies are assigned according to Eq. (13). Each point is an average over 100 network realizations. Error bars are smaller than symbols.

Figure 2

Critical coupling $K_c$ against network size $N$ for UCM networks with power-law exponent (a) $\gamma =2.25$, (b) $\gamma =2.7$, (c) $\gamma =3.5$. All networks have $k_{min}=5$. Insets in (a) and (b) depict the difference between numerical estimation of $K_c$ and mean-field theories. Each point is an average over 100 network realizations. Error bars are smaller than symbols.

Figure 3

Numerical calculation of critical coupling $K_c$ as a function of number of oscillators $N$ of SF networks with $\gamma =2.25$ under different normalizations $\mathcal{N}$. Dashed line marks the result $K_c=2/\left[\pi g\left(0\right)\right]$. Natural frequencies distributed according to $g\left(\omega \right)=1/\pi ({\omega }^2+1)$. Each point is an average over 100 network realizations. Error bars are smaller than symbols.

Figure 4

Synchronization curves considering different normalizations and network topologies. Natural frequencies distributed according to $g\left(\omega \right)=1/\pi ({\omega }^2+1)$. SF networks considered in this figure have $\gamma =2.25$ and $k_{min}=5$. ER were generated with the same average degree as the SF networks with $N=1\times {10}^5$. Each point is an average over 100 network realizations. Error bars are smaller than symbols.