On the onset of synchronization of Kuramoto oscillators in scale-free networks

Abstract

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.

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• Emergent synchrony in oscillator networks with adaptive arbitrary-order interactions
• Global hierarchy vs. local structure: spurious self-feedback in scale-free networks
• Adversarial control of synchronization in complex oscillator networks
• Mean-field theory of vector spin models on networks with arbitrary degree distributions