Relaxation time of the global order parameter on multiplex networks: the role of interlayer coupling in Kuramoto oscillators

TL;DR

This paper investigates how interlayer coupling affects the relaxation times of synchronization in multiplex Kuramoto networks, revealing that stronger coupling accelerates synchronization and that global order converges faster than interlayer synchronization.

Contribution

It provides analytical and numerical insights into the relaxation dynamics of multiplex Kuramoto oscillators, highlighting the role of interlayer coupling and frequency effects.

Findings

Interlayer coupling strength inversely affects synchronization timescales.

Global order parameter converges faster than interlayer synchronization.

Timescale of the global order parameter is at least half of the multiplex's timescale.

Abstract

This work considers the timescales associated with the global order parameter and the interlayer synchronization of coupled Kuramoto oscillators on multiplexes. For the two-layer multiplexes with initially high degree of synchronization in each layer, the difference between the average phases in each layer is analyzed from two different perspectives: the spectral analysis and the non-linear Kuramoto model. Both viewpoints confirm that the prior timescales are inversely proportional to the interlayer coupling strength. Thus, increasing the interlayer coupling always shortens the transient regimes of both the global order parameter and the interlayer synchronization. Surprisingly, the analytical results show that the convergence of the global order parameter is faster than the interlayer synchronization, and the latter is generally faster than the global synchronization of the multiplex. The formalism also outlines the effects of frequencies on the difference between the average phases of each layer, and identifies the conditions for an oscillatory behavior. Computer simulations are in fairly good agreement with the analytical findings and reveal that the timescale of the global order parameter is at least half times smaller than timescale of the multiplex.