The stability of group synchronization in multi-layer networks of coupled oscillators

The synchronization of complex networks is a widespread phenomenon in both natural and engineering systems. However, the underlying mechanism of group synchronization in multi-layer networks remains unclear. In this paper, we establish a multi-layer coupled oscillator model to investigate group synchronization. By employing group representation theory, we perform a dimensionality reduction analysis and establish stability criteria. Our analysis reveals that the stability of group synchronization is governed by the relative strength of intra-layer and inter-layer coupling. We demonstrate how stability depends on coupling strength under cooperative, competitive, inter-mixing, intra-mixing inter-layer dynamics, and discover the different influence of inter-layer connectivity in stability. Then we use eigenvalue analysis to validate the proposed approach by comparing system dynamics before and after network reduction, which reserves the essential dynamical features. The model is further extended to large-scale Erdős–Rényi networks to examine the impact of systemic perturbations. We find that small structural disturbation can enhance the stability because of the decrease of the shortest path in networks. Our study enhances the understanding of synchronization in multi-layer networks and provides a theoretical foundation for controlling neural networks and other complex systems.