When describing complex interconnected systems, one often has to go beyond the standard network description to account for generalized interactions. Here, we establish a unified framework to simplify the stability analysis of cluster synchronization patterns for a wide range of generalized networks, including hypergraphs, multilayer networks, and temporal networks. The framework is based on finding a simultaneous block diagonalization of the matrices encoding the synchronization pattern and the network topology. As an application, we use simultaneous block diagonalization to unveil an intriguing type of chimera states that appear only in the presence of higher-order interactions. The unified framework established here can be extended to other dynamical processes and can facilitate the discovery of emergent phenomena in complex systems with generalized interactions.

在描述复杂的互连系统时，通常必须超越标准网络描述来解释广义交互。在这里，我们建立了一个统一的框架，以简化对各种广义网络（包括超图、多层网络和时间网络）的集群同步模式的稳定性分析。该框架基于找到编码同步模式和网络拓扑的矩阵的同时块对角化。作为一个应用，我们使用同步块对角化来揭示一种有趣的嵌合状态，这种状态仅在存在高阶相互作用时才会出现。