In the study of dynamical systems on networks or graphs, a key theme is how the network topology influences stability for steady states or synchronized states. Ideally, one would like to derive conditions for stability or instability that, instead of microscopic details of the individual nodes or vertices, rather make the influence of the network coupling topology visible. The master stability function is an important such tool to achieve this goal. Here, we generalize the master stability approach to hypergraphs. A hypergraph coupling structure is important as it allows us to take into account arbitrary higher-order interactions between nodes. As, for instance, in the theory of coupled map lattices, we study Laplace-type interaction structures in detail. Since the spectral theory of Laplacians on hypergraphs is richer than on graphs, we see the possibility of different dynamical phenomena. More generally, our arguments provide a blueprint for how to generalize dynamical structures and results from graphs to hypergraphs.

中文翻译：

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超图上的耦合动力学：稳态和同步的主要稳定性。

在研究网络或图形上的动力学系统时，一个关键主题是网络拓扑如何影响稳态或同步状态的稳定性。理想情况下，人们希望得出稳定性或不稳定性的条件，而不是单个节点或顶点的微观细节，而是使网络耦合拓扑的影响可见。主机稳定性功能是实现此目标的重要工具。在这里，我们概括了超图的主稳定性方法。超图耦合结构很重要，因为它允许我们考虑节点之间的任意高阶交互。例如，在耦合地图晶格理论中，我们详细研究了拉普拉斯型相互作用结构。由于拉普拉斯算子在超图上的频谱理论比图上的丰富，我们看到了不同动力学现象的可能性。更笼统地说，我们的论点为如何概括从图到超图的动力学结构和结果提供了一个蓝图。