Mulas Raffaella, Kuehn Christian, Jost Jürgen
Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103 Leipzig, Germany and Santa Fe Institute for the Sciences of Complexity, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA.
Phys Rev E. 2020 Jun;101(6-1):062313. doi: 10.1103/PhysRevE.101.062313.
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.
在对网络或图上的动力系统的研究中，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，人们希望推导出稳定性或不稳定性的条件，这些条件不是基于单个节点或顶点的微观细节，而是能使网络耦合拓扑的影响显现出来。主稳定性函数就是实现这一目标的重要工具。在此，我们将主稳定性方法推广到超图。超图耦合结构很重要，因为它使我们能够考虑节点之间任意的高阶相互作用。例如，在耦合映射格点理论中，我们详细研究拉普拉斯型相互作用结构。由于超图上拉普拉斯算子的谱理论比图上的更丰富，我们看到了出现不同动力学现象的可能性。更一般地说，我们的论证为如何将图的动力学结构和结果推广到超图提供了一个蓝图。
