In real-world complex systems, most networks are interconnected with other networks through interlayer dependencies, forming multilayer interdependent networks. In each system, the interactions between nodes are not limited to pairwise but also exist in a higher-order interaction composed of three or more individuals, thus inducing a multilayer interdependent higher-order network (MIHN). First, we build four types of artificial MIHN models (i.e., chain-like, tree-like, star-like and ring-like), in which the higher-order interactions are described by the simplicial complexes, and the interlayer dependency is built via a one-to-one matching dependency link. Then, we propose a cascading failure model on MIHN and suggest a corresponding percolation-based theory to study the robustness of MIHN by investigating the giant connected components (GCC) and percolation threshold. We find that the density of the simplicial complexes and the number of layers of the network affect its penetration behavior. When the density of simplicial complexes exceeds a certain threshold, the network has a double transition, and the increase in network layers significantly enhances the vulnerability of MIHN. By comparing the simulation results of MIHNs with four types, we observe that under the same density of simplicial complexes, the size of the GCC is independent of the topological structures of MIHN after removing a certain number of nodes. Although the cascading failure process of MIHNs with different structures is different, the final results tend to be the same. We further analyze in detail the cascading failure process of MIHN with different structures and elucidate the factors influencing the speed of cascading failures. Among these four types of MIHNs, the chain-like MIHN has the slowest cascading failure rate and more stable robustness compared to the other three structures, followed by the tree-like MIHN and star-like MIHN. The ring-like MIHN has the fastest cascading failure rate and weakest robustness due to its ring structure. Finally, we give the time required for the MIHN with different structures to reach the stable state during the cascading failure process and find that the closer to the percolation threshold, the more time the network requires to reach the stable state.

中文翻译：

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多层相互依赖的高阶网络的鲁棒性


在现实世界的复杂系统中，大多数网络通过层间依赖关系与其他网络互连，形成多层相互依赖的网络。在每个系统中，节点之间的交互不仅限于成对交互，还存在于由三个或更多个体组成的高阶交互中，从而诱导出多层相互依赖的高阶网络 （MIHN）。首先，我们构建了四种类型的人工 MIHN 模型（即链状、树状、星形和环状），其中高阶交互由简单复合体描述，层间依赖关系通过一对一的匹配依赖关系链接构建。然后，我们提出了一个基于 MIHN 的级联失效模型，并提出了一个相应的基于渗流的理论，通过研究巨连接分量 （GCC） 和渗流阈值来研究 MIHN 的鲁棒性。我们发现简单复合体的密度和网络的层数会影响其穿透行为。当简单复合体的密度超过一定阈值时，网络具有双重跃迁，网络层的增加显著增强了 MIHN 的脆弱性。通过比较四种类型的 MIHN 的模拟结果，我们观察到，在相同密度的简单配合物下，在去除一定数量的节点后，GCC 的大小与 MIHN 的拓扑结构无关。尽管不同结构的 MIHNs 的级联破坏过程不同，但最终结果趋于相同。我们进一步详细分析了不同结构的 MIHN 的级联破坏过程，并阐明了影响级联破坏速度的因素。 在这 4 种 MIHN 类型中，链状 MIHN 的级联失效率最低，与其他三种结构相比，稳健性更稳定，其次是树状 MIHN 和星形 MIHN。环状 MIHN 由于其环形结构，具有最快的级联故障率和最弱的稳健性。最后，我们给出了不同结构的 MIHN 在级联失效过程中达到稳定状态所需的时间，发现越接近渗流阈值，网络达到稳定状态所需的时间就越长。