Interdependent Networks with Higher-Order Structures

With the exploration of real-world systems, we find that a single network can no longer meet our real needs[1,2], Networks in the real world usually interact with other networks to varying degrees[3-5] and jointly function to serve the real world. These networks often take the form of multi-layer interdependent networks[6].

In such a network, the cascading failure process is more worthy of study. It usually manifests as the removal of a few nodes triggering iterations through intra-layer failures or inter-layer failures, ultimately bringing the network to a stable state. However, merely the interdependence relationship is far from sufficient. Networks in the real world usually exhibit complex relationships and dependencies, not just pairwise connections. To address this issue, we introduce hypergraphs to reflect high-order interactions and dependencies[7], providing a more comprehensive representation. For example, in social networks[8,9], we can consider the interaction patterns among multiple users, explore the high-order associations among multiple users. In biological networks[10], we can effectively identify the strong and weak relationships between nodes. 

Further study the robustness of the network. Percolation theory is a commonly used method to study network robustness[11], played a key role in the study of the collective behavior of systems. There are usually two types of percolation, one of which is bond percolation[12], one is point percolation[13]. So usually, the seepage situation is studied to observe the system behavior of the network. Zhou et al[14] analyzing the characteristics of interdependent networks, a related k-core percolation model is proposed. Given any degree distribution of edge-coupled interdependent networks, the relative sizes of the k-core and the corona cluster as well as the percolation threshold can be obtained through the derivation of self-consistent equations, and then the phase transition of k-core percolation can be analyzed. The proposed k-core percolation model and protection strategy not only help to understand the hierarchical structure of the network, but also provide some guidance for enhancing the resilience of the network against attacks. Secondly, the higher-order interactions in interdependent networks can usually be represented by hyperedges or simplicial complexes. For simplicial complexes, Peng et al[6,15] constructed a two-layer partial dependence network theoretical model with a simplicial complex, in which failures between nodes occur through the synergistic effects of pairwise interactions and high-order interactions. In this model, removing a node will cause all other nodes in the same simplex to be removed, and due to the dependence between the two networks, node failures will spread through the dependence links between the two networks. This process will occur recursively, ultimately leading to a cascading process. Four types of artificial MIHN models were further constructed, where high-order interactions are still described by simplicial complexes, and inter-layer dependencies are established through one-to-one matching dependence links. The robustness of MIHN was studied by investigating the largest connected component and the percolation threshold. We found that the density of the simplicial complex and the number of network layers affect its percolation behavior.

For hyperedges, Liu et al[16] proposed a percolation model that takes into account the dependence of hyperedges on their internal nodes, and the research reveals the different impacts of the hyperdegree distribution on the system robustness in single-layer and double-layer hypergraphs. In the real world, not only do nodes have interdependent relationships, but edges do as well. Qian et al[17] proposed an interdependent hypergraph model considering the inter-layer node dependency. The cascading failures in hypergraphs with different inter-layer dependencies were studied. The maximum attack intensity that the network can withstand was determined through theoretical analysis, as well as how its robustness changes under different attack intensities. The multi-layer hypergraph can represent the relationships between nodes more clearly. However, there is less research on identifying important nodes within this framework. Wang et al[18] proposed a method named HCT to fill this gap. The global centrality of nodes in the entire network can be calculated. Compared with other methods, the important nodes identified by HCT exhibit stronger propagation capabilities, and removing these nodes will seriously damage the connectivity and robustness of the network. To further reveal the profound impact of group support on the resilience of the system against cascading failures, Chen et al [19] designed a framework consisting of a two-layer interdependent hypergraph system, where the nodes in one layer can obtain support through the hyperedges in the other layer. The article derived the critical threshold of the initial node survival probability that marks the second-order phase transition point.

There is also an understanding of network resilience. Network resilience measures the degree of performance degradation of a network after being perturbed and its recovery ability, and is closely related to the ability to resist cascading failures. Li et al[20] proposed three resilience enhancement strategies based on the node capacity redundancy at different structural scales, and developed a network resilience evaluation method that considers both the structure and node load. The performance of the enhancement strategies is closely related to the node capacity redundancy. Specifically, when enhancing nodes with larger capacity redundancy, the enhancement efficiency is higher. In addition, the heterogeneity of node load has a profound impact on the enhancement efficiency. Lv et al [21] proposed a resilience assessment model to predict the performance of interdependent networks against cascading failures. This model can accurately monitor the activities of each node during the cascading process.

In the real world, systems often exist in the form of groups, so that the network not only needs to consider the intra-layer dependencies but also the inter-layer dependencies. Furthermore, it is necessary to consider whether this dependence is node dependence or hyperedge dependence. Therefore, we will combine these two aspects and use hypergraphs to study the robustness of two-layer partially interdependent networks.