Robustness of higher-order interdependent networks with reinforced nodes

In reality, pairwise interactions are no longer sufficient to describe the higher-order interactions between nodes, such as brain networks, social networks, etc., which often contain groups of three or more nodes. Since the failure of one node in a high-order network can lead to the failure of all simplices in which it is located and quickly propagates to the whole system through the interdependencies between networks, multilayered high-order interdependent networks are challenged with high vulnerability risks. To increase the robustness of higher-order networks, in this paper, we proposed a theoretical model of a two-layer partial high-order interdependent network, where a proportion of reinforced nodes are introduced that can function and support their simplices and components, even losing connection with the giant component. We study the order parameter of the proposed model, including the giant component and functional components containing at least one reinforced node, via theoretical analysis and simulations. Rich phase transition phenomena can be observed by varying the density of 2-simplices and the proportion of the network's reinforced nodes. Increasing the density of 2-simplices makes a double transition appear in the network. The proportion of reinforced nodes can alter the type of second transition of the network from discontinuous to continuous or transition-free, which is verified on the double random simplicial complex, double scale-free simplicial complex, and real-world datasets, indicating that reinforced nodes can significantly enhance the robustness of the network and can prevent networks from abrupt collapse. Therefore, the proposed model provides insights for designing robust interdependent infrastructure networks.