Robustness of interdependent directed higher-order networks against cascading failures

In the real world, directed networks are not just constructed as pairs of directed interactions, but also occur in groups of three or more nodes that form the higher-order structure of the network. From social networks to biological networks, there is growing evidence that real-world systems connect the functional relationships of multiple systems through interdependence. To understand the robustness of interdependent directed higherorder networks, we propose a new theoretical framework to model and analyze the robustness of such networks under random failures by percolation theory. We find that adding higher-order edges makes the network more vulnerable which quantifies and compares by two criteria: the size of the giant connected components and the percolation threshold. Increasing the hyperdegree distribution of heterogeneity or the hyperedge cardinality distribution of heterogeneity in interdependent directed higher-order networks will also make the network more vulnerable. Interestingly, the phase transition type changes from continuous to discontinuous with the increase of coupling strength, and partially interdependent directed higher-order networks exist hybrid phase transition. Moreover, by applying our theoretical analysis to real interdependent directed higher-order networks further validated our conclusion, it has implications for the design of flexible complex systems.