Robustness of directed higher-order networks

In complex systems, from human social networks to biological networks, pairwise interactions are insufficient to express the directed interactions in higher-order networks since the internal function is not only contained in directed pairwise interactions but rather in directed higher-order interactions. Therefore, researchers adopted directed higher-order networks to encode multinode interactions explicitly and revealed that higher-order interactions induced rich critical phenomena. However, the robustness of the directed higher-order networks has yet to receive much attention. Here, we propose a theoretical percolation model to analyze the robustness of directed higher-order networks. We study the size of the giant connected components and the percolation threshold of our proposed model by the theory and Monte-Carlo simulations on artificial networks and real-world networks. We find that the percolation threshold is affected by the inherent properties of higher-order networks, including the heterogeneity of the hyperdegree distribution and the hyperedge cardinality, which represents the number of nodes in the hyperedge. Increasing the hyperdegree distribution of heterogeneity or the hyperedge cardinality distribution of heterogeneity in higher-order networks will make the network more vulnerable, weakening the higher-order network's robustness. In other words, adding higher-order directed edges enhances the robustness of the systems. Our proposed theory can reasonably predict the simulations for percolation on artificial and real-world directed higher-order networks.