Generalized k-core percolation on higher-order dependent networks

In biology and technology systems, the proper functioning of agents may be mutually interdependent, where the failure of an agent can cause the dysfunctionality of other dependent agents. Usually, we adopt the dependence network to capture such dependency relations among agents; however, the interdependency relations are only pairwise but could be of higher-order. In other words, an agent’s failure can induce the failure of several other nodes in a high-order interaction (e.g., the same group or clique) simultaneously. In this paper, we propose a generalized k-core percolation model to investigate the robustness of the higher-order dependent networks. In particular, we consider higher-order multi-layered dependency networks where both the interlayer and intralayer dependency relations are of a high order. We study the model using percolation theory and numerical simulations and find that the k-core percolation threshold and phase transition type depend on the average degree. Increasing the average degree enhances the system robustness. The system exhibits a discontinuous phase transition with a small k-core percolation threshold for networks with small average degree. Meanwhile, for networks with large average degrees, the system can either shows a continuous phase transition with a small k-core percolation threshold or a discontinuous phase transition with a large percolation threshold. In addition, we find that the intralayer dependency enhances the robustness of the system. Finally, we reveal that the degree heterogeneity makes the network more fragile. The above stated phenomena are well predicted by our developed percolation theory.