Robustness analysis of multi-dependency networks: k-core percolation and deliberate attacks

The k-core percolation is an advantageous method for studying network robustness. Most of the existing research is based on multiplex networks with just one-to-one node dependencies, while in reality, a node may depend on a group of nodes, and there is a lack of research on k-core percolation in multi-dependency networks. To better address these practical needs, the percolation equation of k-core model on multi-dependency networks is derived with the definition of failure tolerance β. It reveals that at the critical point, the phase transition of k-core can be described as a hybrid first-order and continuous singular phase transitions; while when k=1,2 and β=1, the phase transition behavior of k-core is second-order. The correctness of theoretical analysis is verified by performing simulations on ER−ER, SF−SF and ER−SF networks, in which the results indicate that increasing the failure tolerance β can effectively enhance the robustness of k-core structures in multi-dependency networks. Contrary to expectations, as the maximum size of the dependent cluster increasing, the robustness of k-core structures first decreases and then increases. Additionally, the results reveal that the critical points of k-core and the corona clusters are consistent. Based on this finding, an improved edge measurement method has been proposed, which can identify the critical links in corona clusters. By targeting these critical links, the network robustness can be reduced. Simulation results show that the given edge measure is not only superior to some basic methods but also beneficial for suppressing virus propagation. Nevertheless, the given framework can give help in understanding the overall hierarchy structure of networks and provide a foundation for further exploration of complex networks.