Percolation transitions in edge-coupled interdependent networks with directed dependency links

Abstract

We propose a model of edge-coupled interdependent networks with directed dependency links (EINDDLs) and develop the theoretical analysis framework of this model based on the self-consistent probabilities method. The phase transition behaviors and parameter thresholds of this model under random attacks are analyzed theoretically on both random regular (RR) networks and Erdös–Rényi (ER) networks, and computer simulations are performed to verify the results. In this EINDDL model, a fraction β of connectivity links within network B depends on network A and a fraction (1 − β) of connectivity links within network A depends on network B. It is found that randomly removing a fraction (1 − p) of connectivity links in network A at the initial state, network A exhibits different types of phase transitions (first order, second order and hybrid). Network B is rarely affected by cascading failure when β is small, and network B will gradually converge from the first-order to the second-order phase transition as β increases. We present the critical values of β for the phase change process of networks A and B, and give the critical values of p and β for network B at the critical point of collapse. Furthermore, a cascading prevention strategy is proposed. The findings are of great significance for understanding the robustness of EINDDLs.