LAUSR

In this paper, an edge-coupled interdependent networks with directed dependency links (EINDDL) model is proposed, and the theoretical analysis framework of this model based on the self-consistent probabilities method is developed. 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 simulation 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. The critical values of β for the phase changes process of network A and network B, and the critical values of p and β for network B at the critical point of collapse are given. Furthermore, a cascading prevention strategy is proposed. The findings are of great significance for understanding the robustness of EINDDL.