Recovery of contour nodes in interdependent directed networks

Extensive research has focused on studying the robustness of interdependent non-directed networks and the design of mitigation strategies aimed at reducing disruptions caused by cascading failures. However, real systems such as power and communication networks are directed, which underscores the necessity of broadening the analysis by including directed networks. In this work, we develop an analytical framework to study a recovery strategy in two interdependent directed networks in which a fraction q of nodes in each network have single dependencies with nodes in the other network. Following the random failure of nodes that leaves a fraction p intact, we repair a fraction of nodes that are neighbors of the giant strongly connected component of each network with probability or recovery success rate γ. Our analysis reveals an abrupt transition between total system collapse and complete recovery as p is increased. As a consequence, we identify three distinct phases in the (p,γ) parameter space: collapse despite intervention, recovery enabled by the strategy, and resilience without intervention. Moreover, we demonstrate our strategy on a system built from empirical data and find that it can save resources compared to a random recovery strategy. Our findings underscore the potential of targeted recovery strategies to enhance the robustness of real interdependent directed networks against cascading failures.