| Systems composed of dynamical networks --- such as the human body with its biological networks or the global economic network consisting of regional clusters --- often exhibit complicated collective dynamics. Three fundamental processes that are typically present are failure, damage spread, and recovery. Here we develop a model for such systems and find phase diagrams for single and interacting networks. By investigating networks with a small number of nodes, where finite-size effects are pronounced, we describe the spontaneous recovery phenomenon present in these systems. In the case of interacting networks the phase diagram is very rich and becomes increasingly more complex as the number of interacting networks increases. In the simplest example of two interacting networks we find two critical points, four triple points, ten allowed transitions, and two forbidden transitions, as well as complex hysteresis loops. Remarkably, we find that triple points play the dominant role in constructing the optimal repairing strategy in damaged interacting systems. To test our model, we analyze an example of real interacting financial networks and find evidence of rapid dynamical transitions between well-defined states, in agreement with the predictions of our model. |
