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Systems that comprise many interacting dynamical networks, such as the humanbody with its biological networks or the global economic network consisting ofregional clusters, often exhibit complicated collective dynamics. To understandthe collective behavior of such systems, we investigate a model of interactingnetworks exhibiting the fundamental processes of failure, damage spread, andrecovery. We find a very rich phase diagram that becomes exponentially morecomplex as the number of networks is increased. In the simplest example of$n=2$ interacting networks we find two critical points, 4 triple points, 10allowed transitions, and two "forbidden" transitions, as well as complexhysteresis loops. Remarkably, we find that triple points play the dominant rolein constructing the optimal repairing strategy in damaged interacting systems.To support our model, we analyze an example of real interacting financialnetworks and find evidence of rapid dynamical transitions between well-definedstates, in agreement with the predictions of our model.

Multiple Tipping Points and Optimal Repairing in Interacting Networks