  Systems that comprise many interacting 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. To understand
the collective behavior of such systems, we investigate a model of interacting
networks exhibiting the fundamental processes of failure, damage spread, and
recovery. We find a very rich phase diagram that becomes exponentially more
complex as the number of networks is increased. In the simplest example of
$n=2$ interacting networks we find two critical points, 4 triple points, 10
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 support 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.
