  A model of interdependent networks of networks (NoN) has been introduced
recently in the context of brain activation to identify the neural collective
influencers in the brain NoN. Here we develop a new approach to derive an exact
expression for the random percolation transition in Erd\"{o}s-R\'enyi NoN.
Analytical calculations are in excellent agreement with numerical simulations
and highlight the robustness of the NoN against random node failures.
Interestingly, the phase diagram of the model unveils particular patterns of
interconnectivity for which the NoN is most vulnerable. Our results help to
understand the emergence of robustness in such interdependent architectures.
