  Efficient complex systems have a modular structure, but modularity does not
guarantee robustness, because efficiency also requires an ingenious interplay
of the interacting modular components. The human brain is the elemental
paradigm of an efficient robust modular system interconnected as a network of
networks (NoN). Understanding the emergence of robustness in such modular
architectures from the interconnections of its parts is a long-standing
challenge that has concerned many scientists. Current models of dependencies in
NoN inspired by the power grid express interactions among modules with fragile
couplings that amplify even small shocks, thus preventing functionality.
Therefore, we introduce a model of NoN to shape the pattern of brain
activations to form a modular environment that is robust. The model predicts
the map of neural collective influencers (NCIs) in the brain, through the
optimization of the influence of the minimal set of essential nodes responsible
for broadcasting information to the whole-brain NoN. Our results suggest new
intervention protocols to control brain activity by targeting influential
neural nodes predicted by network theory.
