Evidence suggests that the brain is arranged in functionally specialized modules to form a network of networks (NoN). Understanding how functionality emerges from the integration of such modular architectures is one of the greatest scientific challenges. We introduce a model of brain NoN, which is robust against random node failures, captures the integration of functionally specialized modules in the brain, and provides falsifiable predictions about the locations of the most influential nodes, called neural collective influencers, in the brain network—predictions that are impossible in existing but fragile models of interdependent NoN. If confirmed by experiment, our results may pave the way for applications of clinical interest.

Abstract

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 longstanding 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 intervention protocols to control brain activity by targeting influential neural nodes predicted by network theory.