Networks in nature do not act in isolation, but instead exchange information and depend on one another to function properly [1][2][3] . Theory has shown that connecting random networks may very easily result in abrupt failures [3][4][5][6] . This finding reveals an intriguing paradox 7,8 : if natural systems organize in interconnected networks, how can they be so stable? Here we provide a solution to this conundrum, showing that the stability of a system of networks relies on the relation between the internal structure of a network and its pattern of connections to other networks. Specifically, we demonstrate that if interconnections are provided by network hubs, and the connections between networks are moderately convergent, the system of networks is stable and robust to failure. We test this theoretical prediction on two independent experiments of functional brain networks (in task and resting states), which show that brain networks are connected with a topology that maximizes stability according to the theory.The theory of networks of networks relies largely on unstructured patterns of connectivity between networks 3,4,6 . When two stable networks are fully interconnected with one-to-one random connections, such that every node in a network depends on a randomly chosen node in the other network, small perturbations in one network are amplified by the interaction between networks 3,6 . This process leads to cascading failures, which are thought to underpin catastrophic outcomes in man-made infrastructures, such as blackouts in power grids 3,4 . By contrast, many stable living systems, including the brain 9 and cellular networks 10 , are organized in interconnected networks. Random networks are very efficient mathematical constructs to develop theory, but the majority of networks observed in nature are correlated 11,12 . Correlations, in turn, provide structure and are known to influence the dynamical and structural properties of interconnected networks, as has been recently shown 13 . Most natural networks form hubs, increasing the relevance of certain nodes. This adds a degree of freedom to the system, in determining whether hubs broadcast information to other networks or, conversely, whether cross-network communication is governed by nodes with less influence in their own network.We develop a full theory for systems of structured networks, identifying a structural communication protocol that ensures the system of networks is stable (less susceptible to catastrophic failure) and optimized for fast communication across the entire system. The theory establishes concrete predictions of a regime of correlated connectivity between the networks composing the system. We test these predictions with two different systems of brain connectivity based on functional magnetic resonance imaging (fMRI) data. The brain organizes in a series of interacting networks 9,14 , presenting a paradigmatic case study for a theory of connected correlated networks. We show that for two independent experiments of functional networks in ta...

The probability distribution of number of ties of an individual in a social network follows a scale-free power-law. However, how this distribution arises has not been conclusively demonstrated in direct analyses of people's actions in social networks. Here, we perform a causal inference analysis and find an underlying cause for this phenomenon. Our analysis indicates that heavy-tailed degree distribution is causally determined by similarly skewed distribution of human activity. Specifically, the degree of an individual is entirely random - following a “maximum entropy attachment” model - except for its mean value which depends deterministically on the volume of the users' activity. This relation cannot be explained by interactive models, like preferential attachment, since the observed actions are not likely to be caused by interactions with other people.

We investigate the navigation problem in lattices with long-range connections and subject to a cost constraint. Our network is built from a regular two-dimensional (d=2) square lattice to be improved by adding long-range connections (shortcuts) with probability P(ij) approximately r(ij)(-alpha), where r(ij) is the Manhattan distance between sites i and j, and alpha is a variable exponent. We introduce a cost constraint on the total length of the additional links and find optimal transport in the system for alpha=d+1 established here for d=1 and d=2. Remarkably, this condition remains optimal, regardless of the strategy used for navigation, being based on local or global knowledge of the network structure, in sharp contrast with the results obtained for unconstrained navigation using global or local information, where the optimal conditions are alpha=0 and alpha=d, respectively. The validity of our results is supported by data on the U.S. airport network.