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.

As any cognitive task, visual search involves a number of underlying processes that cannot be directly observed and measured. In this way, the movement of the eyes certainly represents the most explicit and closest connection we can get to the inner mechanisms governing this cognitive activity. Here we show that the process of eye movement during visual search, consisting of sequences of fixations intercalated by saccades, exhibits distinctive persistent behaviors. Initially, by focusing on saccadic directions and intersaccadic angles, we disclose that the probability distributions of these measures show a clear preference of participants towards a reading-like mechanism (geometrical persistence), whose features and potential advantages for searching/foraging are discussed. We then perform a Multifractal Detrended Fluctuation Analysis (MF-DFA) over the time series of jump magnitudes in the eye trajectory and find that it exhibits a typical multifractal behavior arising from the sequential combination of saccades and fixations. By inspecting the time series composed of only fixational movements, our results reveal instead a monofractal behavior with a Hurst exponent , which indicates the presence of long-range power-law positive correlations (statistical persistence). We expect that our methodological approach can be adopted as a way to understand persistence and strategy-planning during visual search.

We investigate the emergence of extreme opinion trends in society by employing statistical physics modeling and analysis on polls that inquire about a wide range of issues such as religion, economics, politics, abortion, extramarital sex, books, movies, and electoral vote. The surveys lay out a clear indicator of the rise of extreme views. The precursor is a nonlinear relation between the fraction of individuals holding a certain extreme view and the fraction of individuals that includes also moderates, e.g., in politics, those who are “very conservative” versus “moderate to very conservative” ones. We propose an activation model of opinion dynamics with interaction rules based on the existence of individual “stubbornness” that mimics empirical observations. According to our modeling, the onset of nonlinearity can be associated to an abrupt bootstrap-percolation transition with cascades of extreme views through society. Therefore, it represents an early-warning signal to forecast the transition from moderate to extreme views. Moreover, by means of a phase diagram we can classify societies according to the percolative regime they belong to, in terms of critical fractions of extremists and people’s ties.

We show that logic computational circuits in gene regulatory networks arise from a fibration symmetry breaking in the network structure. From this idea we implement a constructive procedure that reveals a hierarchy of genetic circuits, ubiquitous across species, that are surprising analogues to the emblematic circuits of solid-state electronics: starting from the transistor and progressing to ring oscillators, current-mirror circuits to toggle switches and flip-flops. These canonical variants serve fundamental operations of synchronization and clocks (in their symmetric states) and memory storage (in their broken symmetry states). These conclusions introduce a theoretically principled strategy to search for computational building blocks in biological networks, and present a systematic route to design synthetic biological circuits.

We introduce a cluster growth process that provides a clear connection between equilibrium statistical mechanics and an explosive percolation model similar to the one recently proposed by Achlioptas et al. [Science 323, 1453[Science 323, (2009. We show that the following two ingredients are essential for obtaining an abrupt (first-order) transition in the fraction of the system occupied by the largest cluster: (i) the size of all growing clusters should be kept approximately the same, and (ii) the inclusion of merging bonds (i.e., bonds connecting vertices in different clusters) should dominate with respect to the redundant bonds (i.e., bonds connecting vertices in the same cluster). Moreover, in the extreme limit where only merging bonds are present, a complete enumeration scheme based on tree-like graphs can be used to obtain an exact solution of our model that displays a firstorder transition. Finally, the proposed mechanism can be viewed as a generalization of standard percolation that discloses an entirely new family of models with potential application in growth and fragmentation processes of real network systems. PACS numbers:The second-order critical point of percolation [1, 2] has been successfully used to describe a large variety of phenomena in Nature, including the sol-gel transition [3], or incipient flow through porous media [4], as well as epidemic spreading [5] and network failure [6,7,8,9]. A long standing question of practical interest has been since, how the transition could be made more abrupt and in the limit become even of first-order. In other words, what ingredient must be tuned in the basic model of random percolation to change the order of the transition?Recently Achlioptas et al.[10] proposed a new mechanism on random graphs which they termed "explosive percolation" that exhibits first-order phase transition. Their process takes place in successive steps, with bonds being added to the system in accordance to a selection rule. At each step, a set of two unoccupied bonds are chosen randomly. From these two, only the one with minimum weight becomes occupied. In Ref.[10], the weight is defined as the product of the sizes of the clusters connected by this bond (this is called "product rule"). Importantly, if the bond connects two sites that already belong to the same cluster, the weight is proportional to the square of the cluster size. Since unoccupied bonds connecting vertices in the largest cluster have the largest possible weight, these bonds will become occupied only if two of them are randomly chosen. Thus, this selection rule hinders the inclusion of bonds connecting vertices that already belong to the largest cluster. As a consequence, bonds merging two smaller clusters will be selected more frequently, resulting in the fast growth observed. Their model was then implemented on a fully connected graph, however, it was shown that the same effect takes place on 2D square lattices [11] as well as scale-free networks [12,13].In this letter we investigate what are the basic principles t...

The imposition of a cost constraint for constructing the optimal navigation structure surely represents a crucial ingredient in the design and development of any realistic navigation network. Previous works have focused on optimal transport in small-world networks built from two-dimensional lattices by adding long-range connections with Manhattan length r(ij) taken from the distribution P(ij)~r(ij)(-α), where α is a variable exponent. It has been shown that, by introducing a cost constraint on the total length of the additional links, regardless of the strategy used by the traveler (independent of whether it is based on local or global knowledge of the network structure), the best transportation condition is obtained with an exponent α=d+1, where d is the dimension of the underlying lattice. Here we present further support, through a high-performance real-time algorithm, on the validity of this conjecture in three-dimensional regular as well as in two-dimensional critical percolation clusters. Our results clearly indicate that cost constraint in the navigation problem provides a proper theoretical framework to justify the evolving topologies of real complex network structures, as recently demonstrated for the networks of the US airports and the human brain activity.