Real networks often form interacting parts of larger and more complex systems. Examples can be found in different domains, ranging from the Internet to structural and functional brain networks. Here, we show that these multiplex systems are not random combinations of single network layers. Instead, they are organized in specific ways dictated by hidden geometric correlations interweaving the layers. We find that these correlations are significant in different real multiplexes, and form a key framework for answering many important questions. Specifically, we show that these geometric correlations facilitate: (i) the definition and detection of multidimensional communities, which are sets of nodes that are simultaneously similar in multiple layers; (ii) accurate trans-layer link prediction, where connections in one layer can be predicted by observing the hidden geometric space of another layer; and (iii) efficient targeted navigation in the multilayer system using only local knowledge, which outperforms navigation in the single layers only if the geometric correlations are sufficiently strong. Our findings uncover fundamental organizing principles behind real multiplexes and can have important applications in diverse domains.Real networks are often not isolated entities but instead can be considered constituents of larger systems, called multiplexes or multilayer networks [1][2][3][4][5][6][7][8]. Examples can be found everywhere. One is the multiplex consisting of the different social networks that a person may belong to. Other examples include the Internet's IPv4 and IPv6 topologies, or the structural and functional networks in the brain. Understanding the relations among the networks comprising a multiplex is crucial for understanding the behavior of a wide range of real world systems [9][10][11][12][13]. However, despite the burst of recent research in studying the properties of multiplex networks, e.g., [1,5,14], a universal framework describing the relations among the single networks comprising a multiplex and what implications these relations may have when it comes to applications remains elusive.Here, we show that real multiplexes are not random combinations of single network layers. Instead, we find that their constituent networks exhibit significant hidden geometric correlations. These correlations are called "hidden" as they are not directly observable by looking at each individual network topology. Specifically, each single network can be mapped (or embedded) into a separate hyperbolic space [15][16][17][18], where node coordinates abstract the popularity and similarity of nodes [19,20]. We find that node coordinates are significantly correlated across layers of real multiplexes, meaning that distances between nodes in the underlying hyperbolic spaces of the constituent networks are also significantly correlated.The discovered geometric correlations yield a very powerful framework for answering important questions related to real multiplexes. Specifically, we first show * kajkoljakleineberg@gmail.com †...

We show that real multiplex networks are unexpectedly robust against targeted attacks on high-degree nodes and that hidden interlayer geometric correlations predict this robustness. Without geometric correlations, multiplexes exhibit an abrupt breakdown of mutual connectivity, even with interlayer degree correlations. With geometric correlations, we instead observe a multistep cascading process leading into a continuous transition, which apparently becomes fully continuous in the thermodynamic limit. Our results are important for the design of efficient protection strategies and of robust interacting networks in many domains.

The evolution of cooperation in social dilemmas in structured populations has been studied extensively in recent years. Whereas many theoretical studies have found that a heterogeneous network of contacts favors cooperation, the impact of spatial effects in scale-free networks is still not well understood. In addition to being heterogeneous, real contact networks exhibit a high mean local clustering coefficient, which implies the existence of an underlying metric space. Here we show that evolutionary dynamics in scale-free networks self-organize into spatial patterns in the underlying metric space. The resulting metric clusters of cooperators are able to survive in social dilemmas as their spatial organization shields them from surrounding defectors, similar to spatial selection in Euclidean space. We show that under certain conditions these metric clusters are more efficient than the most connected nodes at sustaining cooperation and that heterogeneity does not always favor—but can even hinder—cooperation in social dilemmas.

Online social networks (OSNs) enable researchers to study the social universe at a previously unattainable scale. The worldwide impact and the necessity to sustain their rapid growth emphasize the importance to unravel the laws governing their evolution. We present a quantitative two-parameter model which reproduces the entire topological evolution of a quasi-isolated OSN with unprecedented precision from the birth of the network. This allows us to precisely gauge the fundamental macroscopic and microscopic mechanisms involved. Our findings suggest that the coupling between the real pre-existing underlying social structure, a viral spreading mechanism, and mass media influence govern the evolution of OSNs. The empirical validation of our model, on a macroscopic scale, reveals that virality is four to five times stronger than mass media influence and, on a microscopic scale, individuals have a higher subscription probability if invited by weaker social contacts, in agreement with the "strength of weak ties" paradigm.

The overwhelming success of Web 2.0, within which online social networks are key actors, has induced a paradigm shift in the nature of human interactions. The user-driven character of Web 2.0 services has allowed researchers to quantify large-scale social patterns for the first time. However, the mechanisms that determine the fate of networks at the system level are still poorly understood. For instance, the simultaneous existence of multiple digital services naturally raises questions concerning which conditions these services can coexist under. Analogously to the case of population dynamics, the digital world forms a complex ecosystem of interacting networks. The fitness of each network depends on its capacity to attract and maintain users’ attention, which constitutes a limited resource. In this paper, we introduce an ecological theory of the digital world which exhibits stable coexistence of several networks as well as the dominance of an individual one, in contrast to the competitive exclusion principle. Interestingly, our theory also predicts that the most probable outcome is the coexistence of a moderate number of services, in agreement with empirical observations.

We present a model that takes into account the coupling between evolutionary game dynamics and social influence. Importantly, social influence and game dynamics take place in different domains, which we model as different layers of a multiplex network. We show that the coupling between these dynamical processes can lead to cooperation in scenarios where the pure game dynamics predicts defection. In addition, we show that the structure of the network layers and the relation between them can further increase cooperation. Remarkably, if the layers are related in a certain way, the system can reach a polarized metastable state. These findings could explain the prevalence of polarization observed in many social dilemmas.

We present a framework to simulate SIR processes on networks using weighted shortest paths. Our framework maps the SIR dynamics to weights assigned to the edges of the network, which can be done for Markovian and non-Markovian processes alike. The weights represent the propagation time between the adjacent nodes for a particular realization. We simulate the dynamics by constructing an ensemble of such realizations, which can be done by using a Markov Chain Monte Carlo method or by direct sampling. The former provides a runtime advantage when realizations from all possible sources are computed as the weighted shortest paths can be re-calculated more efficiently. We apply our framework to three empirical networks and analyze the expected propagation time between all pairs of nodes. Furthermore, we have employed our framework to perform efficient source detection and to improve strategies for time-critical vaccination.

The overwhelming success of online social networks, the key actors in the Web 2.0 cosmos, has reshaped human interactions globally. To help understand the fundamental mechanisms which determine the fate of online social networks at the system level, we describe the digital world as a complex ecosystem of interacting networks. In this paper, we study the impact of heterogeneity in network fitnesses on the competition between an international network, such as Facebook, and local services. The higher fitness of international networks is induced by their ability to attract users from all over the world, which can then establish social interactions without the limitations of local networks. In other words, inter-country social ties lead to increased fitness of the international network. To study the competition between an international network and local ones, we construct a 1:1000 scale model of the digital world, consisting of the 80 countries with the most Internet users. Under certain conditions, this leads to the extinction of local networks; whereas under different conditions, local networks can persist and even dominate completely. In particular, our model suggests that, with the parameters that best reproduce the empirical overtake of Facebook, this overtake could have not taken place with a significant probability.