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 † f.papadopoulos@cut.ac.cy that these correlations suggest the existence of multidimensional communities, which are sets of nodes that are similar (close in the underlying space) in multiple layers. Further, we show that strong geometric correlations imply accurate trans-layer link prediction, where connections in one layer can be predicted by knowing the hyperbolic distances among nodes in another layer. This is important for applications where we only know the connections among nodes in one context, e.g., structural connections between brain regions, and we want to predict connections between the same nodes in some other context, e.g., likelihood of functional connections between the same brain regions. Finally, to study the effects of geometric correlations on dynamical processes, we consider targeted navigation. Targeted navigation is a key function of many real networks, where either goods, people, or information is transferred from a source to a destination using the connections of the network. It has been shown that single complex networks, like the IPv4 Internet or the network of airport connections, can be navigated efficiently by performing greedy routing in their underlying geometric spaces [15, 16, 21, 22] . In greedy routing, nodes forward incoming messages to their neighbors that are closest to the destination in the geometric space so that they only need local knowledge about the coordinates of their neighbors. In multilayer systems, messages can switch between layers and so a node forwards a message to its neighbor that is closest to the destination in any of the layers comprising the system. We call this process mutual greedy routing (MGR). Mutual greedy routing follows the same line of reasoning as greedy routing in Milgram's experiment [23] , which was indeed performed using multiple domains. For example, in the case of a single network, to reach a lawyer in Boston one might want to forward a message to a judge
Recently, the work in REF has shown that connections in one layer of a multiplex can be accurately predicted by utilizing the hyperbolic distances between nodes from another layer in a hidden geometric space.
Hidden geometric correlations in real multiplex networks
Nature Physics
Physics