A network representation is useful for describing the structure of a large variety of complex systems. However, most real and engineered systems have multiple subsystems and layers of connectivity, and the data produced by such systems is very rich. Achieving a deep understanding of such systems necessitates generalizing "traditional" network theory, and the newfound deluge of data now makes it possible to test increasingly general frameworks for the study of networks. In particular, although adjacency matrices are useful to describe traditional single-layer networks, such a representation is insufficient for the analysis and description of multiplex and time-dependent networks. One must therefore develop a more general mathematical framework to cope with the challenges posed by multi-layer complex systems. In this paper, we introduce a tensorial framework to study multi-layer networks, and we discuss the generalization of several important network descriptors and dynamical processes-including degree centrality, clustering coefficients, eigenvector centrality, modularity, Von Neumann entropy, and diffusion-for this framework. We examine the impact of different choices in constructing these generalizations, and we illustrate how to obtain known results for the special cases of single-layer and multiplex networks. Our tensorial approach will be helpful for tackling pressing problems in multi-layer complex systems, such as inferring who is influencing whom (and by which media) in multichannel social networks and developing routing techniques for multimodal transportation systems.

Assessing the navigability of interconnected networks (transporting information, people, or goods) under eventual random failures is of utmost importance to design and protect critical infrastructures. Random walks are a good proxy to determine this navigability, specifically the coverage time of random walks, which is a measure of the dynamical functionality of the network. Here, we introduce the theoretical tools required to describe random walks in interconnected networks accounting for structure and dynamics inherent to real systems. We develop an analytical approach for the covering time of random walks in interconnected networks and compare it with extensive Monte Carlo simulations. Generally speaking, interconnected networks are more resilient to random failures than their individual layers per se, and we are able to quantify this effect. As an application--which we illustrate by considering the public transport of London--we show how the efficiency in exploring the multiplex critically depends on layers' topology, interconnection strengths, and walk strategy. Our findings are corroborated by data-driven simulations, where the empirical distribution of check-ins and checksout is considered and passengers travel along fastest paths in a network affected by real disruptions. These findings are fundamental for further development of searching and navigability strategies in real interconnected systems.N etwork theory has been revealed to be a perfect instrument to model the structure of complex systems and the dynamical process in which they are involved. However, the classical approach does not take into account the possibility that agents can be networked in different ways, and with different intensity, on multiple layers simultaneously. As an example, in the case of social sciences the same user might choose to subscribe to two or more online social networks and to build different social relationships with different users on each social platform (e.g., LinkedIn for the network of professional contacts, Facebook for the network of friends, etc.). Another example is represented by transportation networks in a city: the network of bus stops, the first layer, is different from the tube network, the second layer, but people make use of both by combining paths to move from one place to another within the city. The cases where one vertex is not present in the full multiplex can be accounted for by including it as an isolated vertex in the layers where it is missing, without altering either topological or dynamical properties of the interconnected network.The existence of such multiple connections on different layers invites a generalization of the theory of complex networks to cope with multilayer interconnected networked systems. More specifically, very recent studies focused on a particular type of multilayer network, the multiplex, where each agent participates in different layers simultaneously, like our previous example in the case of online social networks. Indeed, the actors (vertices) in every layer are t...

The determination of the most central agents in complex networks is important because they are responsible for a faster propagation of information, epidemics, failures and congestion, among others. A challenging problem is to identify them in networked systems characterized by different types of interactions, forming interconnected multilayer networks. Here we describe a mathematical framework that allows us to calculate centrality in such networks and rank nodes accordingly, finding the ones that play the most central roles in the cohesion of the whole structure, bridging together different types of relations. These nodes are the most versatile in the multilayer network. We investigate empirical interconnected multilayer networks and show that the approaches based on aggregating-or neglecting-the multilayer structure lead to a wrong identification of the most versatile nodes, overestimating the importance of more marginal agents and demonstrating the power of versatility in predicting their role in diffusive and congestion processes.

One of the more challenging tasks in the understanding of dynamical properties of models on top of complex networks is to capture the precise role of multiplex topologies. In a recent paper, Gómez et al. [Phys. Rev. Lett. 110, 028701 (2013)], some of the authors proposed a framework for the study of diffusion processes in such networks. Here, we extend the previous framework to deal with general configurations in several layers of networks and analyze the behavior of the spectrum of the Laplacian of the full multiplex. We derive an interesting decoupling of the problem that allow us to unravel the role played by the interconnections of the multiplex in the dynamical processes on top of them. Capitalizing on this decoupling we perform an asymptotic analysis that allow us to derive analytical expressions for the full spectrum of eigenvalues. This spectrum is used to gain insight into physical phenomena on top of multiplex, specifically, diffusion processes and synchronizability.

As new instances of nested organization-beyond ecological networks-are discovered, scholars are debating the coexistence of two apparently incompatible macroscale architectures: nestedness and modularity. The discussion is far from being solved, mainly for two reasons. First, nestedness and modularity appear to emerge from two contradictory dynamics, cooperation and competition. Second, existing methods to assess the presence of nestedness and modularity are flawed when it comes to the evaluation of concurrently nested and modular structures. In this work, we tackle the latter problem, presenting the concept of in-block nestedness, a structural property determining to what extent a network is composed of blocks whose internal connectivity exhibits nestedness. We then put forward a set of optimization methods that allow us to identify such organization successfully, in synthetic and in a large number of real networks. These findings challenge our understanding of the topology of ecological and social systems, calling for new models to explain how such patterns emerge.

Multiplex networks are representations of multilayer interconnected complex networks where the nodes are the same at every layer. They turn out to be good abstractions of the intricate connectivity of multimodal transportation networks, among other types of complex systems. One of the most important critical phenomena arising in such networks is the emergence of congestion in transportation flows. Here, we prove analytically that the structure of multiplex networks can induce congestion for flows that otherwise would be decongested if the individual layers were not interconnected. We provide explicit equations for the onset of congestion and approximations that allow us to compute this onset from individual descriptors of the individual layers. The observed cooperative phenomenon is reminiscent of Braess' paradox in which adding extra capacity to a network when the moving entities selfishly choose their route can in some cases reduce overall performance. Similarly, in the multiplex structure, the efficiency in transportation can unbalance the transportation loads resulting in unexpected congestion.

Real-world complex systems exhibit multiple levels of relationships. In many cases they require to be modeled as interconnected multilayer networks, characterizing interactions of several types simultaneously. It is of crucial importance in many fields, from economics to biology and from urban planning to social sciences, to identify the most (or the less) influential nodes in a network using centrality measures. However, defining the centrality of actors in interconnected complex networks is not trivial. In this paper, we rely on the tensorial formalism recently proposed to characterize and investigate this kind of complex topologies, and extend two well known random walk centrality measures, the random walk betweenness and closeness centrality, to interconnected multilayer networks. For each of the measures we provide analytical expressions that completely agree with numerically results.Comment: 18 pages, 4 figures. arXiv admin note: substantial text overlap with arXiv:1311.290

Recent advances in the study of networked systems have highlighted that our interconnected world is composed of networks that are coupled to each other through different "layers" that each represent one of many possible subsystems or types of interactions. Nevertheless, it is traditional to aggregate multilayer networks into a single weighted network in order to take advantage of existing tools. This is admittedly convenient, but it is also extremely problematic, as important information can be lost as a result. It is therefore important to develop multilayer generalizations of network concepts. In this paper, we analyze triadic relations and generalize the idea of transitivity to multiplex networks. By focusing on triadic relations, which yield the simplest type of transitivity, we generalize the concept and computation of clustering coefficients to multiplex networks. We show how the layered structure of such networks introduces a new degree of freedom that has a fundamental effect on transitivity. We compute multiplex clustering coefficients for several real multiplex networks and illustrate why one must take great care when generalizing standard network concepts to multiplex networks. We also derive analytical expressions for our clustering coefficients for ensemble averages of networks in a family of random multiplex networks. Our analysis illustrates that social networks have a strong tendency to promote redundancy by closing triads at every layer and that they thereby have a different type of multiplex transitivity from transportation networks, which do not exhibit such a tendency. These insights are invisible if one only studies aggregated networks.