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.

We study the time scales associated with diffusion processes that take place on multiplex networks, i.e., on a set of networks linked through interconnected layers. To this end, we propose the construction of a supra-laplacian matrix, which consists of a dimensional lifting of the laplacian matrix of each layer of the multiplex network. We use perturbative analysis to reveal analytically the structure of eigenvectors and eigenvalues of the complete network in terms of the spectral properties of the individual layers. The spectrum of the supra-laplacian allows us to understand the physics of diffusionlike processes on top of multiplex networks.

We present the analysis of the interrelation between two processes accounting for the spreading of an epidemic, and the information awareness to prevent its infection, on top of multiplex networks. This scenario is representative of an epidemic process spreading on a network of persistent real contacts, and a cyclic information awareness process diffusing in the network of virtual social contacts between the same individuals. The topology corresponds to a multiplex network where two diffusive processes are interacting affecting each other. The analysis using a microscopic Markov chain approach reveals the phase diagram of the incidence of the epidemics and allows us to capture the evolution of the epidemic threshold depending on the topological structure of the multiplex and the interrelation with the awareness process. Interestingly, the critical point for the onset of the epidemics has a critical value (metacritical point) defined by the awareness dynamics and the topology of the virtual network, from which the onset increases and the epidemics incidence decreases.

Explosive collective phenomena have attracted much attention since the discovery of an explosive percolation transition. In this Letter, we demonstrate how an explosive transition shows up in the synchronization of scale-free networks by incorporating a microscopic correlation between the structural and the dynamical properties of the system. The characteristics of the explosive transition are analytically studied in a star graph reproducing the results obtained in synthetic networks. Our findings represent the first abrupt synchronization transition in complex networks and provide a deeper understanding of the microscopic roots of explosive critical phenomena.

Modular structure is ubiquitous in real-world complex networks, and its detection is important because it gives insights in the structure-functionality relationship. The standard approach is based on the optimization of a quality function, modularity, which is a relative quality measure for a partition of a network into modules. Recently some authors [1,2] have pointed out that the optimization of modularity has a fundamental drawback: the existence of a resolution limit beyond which no modular structure can be detected even though these modules might have own entity. The reason is that several topological descriptions of the network coexist at different scales, which is, in general, a fingerprint of complex systems. Here we propose a method that allows for multiple resolution screening of the modular structure. The method has been validated using synthetic networks, discovering the predefined structures at all scales. Its application to two real social networks allows to find the exact splits reported in the literature, as well as the substructure beyond the actual split.