**What is a Multilayer Network**? A Multilayer Network -of which Multiplex and Interdependent networks are particular cases- is a network made up by multiple layers, each of which represents a given operation mode, social circle, or temporal instance. This new paradigm in Network Science is believed to be the next step towards a better and full comprehension of modern cyber, social and physical systems such as online social networks, transportation systems, metabolic and regulatory networks, etc. Below, we develop a bit more the need to study these networks and what are the current challenges. We also briefly describe our work in this line of research.

**Disease Localization in Multilayer Networks**.

G. Ferraz de Arruda, E. Cozzo, T. P. Peixoto, F. A. Rodrigues, and Y. Moreno,

**Physical Review X**

**7**, 011014 (2017).

We present a continuous formulation of epidemic spreading on multilayer networks using a tensorial representation, extending the models of monoplex networks to this context. We derive analytical expressions for the epidemic threshold of the susceptible-infected-susceptible (SIS) and susceptible-infected-recovered dynamics, as well as upper and lower bounds for the disease prevalence in the steady state for the SIS scenario. Using the quasistationary state method, we numerically show the existence of disease localization and the emergence of two or more susceptibility peaks, which are characterized analytically and numerically through the inverse participation ratio. At variance with what is observed in single-layer networks, we show that disease localization takes place on the layers and not on the nodes of a given layer. Furthermore, when mapping the critical dynamics to an eigenvalue problem, we observe a characteristic transition in the eigenvalue spectra of the supra-contact tensor as a function of the ratio of two spreading rates: If the rate at which the disease spreads within a layer is comparable to the spreading rate across layers, the individual spectra of each layer merge with the coupling between layers. Finally, we report on an interesting phenomenon, the barrier effect; i.e., for a three-layer configuration, when the layer with the lowest eigenvalue is located at the center of the line, it can effectively act as a barrier to the disease. The formalism introduced here provides a unifying mathematical approach to disease contagion in multiplex systems, opening new possibilities for the study of spreading processes.

**Characterization of multiple topological scales in multiplex networks through supra-Laplacian eigengaps**.

E. Cozzo and Y. Moreno,

**Physical Review E**

**94**, 052318 (2016).

Multilayer networks have been the subject of intense research during the past few years, as they represent better the interdependent nature of many real-world systems. Here, we address the question of describing the three different structural phases in which a multiplex network might exist. We show that each phase can be characterized by the presence of gaps in the spectrum of the supra-Laplacian of the multiplex network. We therefore unveil the existence of different topological scales in the system, whose relation characterizes each phase. Moreover, by capitalizing on the coarse-grained representation that is given in terms of quotient graphs, we explain the mechanisms that produce those gaps as well as their dynamical consequences.

**A multilayer perspective for the analysis of urban transportation systems**.

A. Aleta, S. Meloni, and Y. Moreno, submitted for publication (2017).

Public urban mobility systems are composed by several transportation modes connected together. Most studies in urban mobility and planning often ignore the multi-layer nature of transportation systems considering only aggregated versions of this complex scenario. In this work we present a model for the representation of the transportation system of an entire city as a multiplex network. Using two different perspectives, one in which each line is a layer and one in which lines of the same transportation mode are grouped together, we study the interconnected structure of 9 different cities in Europe raging from small towns to mega-cities like London and Berlin highlighting their vulnerabilities and possible improvements. Finally, for the city of Zaragoza in Spain, we also consider data about service schedule and waiting times, which allow us to create a simple yet realistic model for urban mobility able to reproduce real-world facts and to test for network improvements.

**Scaling Properties of Multilayer Random Networks**.

J. A. Méndez-Bermúdez, G. Ferraz de Arruda, F. A. Rodrigues, and Y. Moreno, submitted for publication (2017).

Multilayer networks are widespread in natural and manmade systems. Key properties of these networks are their spectral and eigenfunction characteristics, as they determine the critical properties of many dynamics occurring on top of them. In this paper, we numerically demonstrate that the normalized localization length of the eigenfunctions of multilayer random networks follows a simple scaling law. The reported scaling law might help to better understand criticality in multilayer networks as well as to predict the eigenfunction localization properties of them.

**Lévy random walks on multiplex networks**.

Q. Guo, E. Cozzo, Z. Zheng, and Y. Moreno, ** Scientific Reports** **6**:37641 (2016).

Random walks constitute a fundamental mechanism for many dynamics taking place on complex networks. Besides, as a more realistic description of our society, multiplex networks have been receiving a growing interest, as well as the dynamical processes that occur on top of them. Here, inspired by one specific model of random walks that seems to be ubiquitous across many scientific fields, the Lévy flight, we study a new navigation strategy on top of multiplex networks. Capitalizing on spectral graph and stochastic matrix theories, we derive analytical expressions for the mean first passage time and the average time to reach a node on these networks. Moreover, we also explore the efficiency of Lévy random walks, which we found to be very different as compared to the single layer scenario, accounting for the structure and dynamics inherent to the multiplex network. Finally, by comparing with some other important random walk processes defined on multiplex networks, we find that in some region of the parameters, a Lévy random walk is the most efficient strategy. Our results give us a deeper understanding of Lévy random walks and show the importance of considering the topological structure of multiplex networks when trying to find efficient navigation strategies.

**On degree-degree correlations in multilayer networks**.

G. Ferraz de Arruda, E. Cozzo, Y. Moreno, and F. A. Rodrigues,** Physica D ****323-324**, 5 (2016).

We propose a generalization of the concept of assortativity based on the tensorial representation of multilayer networks, covering the definitions given in terms of Pearson and Spearman coefficients. Our approach can also be applied to weighted networks and provides information about correlations considering pairs of layers. By analyzing the multilayer representation of the airport transportation network, we show that contrasting results are obtained when the layers are analyzed independently or as an interconnected system. Finally, we study the impact of the level of assortativity and heterogeneity between layers on the spreading of diseases. Our results highlight the need of studying degree–degree correlations on multilayer systems, instead of on aggregated networks.

**Multilayer networks: metrics and spectral properties**.

E. Cozzo, G. Ferraz de Arruda, F. A. Rodrigues, and Y. Moreno, **Chapter contribution** to the book **“Interconnected networks”**, edited by F. Schweitzer and A. Garas. ISBN 978-3-319-23947-7, Springer (2016).

Multilayer networks represent systems in which there are several topological levels each one representing one kind of interaction or interdependency between the systems’ elements. These networks have attracted a lot of attention recently be- cause their study allows considering different dynamical modes concurrently. Here, we revise the main concepts and tools developed up to date. Specifically, we focus on several metrics for multilayer network characterization as well as on the spectral properties of the system, which ultimately enable for the dynamical characterization of several critical phenomena. The theoretical framework is also applied for description of real-world multilayer systems.