Complexity Forum: Alex Arenas (Universitat Rovira i Virgili)
Speaker: Alex Arenas (Universitat Rovira i Virgili)
Title: Multiplex networks, structure and dynamics
Abstract: Modern theory of complex networks is facing new challenges that arise from the necessity of understanding properly the dynamical evolution of real systems. One of such open problems concerns the topological and dynamical characterization of systems made up by two or more interconnected networks. The standard approach in network modeling assumes that every edge (link) is of the same type and consequently considered at the same temporal and topological scale. This is clearly an abstraction of any real topological structure and represents either instantaneous or aggregated interactions over a certain time window. Therefore, to understand the intricate variability of real complex systems, where many different time scales and structural patterns coexist we need a new scenario, a new level of description. We present the time scales associated to 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 diffusion-like processes on top of multiplex networks. Although adjacency matrices are useful to describe traditional single-layer networks, such a representation is sometimes insufficient for the analysis and description of multiplex and time-dependent networks. Such evidences of the importance of multi-level relationships pushed the study of a unified mathematical formulation of multiplexes. We define a unified and self-consistent language, based on a tensorial formulation, allowing to extend the well-known algebra of "mono-plexes" (i.e., single-layer networks) to the realm of interconnected multi-layer networks.