The elementary units of a many real-world complex systems, from neural networks to trasportation systems, interact through several different kind of relationships at the same time, beyond single-layer networks. In the multiplex metaphor the interactions among the nodes of a system are represented by a multi-layer graph, where all the edges of a certain kind are grouped in a separate layer.
By studying five multi-dimensional data sets of social, technological and biological systems we show that real-world multi-layer networks are characterised by several different kinds of non-trivial correlations, suggesting that multiplexity might introduce new levels of complexity. We discuss a few null-models which can be used to assess the significance of the observed correlation patterns.
Second part (Federico Battiston): Random walks represent a paradigmatic model to study the diffusion properties of complex networks, and have also been employed as a tool to characterise the centrality of nodes and to identify densely connected subgraphs or communities. Biased random walks are a particularly interesting class of walks for which the probability to jump from one node to one of its neighbour depends on a function of a chosen topological property of the destination node, and can be therefore tuned at will in order to systematically prefer (or avoid) to move toward nodes having certain characteristics. In this talk I will present an analytical treatment for biased random walks on multiplex networks, derive the stationary occupation probability distribution independently from the bias function and analyse the effects of different motion rules on the equilibrium distribution and the entropy rate of the processes.