# Mathematics of Networks, Hyper-Networks and Multiplex Networks

Our interest is in developing new mathematical concepts that permit us to understand the organization and function of complex systems. We combine techniques and tools from Graph Theory, Linear Algebra and Statistical Mechanics to characterise the structural and dynamical properties of networks, hypernetworks and multiplexes. In particular, we are interested in the use of matrix functions, operator theory, geometric embedding, and algebraic topology, among others to characterise properties such as expansibility, topological and functional bottlenecks, organization of clusters, global communicability, “clumpiness”, returnability, embedding into Euclidean spaces, random rectangular graphs, etc. In addition, we study dynamical properties of networks, hypernetworks and multiplexes such as diffusion processes, synchronization and epidemic spreading and relate these dynamical properties to the structural characteristics of these networks via algebraic graph theory.