Multiplex networks are used to model complex systems from myriad applications. They generalize classical complex networks by recording different types of relationships, different interactions, or changing interactions over time between the same entities in different layers. They possess natural linear algebraic representations in terms of structured matrices, which makes efficient numerical linear algebra techniques a valuable tool for their analysis. In this talk, we give an overview over several network science problems that can be formulated in terms of matrix function expressions, which we approximate by polynomial and rational Krylov methods. We discuss centrality measures, the solution of stiff systems of non-linear differential equations with exponential Runge--Kutta integrators, as well as un- and semi-supervised community detection. Additionally, we present a nonlinear spectral method for core-periphery detection in multiplex networks. All presented methods have a linear runtime scaling, which allows the treatment of large-scale multiplex networks and we present numerical experiments for all considered problems.