Who are the most influential players in a social network? What's the origin of an epidemic outbreak? The answer to simple questions like these can hide incredibly difficult computational problems, that require powerful methods for the inference, optimization, and control of dynamical processes on large networks. I will present a statistical mechanics approach to inverse dynamical problems in the idealized framework provided by simple models of irreversible contagion and diffusion on networks (linear threshold model, susceptible-infected-removed epidemic model). Using the cavity method (belief propagation), it is possible to explore the large-deviation properties of the dynamics and develop efficient message-passing algorithms to solve optimization and inference problems even on large networks.