Dynamic processes on graphs are fundamental to modeling a wide range of real-world phenomena, including the spread of epidemics, information diffusion in social networks and neural cascades. In such cases, the parameters governing these processes are often unknown, and for an accurate description of the processes they must be deduced from observations that are often incomplete or even erroneous. This thesis addresses the challenge of inferring the governing parameters of discrete-time Markov processes on graphs using observations of the time series. A primary focus is placed on the notoriously difficult task of inferring the network topology itself from partial observations. In the past, inference methods have been proposed for non-recurrent models, i.e.