Network structures appear naturally in a world increasingly connected by digital technologies. Accordingly, methods to study networks are becoming more important.</br> In this thesis, we describe multiple different application scenarios involving networks and develop new methods to work on them.</br> As a first method we investigate pattern distributions as a way to characterize networks.</br> Pattern distributions essentially encode the frequency of small subgraphs in a larger host graph. We describe methods to efficiently approximate pattern distributions for larger graphs and evaluate them on a set of real world examples.</br> Next, we study main path analysis, commonly used to extract representative paths from a directed, acyclic graph. Additionally to providing a formally rigorous review of existing variants, we describe new variants and perform an empirical evaluation using random graphs.</br> Apart from analyzing given networks we also study dynamic models based on networks. In one instance we simulate the opinion formation process on the internet and investigate how social bots might be able to influence the opinion climate.</br> We also develop a model to encode probabilistic knowledge on dynamical systems using Bayesian networks.</br> As part of this work we develop a generalized notion of Bayesian networks based on the categorical concepts of PROPs and string diagrams.