The goal of this project is to develop new mathematical formulas for analyzing and predicting how far and how fast messages will spread in a large contact networks, and for inferring the structure of an unknown network by observing message transmission times. Methodologically this project overlaps the areas of probability theory, graph theory, and statistics. The main approach is to derive and apply probabilistic limit theorems to large network models where the number of nodes tends to infinity. This allows to derive new and more accurate formulas for modeling information diffusion dynamics and the comparison of statistical estimators. The estimators under study provide valuable knowledge for users of contact networks willing to maximize their visibility, for service providers to optimize their technological platform, and for third parties to detect anomalies in network data.