We study methods for computing two network features with topological underpinnings: the Rips and Dowker Persistent Homology Diagrams. Our formulations work for general networks, which may be asymmetric and may have any real number as an edge weight. We study the sensitivity of Dowker persistence diagrams to intrinsic asymmetry in the data, and investigate the theoretical stability properties of both the Dowker and Rips persistence diagrams. We show experimental results on a variety of simulated and real world datasets using our methods. In particular, we apply both methods to a classification task on a database of networks.