I'll discuss recent work on trying to adapt persistent homology methods to datasets that exhibit asymmetry. Natural candidates are the Rips and Cech filtrations. Whereas the Rips filtration can unambiguously be generalized directly, generalizing the Cech filtration gives rise to two different versions: the sink and the source filtrations. It turns out that the Rips filtration imposes a symmetrization on the data whereas the Cech filtrations do not, thus making them more suitable for the analysis of intrinsically asymmetric data. By generalizing a theorem of Dowker we can prove that the persistent homologies of these two Cech filtrations are isomorphic. We establish the stability of these constructions under a metric between networks that generalizes the Gromov-Hausdorff distance. I'll also describe some results ve that characterize the persistence diagrams of some likely "motifs" in real (e.g. biological) networks: cycle-networks, which are directed analogues of the standard (discrete) circle.