Abstract:
We'll discuss recent work on trying to adapt PH methods to datasets that exhibit asymmetry. Natural candidates are the Rips and Cech filtrations. Whereas the Rips filtration can be generalized directly, generalizing the Cech filtration gives rise to two different versions: the sink and the source filtrations. It turns out that whereas the Rips filtration imposes a max-symmetrization on the data, the Cech filtrations does not, thus making it 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. Stability of these constructions takes place under a metric between networks that generalizes the Gromov-Hausdorff distance. I'll describe some results we have 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. Finally, as an application, we'll show computational results about classifying simulated networks arising from ensembles of hippocampal cells. |