Distances between two complexes when using Persistence Homology I am using Persistence Homology to look at two different facebook networks. I can generate a distance matrix between individuals and then create the usual barcodes and persistence diagrams according to topological data analysis methods.
Usually I create an alpha complex or Rips complex from the data and then compute the corresponding topological summaries. But I was wondering what methods exist to compute the distance between two different complexes? So say I compute the complexes for two very similar networks, I would need a metric to define which networks are more similar and which ones are less similar.
Of course trying to match topological complexes might be hard, because that is like an isomorphism problem. But are there other good ways, or consensus ways to describe the distance between two objects or observations based upon persistence homology? I can google for references to this question, but since I don't come from a pure math background it can be hard to tell which papers are better than others. If anyone has any good references on this question, that would be helpful as well.
 A: Often data come in the form of point clouds (i.e., finite metric spaces), which can be used to construct simplicial complexes later.  One theoretically-attractive metric one can put on the collection of compact metric spaces is the Gromov-Hausdorff distance.  Many of the stability results concerning the persistent homology pipeline use this distance on the input metric spaces.  More generally, one can consider Gromov-Wasserstein distances from optimal transport by equipping point clouds with some prior probability distribution (e.g., uniform).  Wasserstein metrics and bottleneck distance are also used to compare persistence diagrams/barcodes, so if you want to measure the distance between two point clouds by their barcode summary, this is the typical use-case.
Also, many of the objects in TDA come as filtered objects, including the filtered complexes that are used to compute persistent homology.  In this case, the interleaving distance may be another option to compare two filtered objects, though this is more traditionally used to compare persistence modules specifically.
