Are there any methods within Topological Data Analysis that would allow me to check for the existence of similar topological substractures within both point clouds? Example (a bit artifficial): I have big data sets of student's results in 15 exams from 20 succeeding years (the sets of students every year don't intersect). I would expect some linkages in each case. E.g. students with good grades in mathematics and students with good grades sciences could in every case form two separate clusters, but in every case there would be some similar linkages: students interested in applications of informatics in genetics or students interested in applications of mathematics to physics would 'topologically connect' the two groups through distinct chains.
Similar, probably easier question: if the same group of students would write the same exams two years in a row - is there a way to show that there are some topological substructures that connect results from both years?
I guess standard approach to estimating the proximity of topological representation would be Wasserstein or bottleneck metric, but as far as I fear they won't allow me to reconstruct what common substructures really appear. I am also not sure if they could be used in this context.