@Anton: So, generally speaking, which types of questions are should be make community wikis? Asking for advice and such? Anything where there's no one right answer?

I take it you're not just talking about matrices that are eventually 1 or 0 on the diagonal.

Yes, that's true.

I don't think persistent homology actually would help with this. The question doesn't seem to be about data analysis, but clustering of data.

Yes, that's why I was confused. It sounded (to me) like you might have been asking something like if two finitely presented groups are given in different alphabets but with the same number of generators and "identical" relations.