Hi Nicolas, thanks for your great question - and sorry for my late answer.

I think that Cauchy sequences can be used as powerful similarity for Graphs, even if they are not always the same thing. Eventually, both Cauchy and Graphs will converge, however I am sure there might be some exceptions about this. I think this could be an interesting reading: https://arxiv.org/pdf/math/0702004.pdf where authors define convergence properties for dense graphs.