Effective resistance in metric spaces
Robi Bhattacharjee Alexander Cloninger Yoav Freund Andreas Oslandsbotn
Effective resistance (ER) is an attractive way to interrogate the structure
of graphs. It is an alternative to computing the eigenvectors of the graph
Laplacian.
  One attractive application of ER is to point clouds, i.e. graphs whose
vertices correspond to IID samples from a distribution over a metric space.
Unfortunately, it was shown that the ER between any two points converges to a
trivial quantity that holds no information about the graph's structure as the
size of the sample increases to infinity.
  In this study, we show that this trivial solution can be circumvented by
considering a region-based ER between pairs of small regions rather than pairs
of points and by scaling the edge weights appropriately with respect to the
underlying density in each region. By keeping the regions fixed, we show
analytically that the region-based ER converges to a non-trivial limit as the
number of points increases to infinity. Namely the ER on a metric space. We
support our theoretical findings with numerical experiments.