  Effective resistance (ER) is an attractive way to interrogate the structure
of graphs. It is an alternative to computing the eigen-vectors of the graph
Laplacian. Graph laplacians are used to find low dimensional structures in high
dimensional data. Here too, ER based analysis has advantages over eign-vector
based methods. Unfortunately Von Luxburg et al. (2010) show that, when vertices
correspond to a sample from a distribution over a metric space, the limit of
the ER between distant points converges to a trivial quantity that holds no
information about the structure of the graph. We show that by using scaling
resistances in a graph with $n$ vertices by $n^2$, one gets a meaningful limit
of the voltages and of effective resistances. We also show that by adding a
"ground" node to a metric graph one gets a simple and natural way to compute
all of the distances from a chosen point to all other points.
