  Phylogenetic networks are notoriously difficult to reconstruct. Here we
suggest that it can be useful to view unknown genetic distance along edges in
phylogenetic networks as analogous to unknown resistance in electric circuits.
This resistance distance, well known in graph theory, turns out to have nice
mathematical properties which allow the precise reconstruction of networks.
Specifically we show that the resistance distance for a weighted 1-nested
network is Kalmanson, and that the unique associated circular split network
fully represents the splits of the original phylogenetic network (or circuit).
In fact, this full representation corresponds to a face of the balanced minimal
evolution polytope for level-1 networks. Thus the unweighted class of the
original network can be reconstructed by either the greedy algorithm
neighbor-net or by linear programming over a balanced minimal evolution
polytope. We begin study of 2-nested networks with both minimum path and
resistance distance, and include some counting results for 2-nested networks.
