Phylogenetic networks as circuits with resistance distance

TL;DR

This paper introduces a novel approach to reconstruct phylogenetic networks by modeling genetic distances as resistance distances in electric circuits, enabling precise network reconstruction using graph theory properties.

Contribution

It establishes that resistance distance for 1-nested networks is Kalmanson and that the associated circular split network fully captures the original network's splits.

Findings

Resistance distance for 1-nested networks is Kalmanson.

The associated circular split network fully represents the original network.

Reconstruction can be done via neighbor-net or linear programming.

Abstract

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.