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.
Highlights
Consider an electrical circuit: a network made of wires joining resistors in parallel and in sequence, with some portion hidden inside an opaque box
The upshot is that when the distances between taxa are effective resistances based on unknown connections, using well known methods we can recover an unweighted circular split network, which gives us the precise class of 1nested phylogenetic network
Several features of the resistance distance seem exactly suited to phylogenetic networks with weighted edges
Summary
Consider an electrical circuit: a network made of wires joining resistors in parallel and in sequence, with some portion hidden inside an opaque box. In Curtis et al (1998) and Curtis and Morrow (1990, 1991), the authors study circular planar graphs with boundary nodes that are analogous to the leaves of our phylogenetic networks. They consider resistance values (or conductivity) on the edges. In Ejov et al (2019), the authors consider the entire set of resistance distances (again using unit values for edges), between any pair of nodes ( leaves.) They show that using this metric is useful for discovering Hamiltonian cycles via algorithms for the Traveling Salesman problem. There is a close connection to our applications, since the algorithm neighbor-net can be used as a greedy approach to the Traveling Salesman problem as shown in Levy and Pachter (2011)
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