Phylogenetic Networks as Circuits With Resistance Distance

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.

Structure from Voltage

“…ER-based methods have several advantages over Eigenmap methods including the ability distribute the data and the computation across computers Gillani and Bagchi (2021). Applications of effective resistance include phylogenetic networks Forcey and Scalzo (2020), detecting community structure Zhang and Bu (2019), distributed control Barooah and Hespanha (2006), graph edge sparsification Spielman and Srivastava (2011), and measuring cascade effects Tauch et al (2015), especially in power grids Koc ¸et al (2014); Wang et al (2015); Cavraro and Kekatos (2018).…”