Circular Planar Electrical Networks, Split Systems, and Phylogenetic Networks

Abstract

Two well-known generalizations of phylogenetic trees are split systems and phylogenetic networks. Distances for both are typically calculated as the minimum path weight. Relatedly, a tree-shaped electrical network has effective resistance between its leaf nodes found as the minimum path resistance. This latter fact raises the question of comparing the effective resistance on electrical networks to the minimum path length on split systems. Surprisingly, it turns out that circular planar electrical networks correspond to unique weighted circular split systems. This crucial result is found precisely by showing that the resistance metric of the Kron reduction of a circular planar electrical network obeys the Kalmanson condition. We demonstrate how this fact allows interchange of methods: primarily circuit reconstruction using phylogenetic techniques.