Circular planar electrical networks, Split systems, and Phylogenetic networks

TL;DR

This paper explores the connection between circular planar electrical networks and phylogenetic split systems, revealing that response matrices correspond to unique resistance metrics and split systems, enabling cross-disciplinary methods for network and circuit reconstruction.

Contribution

It establishes a novel link between electrical network invariants and phylogenetic split systems, allowing interchange of analytical techniques between the two fields.

Findings

Response matrices correspond to unique resistance metrics obeying the Kalmanson condition.

Weighted circular split systems are uniquely determined by electrical network invariants.

Methods from phylogenetics and electrical network analysis can be interchanged for network reconstruction.

Abstract

We study a new invariant of circular planar electrical networks, well known to phylogeneticists: the circular split system. We use our invariant to answer some open questions about levels of complexity of networks and their related Kalmanson metrics. The key to our analysis is the realization that certain matrices arising from weighted split systems are studied in another guise: the Kron reductions of Laplacian matrices of planar electrical networks. Specifically we show that a response matrix of a circular planar electrical network corresponds to a unique resistance metric obeying the Kalmanson condition, and thus a unique weighted circular split system. Our results allow interchange of methods: phylogenetic reconstruction using theorems about electrical networks, and circuit reconstruction using phylogenetic techniques.