Electrical networks and data analysis in phylogenetics

TL;DR

This paper explores the relationship between electrical network metrics and phylogenetic data analysis, providing a geometric characterization of certain metrics and suggesting new methods for reconstructing phylogenetic networks.

Contribution

It offers a complete description of electrical Kalmanson metrics using the geometry of the positive Isotropic Grassmannian, linking electrical network theory with phylogenetics and cluster algebras.

Findings

Electrical resistance matrices define metrics on network nodes.

Electrical Kalmanson metrics are characterized geometrically via the Isotropic Grassmannian.

The results enable new data analysis methods for phylogenetic networks.

Abstract

A classic problem in data analysis is studying the systems of subsets defined by either a similarity or a dissimilarity function on $X$ which is either observed directly or derived from a data set. For an electrical network there are two functions on the set of the nodes defined by the resistance matrix and the response matrix either of which defines the network completely. We argue that these functions should be viewed as a similarity and a dissimilarity function on the set of the nodes moreover they are related via the covariance mapping also known as the Farris transform or the Gromov product. We will explore the properties of electrical networks from this point of view. It has been known for a while that the resistance matrix defines a metric on the nodes of the electrical networks. Moreover for a circular electrical network this metric obeys the Kalmanson property as it was shown recently. We will call such a metric an electrical Kalmanson metric. The main results of this paper is a complete description of the electrical Kalmanson metrics in the set of all Kalmanson metrics in terms of the geometry of the positive Isotropic Grassmannian whose connection to the theory of electrical networks was discovered earlier. One important area of applications where Kalmanson metrics are actively used is the theory of phylogenetic networks which are a generalization of phylogenetic trees. Our results allow us to use in phylogenetics the powerful methods of reconstruction of the minimal graphs of electrical networks and possibly open the door into data analysis for the methods of the theory of cluster algebras.