Electrical networks and data analysis in phylogenetics

TL;DR

The paper establishes a geometric bridge between electrical network theory and phylogenetic-like metrics by showing that resistance distances on circular networks yield Kalmanson metrics that sit inside the positive isotropic Grassmannian via Lam's embedding. It provides a precise, Grassmannian-based criterion for when a Kalmanson metric arises from an electrical network and offers an explicit reconstruction workflow from resistance data to a minimal circular network. By linking impedance/response data to the positivity framework of Grassmannians and circular split systems, the work opens avenues for applications in phylogenetics and cluster algebra positivity, as well as practical topology recovery of networks. Overall, it integrates network theory, metric geometry, and algebraic geometry to enable both theoretical characterizations and algorithmic topology inference.

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.

Figures (11)

• Figure 1: Electrical transformations
• Figure 2: A cactus electrical network with 4 nodes
• Figure 3: $4$-point Kalmanson property
• Figure 4: Circular split system and their polygon representations. Removing any set of parallel edges defines a split of the set of leaves of the graph on the right which coincides with the split of the set of the sides of a polygon on the left obtained by removing an appropriate diagonal.
• Figure 5: Labeling in the formula \ref{['formdual']}

Theorems & Definitions (51)

• Definition 2.1
• Theorem 2.2
• Definition 2.3
• Theorem 2.4
• Definition 2.5
• Definition 2.6
• Theorem 2.7
• Theorem 2.8
• Example 2.9
• Definition 3.1