Metric properties of electrical networks and the graph reconstruction problems

TL;DR

This paper explores the mathematical properties of electrical networks, demonstrating their embedding into Grassmannians, characterizing certain metrics, and presenting algorithms for network reconstruction with applications in phylogenetics and inverse problems.

Contribution

It introduces a novel embedding of electrical networks into Grassmannians and provides a comprehensive characterization of planar electrical Kalmanson metrics.

Findings

Effective resistances satisfy the Kalmanson property

Complete characterization of planar electrical Kalmanson metrics

A new graph reconstruction algorithm for phylogenetics and inverse problems

Abstract

Using the generalized Temperley trick, we demonstrate the explicit embedding of circular electrical networks into totally non-negative Grassmannians. Building on this result, we show that the effective resistances between boundary nodes of circular electrical networks satisfy the Kalmanson property, and we provide the full characterization of planar electrical Kalmanson metrics. Additionally, we present a graph reconstruction algorithm with applications in phylogenetic network analysis as well as the numerical solution of the Calderon problem.