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Metric properties of electrical networks and the graph reconstruction problems

V. G. Gorbounov, A. A. Kazakov

TL;DR

This work links electrical network theory with the geometry of the totally non-negative Grassmannian by embedding circular networks into $Gr_{\ge 0}(n-1,2n)$ using the generalized Temperley trick. It proves that effective resistances $R_{ij}$ form a Kalmanson metric on boundary nodes and provides a complete planar characterization via $\Omega_D$ and Plücker coordinates, connecting $R_{ij}$ to a Grassmannian point and to dual-network relations. A reconstruction framework is developed: minimal networks are recoverable from column rank-patterns of $\Omega_R(\mathcal{E})$ with a rank permutation tied to the strand permutation, and the generalized chamber ansatz recovers conductivities up to electrical transformations. The results enable topology inference in phylogenetic networks and Calderón-type inverse problems by exploiting the Grassmannian viewpoint and the Kalmanson structure of the resistance metric.

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.

Metric properties of electrical networks and the graph reconstruction problems

TL;DR

This work links electrical network theory with the geometry of the totally non-negative Grassmannian by embedding circular networks into using the generalized Temperley trick. It proves that effective resistances form a Kalmanson metric on boundary nodes and provides a complete planar characterization via and Plücker coordinates, connecting to a Grassmannian point and to dual-network relations. A reconstruction framework is developed: minimal networks are recoverable from column rank-patterns of with a rank permutation tied to the strand permutation, and the generalized chamber ansatz recovers conductivities up to electrical transformations. The results enable topology inference in phylogenetic networks and Calderón-type inverse problems by exploiting the Grassmannian viewpoint and the Kalmanson structure of the resistance metric.

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.
Paper Structure (4 sections, 8 theorems, 9 equations, 1 figure)

This paper contains 4 sections, 8 theorems, 9 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Applications

Key Result

Theorem 2.2

CIW Consider an electrical network $\mathcal{E}(G, w).$ Then, there is a matrix $M_R(\mathcal{E})=(x_{ij}) \in Mat_{n \times n}(\mathbb{R})$ such that the following holds: This matrix is called the response matrix of a network $\mathcal{E}(G, \omega).$

Figures (1)

  • Figure 1: Star-shape network, its median graph and the strand permutation $\tau(\mathcal{E})=(14)(36)(25)$

Theorems & Definitions (12)

  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Theorem 2.4
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Definition 4.2
  • Theorem 4.3
  • ...and 2 more