Layering $\partial$-Graphs and Networks

TL;DR

This paper develops a layer-stripping method using harmonic continuation and scaffolds to solve the inverse problem for countable electrical networks, generalizing previous results and connecting to algebraic frameworks.

Contribution

It formalizes the concept of recoverability by scaffolds and demonstrates its preservation under various graph operations, extending inverse problem solutions to broader classes of networks.

Findings

Recoverability by scaffolds is sufficient for solving the inverse problem.

The method applies to critical circular planar graphs using medial graphs.

Connections between harmonic continuation and algebraic frameworks are established.

Abstract

We consider the inverse problem for countable, locally finite electrical networks with edge weights in an arbitrary field. The electrical inverse problem seeks to determine the weights of the edges knowing only the potential and current data of harmonic functions on a set of boundary nodes. Motivated by the results of Curtis-Ingerman-Morrow and de-Verdiere-Gitler-Vertigan and others, we formalize the idea of using layer-stripping and harmonic continuation to solve the inverse problem. Our strategy is to iteratively recover "vulnerable" edges near the boundary, then remove them by deletion or contraction. To recover the vulnerable edge, we set up a clever boundary value problem and solve it using discrete harmonic continuation. We define "scaffolds," a set of oriented edges that models the flow of information in harmonic continuation. We formulate a sufficient geometric condition ("recoverability by scaffolds") for the inverse problem to be solvable using the layer-stripping strategy. Recoverability by scaffolds is preserved under box products, harmonic subgraphs, covering graphs, and more generally under preimages by unramified harmonic morphisms. For critical circular planar graphs, we prove recoverability by scaffolds using the medial graph. We also connect the harmonic continuation process to Baez-Fong's compositional framework for networks and Lam-Pylyavskyy's electrical linear group. We use this to generalize results of Curtis-Ingerman-Morrow and de-Verdiere-Gitler-Vertigan relating the size of connections through the graph and the rank of submatrices of the response matrix. We give a symplectic characterization of the boundary behavior for networks and the electrical linear group, valid for fields other than $\mathbb{F}_2$. Many of our results also generalize to the nonlinear networks such as those of Johnson.