Layering $\partial$-Graphs and Networks

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.