  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.
