  We study the problem of comparing a pair of geometric networks that may not
be similarly defined, i.e., when they do not have one-to-one correspondences
between their nodes and edges. Our motivating application is to compare power
distribution networks of a region. Due to the lack of openly available power
network datasets, researchers synthesize realistic networks resembling their
actual counterparts. But the synthetic digital twins may vary significantly
from one another and from actual networks due to varying underlying assumptions
and approaches. Hence the user wants to evaluate the quality of networks in
terms of their structural similarity to actual power networks. But the lack of
correspondence between the networks renders most standard approaches, e.g.,
subgraph isomorphism and edit distance, unsuitable.
  We propose an approach based on the multiscale flat norm, a notion of
distance between objects defined in the field of geometric measure theory, to
compute the distance between a pair of planar geometric networks. Using a
triangulation of the domain containing the input networks, the flat norm
distance between two networks at a given scale can be computed by solving a
linear program. In addition, this computation automatically identifies the 2D
regions (patches) that capture where the two networks are different. We
demonstrate through 2D examples that the flat norm distance can capture the
variations of inputs more accurately than the commonly used Hausdorff distance.
As a notion of stability, we also derive upper bounds on the flat norm distance
between a simple 1D curve and its perturbed version as a function of the radius
of perturbation for a restricted class of perturbations. We demonstrate our
approach on a set of actual power networks from a county in the USA.
