  We develop the theoretical foundations of a generalized Gromov-Hausdorff
distance between functions on networks that has recently been applied to
various subfields of topological data analysis and optimal transport. These
functional representations of networks, or networks for short, specialize in
the finite setting to (possibly asymmetric) adjacency matrices and derived
representations such as distance or kernel matrices. Existing literature
utilizing these constructions cannot, however, benefit from continuous
formulations because the continuum limits of finite networks under this
distance are not well-understood. For example, while there are currently
numerous persistent homology methods on finite networks, it is unclear if these
methods produce well-defined persistence diagrams in the infinite setting. We
resolve this situation by introducing the collection of compact networks that
arises by taking continuum limits of finite networks and developing sampling
results showing that this collection admits well-defined persistence diagrams.
Compared to metric spaces, the isomorphism class of the generalized
Gromov-Hausdorff distance over networks is rather complex, and contains
representatives having different cardinalities and different topologies. We
provide an exact characterization of a suitable notion of isomorphism for
compact networks as well as alternative, stronger characterizations under
additional topological regularity assumptions. Toward data applications, we
describe a unified framework for developing quantitatively stable network
invariants, provide basic examples, and cast existing results on the stability
of persistent homology methods in this extended framework. To illustrate our
theoretical results, we introduce a model of directed circles with finite
reversibility and characterize their Dowker persistence diagrams.
