Distances and isomorphism between networks: stability and convergence of network invariants

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