Persistent Homology based Wasserstein distance for graph networks

The technique of measuring similarity between topological spaces using Wasserstein distance between persistence diagrams is extended to graph networks in this paper. A relationship between the Wasserstein distance of the Cartesian product of topological spaces and the Wasserstein distance of individual spaces is found to ease the comparative study of the Cartesian product of topological spaces. The Cartesian product and the strong product of weighted graphs are defined, and the relationship between the Wasserstein distance between graph products and the Wasserstein distance between individual graphs is determined. For this, clique complex filtration and the Vietoris- Rips filtration are used.