Extended persistent homology of filtration of graded subgroups

In this work, we presented an extended persistence for filtration of graded subgroups by defining a relative homology in this setting. Our work provided a more comprehensive and flexible approach to obtaining an algebraic invariant and overcoming the limitations of the standard approach. As the application of mathematical tools in data analysis requires stability—meaning small perturbations in the input data should induce only small changes in the output—our main contribution was the development of a stability theorem for of filtration of graded subgroups. This theorem was established using an extension of the definition of interleaving, along with the rectangle measure and functor extension. We demonstrated the theorem's utility by applying it to extended persistence modules obtained from the path homology of directed graphs and the homology of hypergraphs, two important examples in topological data analysis.