Artificial Intelligence integrated framework for stability of functions in persistent homology

Explaining the spatial properties in point set topological spaces is a tough task. Experts in TDA have sought to discover if it is possible to find a strong intuition about the geometry and topology in big datasets, easily seen when dealing with all of them at the same time. This point also notes that the estimates will stay useful if we can detect whether the constants remain stable as the data changes, for instance, the Hausdorff distance function when the data exhibits noise, or even when a little noise is added to the point-cloud datapoints. This could happen if these properties are considered in topologically invariant compact subsets of X, which requires very stringent and restrictive assumptions to obtain well-defined shapes that can be drawn from the data in the compact subsets. The aim of this study is to outline factors that make functions in persistent homology stable. The results show that factors like triangularability affect stability of functions. Moreover, we have given an integrated artificial intelligence (AI) framework for stability of functions, to trace the accuracy levels of algorithms in cyber-threat identification and cyber-threat attacks using Persistent Homology.