On Topological Data Analysis Techniques for stability of functions in persistent homology

Describing topological and geometric properties in point set topological spaces and metric spaces has remained inherently difficult over a long period of time to mathematicians. Studies in topological data analysis (TDA) have revolved around the question that asks: Is there a possibility of finding a robust estimate of the properties of geometry and topology of Big data sets within a global setting that preserves its structure? This question further posses the problem, that the estimates can only remain of use provided that we could determine the stability of constants given certain variations within the data inputs are unchanged. In this regard, this study seeks to characterize stability of functions in persistent homology (PH) and their applications in cyber-threat detection and cyber-attack prediction within the cybersecurity data sets, using persistent homology.