Some Statistical Properties of Functional Norms of Persistence Topological Landscapes

We study the relationships between the mean of functional L^p-norms of persistence topological landscapes of noisy datasets and statistical properties of the underlying probability distributions. We establish a functional relationship of the average L^p-norms to variability of datasets sampled from a broad class of distributions. Furthermore, we conduct numerical experiments with bivariate time series generated by stochastic processes with known properties, which reveal that increase of L^p-norms due to rising variability can be suppressed by a growing covariance in the system. These general results help to understand the puzzling behavior of these topological summaries, which we observed on US and European stock markets during the early phase of the global meltdown caused by the COVID-19 pandemic, and could aid in a topological identification of the approaching regime changes in complex systems