  Persistent homology offers a powerful tool for extracting hidden topological
signals from brain networks. It captures the evolution of topological
structures across multiple scales, known as filtrations, thereby revealing
topological features that persist over these scales. These features are
summarized in persistence diagrams, and their dissimilarity is quantified using
the Wasserstein distance. However, the Wasserstein distance does not follow a
known distribution, posing challenges for the application of existing
parametric statistical models.To tackle this issue, we introduce a unified
topological inference framework centered on the Wasserstein distance. Our
approach has no explicit model and distributional assumptions. The inference is
performed in a completely data driven fashion. We apply this method to
resting-state functional magnetic resonance images (rs-fMRI) of temporal lobe
epilepsy patients collected from two different sites: the University of
Wisconsin-Madison and the Medical College of Wisconsin. Importantly, our
topological method is robust to variations due to sex and image acquisition,
obviating the need to account for these variables as nuisance covariates. We
successfully localize the brain regions that contribute the most to topological
differences. A MATLAB package used for all analyses in this study is available
at https://github.com/laplcebeltrami/PH-STAT.
