Spectral permutation test on persistence diagrams

Brain networks constructed from diffusion and functional magnetic resonance imaging (dMRI and fMRI) are typically investigated through graph theoretic models. It has recently been noted that the complexity of brain connectivity may not be sufficiently captured by single-scale models and multi-scale models are needed. Persistent homology (PH) is an algorithm that extracts multi-scale features in brain networks that cannot be easily decoded by standard network analysis. It summarizes topological structures in a network through multi-scale descriptors such as persistence diagram (PD). Various statistical inference procedures have been developed for PDs. In this study, we propose a novel spectral permutation test on PDs by permuting Fourier coefficients from heat kernel estimation of the PDs. The method is applied to test if the connectivity of diffusion and resting-state functional networks within two types of post-stroke aphasia undergo changes across baseline and first treatment visits.