Topological inference on brain networks with application to lesion symptom mapping

Persistent homology (PH) characterizes the shape of brain networks through persistence features. Group comparison of persistence features from brain networks can be challenging as they are inherently heterogeneous. A recent scale-space representation of persistence diagrams (PDs) through heat diffusion reparameterizes them using a finite number of Fourier coefficients with respect to the Laplace--Beltrami (LB) eigenfunction expansion of the domain, providing a powerful vectorized algebraic representation for group comparisons. In this study, we develop a transposition-based permutation test for comparing multiple groups of PDs using heat-diffusion estimates. We evaluate the empirical performance of the spectral transposition test in capturing within- and between-group similarity and dissimilarity under varying levels of topological noise and cycle location variability. In application, we propose a topological lesion symptom mapping (TLSM) method based on the proposed framework. The method is applied to resting-state functional brain networks of individuals with post-stroke aphasia to identify characteristic cycles associated with varying levels of speech-language impairment.