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 diagram (PD) through heat diffusion reparameterizes using a finite number of Fourier coefficients with respect to the Laplace–Beltrami (LB) eigenfunction expansion of the domain, thus providing a powerful vectorized algebraic representation for group comparisons of PDs. In this study, we advance a transposition-based permutation test for comparing multiple groups of PDs using their heat-diffusion estimates of the PDs. We evaluate the empirical performance of the spectral transposition test in capturing within-and between-group similarity and dissimilarity under statistical variation in topological noise and cycle location. In application, we introduce a topological lesion symptom mapping (TLSM) method based on the proposed topological inference framework. The method is applied to resting-state functional brain networks from individuals with post-stroke aphasia to identify characteristic cycles associated with varying degrees of speech-language impairment, as measured by behavioral test scores. © Institute of Mathematical Statistics, 2026.