  Persistent homology (PH) characterizes the shape of brain networks through
the 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 the finite number of Fourier coefficients with respect to
the Laplace-Beltrami (LB) eigenfunction expansion of the domain, which provides
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 through the 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 with respect
to statistical variation of topological noise and hole location. We also
illustrate how the method extends naturally into a clustering scheme by
subtyping individuals with post-stroke aphasia through the PDs of their
resting-state functional brain networks.
