  Graph theoretical methods have proven valuable for investigating alterations
in both anatomical and functional brain connectivity networks during
Alzheimer's disease (AD). Recent studies suggest that representing brain
networks in a suitable geometric space can better capture their connectivity
structure. This study introduces a novel approach to characterize brain
connectivity changes using low-dimensional, informative representations of
networks in a latent geometric space. Specifically, the networks are embedded
in the Poincar\'e disk model of hyperbolic geometry. Here, we define a local
measure of distortion of the geometric neighborhood of a node following a
perturbation. The method is applied to a brain networks dataset of patients
with AD and healthy participants, derived from DWI and fMRI scans. We show
that, compared with standard graph measures, our method identifies more
accurately the brain regions most affected by neurodegeneration. Notably, the
abnormality detection in memory-related and frontal areas are robust across
multiple brain parcellation scales. Finally, our findings suggest that the
geometric perturbation score could serve as a potential biomarker for
characterizing disease progression.
