Hyperbolic embedding of brain networks detects regions disrupted by neurodegeneration in Alzheimer's disease

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 a polar representation of the hyperbolic plane, the hyperbolic disk. Here, we used a geometric score, entirely based on the computation of distances between nodes in the latent space, to measure the effect of a perturbation on the nodes. Precisely, the score is a local measure of distortion in the geometric neighborhood of a node following a perturbation. The method is applied to a brain network dataset of patients with AD and healthy participants, derived from diffusion-weighted (DWI) and functional magnetic resonance (fMRI) imaging scans. We show that, compared with standard graph measures, our method more accurately identifies the brain regions most affected by neurodegeneration. Notably, the abnormalities detected 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 the progression of the disease.