A multi-scale geometric approach to compare healthy and pathological brain networks

Structural connectivity of the human brain is studied at a multi-scale level applying a geometric renormalization protocol, by coarse-graining and averaging over short similarity distances. I apply the transformation to networks embedded in a geometric space according to a model in which distances are hyperbolic, that was found in the literature to successfully riproduce the multi-scale properties of brain networks at different anatomical resolutions. I tested the difference in many global and local topological properties (at each scale) between two groups, composed by healthy subjects and Alzheimer diseased patients. This simple analysis is meant to apply, for the first time, a framework for the multi-scale unfolding of complex networks to the identification and description of distintive features of brains affected by this pathology. For some of the properties analyzed, the two groups are found to be indistinguishable at the original scale, while not inditinguishable anymore on down-scaled networks, suggesting the possibility for a theoretical description of how this disease is affecting brain functionality at multiple, interconnected scales.