The spatial distribution of atrophy in neurodegenerative diseases suggests that brain connectivity mediates disease propagation. Different descriptors of the connectivity graph potentially relate to different underlying mechanisms of propagation. Previous approaches for evaluating the influence of connectivity on neurodegeneration consider each descriptor in isolation and match predictions against late-stage atrophy patterns. We introduce the notion of a topological profile - a characteristic combination of topological descriptors that best describes the propagation of pathology in a particular disease. By drawing on recent advances in disease progression modeling, we estimate topological profiles from the full course of pathology accumulation, at both cohort and individual levels. Experimental results comparing topological profiles for Alzheimer's disease, multiple sclerosis and normal ageing show that topological profiles explain the observed data better than single descriptors. Within each condition, most individual profiles cluster around the cohort-level profile, and individuals whose profiles align more closely with other cohort-level profiles show features of that cohort. The cohort-level profiles suggest new insights into the biological mechanisms underlying pathology propagation in each disease.