Cognitive and behavioral deficits in Alzheimer's disease (AD) and frontotemporal dementia (FTD) result from brain atrophy and altered functional connectivity. However, it is unclear how atrophy relates to functional connectivity disruptions across dementia subtypes and stages. We addressed this question using structural and functional MRI from 221 patients with AD (n=82), behavioral variant FTD (n=41), corticobasal syndrome (n=27), nonfluent (n=34) and semantic (n=37) variant primary progressive aphasia, and 100 cognitively normal individuals. Using partial least squares regression, we identified three principal structure-function components. The first component showed overall atrophy correlating with primary cortical hypo-connectivity and subcortical/association cortical hyper-connectivity. Components two and three linked focal syndrome-specific atrophy to peri-lesional hypo-connectivity and distal hyper-connectivity. Structural and functional component scores predicted global and domain-specific cognitive deficits. Anatomically, functional connectivity changes reflected alterations in specific brain activity gradients. Eigenmode analysis identified temporal phase and amplitude collapse as an explanation for atrophy-driven functional connectivity changes.