Neurodegeneration has been hypothesized to follow predetermined large-scale networks through the trans-synaptic spread of toxic proteins from a syndrome-specific epicentre. To date, no longitudinal neuroimaging study has tested this hypothesis in vivo in frontotemporal dementia spectrum disorders. The aim of this study was to demonstrate that longitudinal progression of atrophy in non-fluent/agrammatic variant primary progressive aphasia spreads over time from a syndrome-specific epicentre to additional regions, based on their connectivity to the epicentre in healthy control subjects. The syndrome-specific epicentre of the non-fluent/agrammatic variant of primary progressive aphasia was derived in a group of 10 mildly affected patients (clinical dementia rating equal to 0) using voxel-based morphometry. From this region, the inferior frontal gyrus (pars opercularis), we derived functional and structural connectivity maps in healthy controls (n = 30) using functional magnetic resonance imaging at rest and diffusion-weighted imaging tractography. Graph theory analysis was applied to derive functional network features. Atrophy progression was calculated using voxel-based morphometry longitudinal analysis on 34 non-fluent/agrammatic patients. Correlation analyses were performed to compare volume changes in patients with connectivity measures of the healthy functional and structural speech/language network. The default mode network was used as a control network. From the epicentre, the healthy functional connectivity network included the left supplementary motor area and the prefrontal, inferior parietal and temporal regions, which were connected through the aslant, superior longitudinal and arcuate fasciculi. Longitudinal grey and white matter changes were found in the left language-related regions and in the right inferior frontal gyrus. Functional connectivity strength in the healthy speech/language network, but not in the default network, correlated with longitudinal grey matter changes in the non-fluent/agrammatic variant of primary progressive aphasia. Graph theoretical analysis of the speech/language network showed that regions with shorter functional paths to the epicentre exhibited greater longitudinal atrophy. The network contained three modules, including a left inferior frontal gyrus/supplementary motor area, which was most strongly connected with the epicentre. The aslant tract was the white matter pathway connecting these two regions and showed the most significant correlation between fractional anisotropy and white matter longitudinal atrophy changes. This study showed that the pattern of longitudinal atrophy progression in the non-fluent/agrammatic variant of primary progressive aphasia relates to the strength of connectivity in pre-determined functional and structural large-scale speech production networks. These findings support the hypothesis that the spread of neurodegeneration occurs by following specific anatomical and functional neuronal network architectures.