The neurodegenerative disorder amyotrophic lateral sclerosis (ALS) is characterized by the progressive loss of upper and lower motor neurons, with pathological involvement of cerebral motor and extra-motor areas in a clinicopathological spectrum with frontotemporal dementia (FTD). A key unresolved issue is how the non-random distribution of pathology in ALS reflects differential network vulnerability, including molecular factors such as regional gene expression, or preferential spread of pathology via anatomical connections. A system of histopathological staging of ALS based on the regional burden of TDP-43 pathology observed in postmortem brains has been supported to some extent by analysis of distribution of in vivo structural MRI changes. In this paper, computational modeling using a Network Diffusion Model (NDM) was used to investigate whether a process of focal pathological 'seeding' followed by structural network-based spread recapitulated postmortem histopathological staging and, secondly, whether this had any correlation to the pattern of expression of a panel of genes implicated in ALS across the healthy brain. Regionally parcellated T1-weighted MRI data from ALS patients (baseline n=79) was studied in relation to a healthy control structural connectome and a database of associated regional cerebral gene expression. The NDM provided strong support for a structural network-based basis for regional pathological spread in ALS, but no simple relationship to the spatial distribution of ALS-related genes in the healthy brain. Interestingly, OPTN gene was identified as a significant but a weaker non-NDM contributor within the network-gene interaction model (LASSO). Intriguingly, the critical seed regions for spread within the model were not within the primary motor cortex but basal ganglia, thalamus and insula, where NDM recapitulated aspects of the postmortem histopathological staging system. Within the ALS-FTD clinicopathological spectrum, non-primary motor structures may be among the earliest sites of cerebral pathology.