Structure determines function. However, this universal theme in biology has been surprisingly difficult to observe in human brain neuroimaging data. Here, we link structure to function by hypothesizing that brain signals propagate as a Markovian process on an underlying structure. We focus on a metric called commute time: the average number of steps for a random walker to go from region A to B and then back to A. Commute times based on white matter tracts from diffusion MRI exhibit an average ± standard deviation Spearman correlation of -0.26 ± 0.08 with functional MRI connectivity data across 434 UK Biobank individuals and -0.24 ± 0.06 across 400 HCP Young Adult brain scans. The correlation increases to -0.36 ± 0.14 and to -0.32 ± 0.12 when the principal contributions of both commute time and functional connectivity are compared for both datasets. The observed weak but robust correlations provide evidence of a relationship, albeit restricted, between neuronal connectivity and brain function. The correlations are stronger by 33% compared to broadly used communication measures such as search information and communicability. The difference further widens to a factor of 5 when commute times are correlated to the principal mode of functional connectivity from its eigenvalue decomposition. Overall, the study points to the utility of commute time to account for the role of polysynaptic (indirect) connectivity underlying brain function.