Bridging the gap between symmetric, direct white matter brain connectivity and neural dynamics that are often asymmetric and polysynaptic may offer insights into brain architecture, but this remains an unresolved challenge in neuroscience. Here, we used the graph Laplacian matrix to simulate symmetric and asymmetric high-order diffusion processes akin to particles spreading through white matter pathways. The simulated indirect structural connectivity outperformed direct as well as absent anatomical information in sculpting effective connectivity, a measure of causal and directed brain dynamics. Crucially, an asymmetric diffusion process determined by the sensitivity of the network nodes to their afferents best predicted effective connectivity. The outcome is consistent with brain regions adapting to maintain their sensitivity to inputs within a dynamic range. Asymmetric network communication models offer a promising perspective for understanding the relationship between structural and functional brain connectomes, both in normalcy and neuropsychiatric conditions. Author Summary Measures of white matter connectivity can usefully inform models of causal and directed brain communication (i.e., effective connectivity). However, due to the inherent differences in biophysical correlates, recording techniques and analytic approaches, the relationship between anatomical and effective brain connectivity is complex and not fully understood. In this study, we use simulation of heat diffusion constrained by the anatomical connectivity of the network to model polysynaptic (high-order) anatomical connectivity. The outcomes afford more useful constraints on effective connectivity than conventional, typically monosynaptic white matter connectivity. Furthermore, asymmetric network diffusion best predicts effective connectivity. In conclusion, the data provide insights into how anatomical connectomes give rise to asymmetric neuronal message passing and brain communication.