**a.** Direct edges in a network (solid blue arrows) can lead to indirect relationships (dashed red arrows) as a result of transitive information flow. These indirect contributions can be of length two (e.g. 1→2→3), three (e.g. 1→2→3→5) or higher, and can combine both direct and indirect effects (e.g. 2→4), and multiple indirect effects along varying paths (e.g. 2→3→5, 2→4→5). Self-loops are excluded from networks. Network deconvolution seeks to reverse the effect of transitive information flow across all indirect paths, in order to recover the true direct network (blue edges, G_{dir}) based on the observed network (combined blue and red edges, G_{obs}). **b.** Algebraically, the transitive closure of a network can be expressed as an infinite sum of the true direct network and all indirect effects along paths of increasing lengths, which can be written in a closed form as an infinite-series sum. Network deconvolution exploits this closed form to express the direct network G_{dir} as a function of the observed network G_{obs}. **c.** To efficiently compute this inverse operation, we express both the true and observed networks G_{dir} and G_{obs} by decomposition into their eigenvectors and eigenvalues, which enables each eigenvalue λ_{i} ^{dir} of the original network to be expressed as a nonlinear function of a single corresponding eigenvalue λ_{i} ^{obs} of the convolved observed network. **d,e.** Network deconvolution assumes that indirect flow weights can be approximated as the product of direct edge weights and that observed edge weights are the sum of direct and indirect flows. When these assumptions hold **(d)**, network deconvolution removes all indirect flow effects and infers all direct interactions and weights exactly. Even when these assumptions do not hold **(e)**, ND infers 87% of direct edges, showing robustness to non-linear effects.

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