On network deconvolution for undirected graphs

Network deconvolution (ND) is a method to reconstruct a direct-effect network describing direct (or conditional) effects (or associations) between any two nodes from a given network depicting total (or marginal) effects (or associations). Its key idea is that, in a directed graph, a total effect can be decomposed into the sum of a direct and an indirect effects, with the latter further decomposed as the sum of various products of direct effects. This yields a simple closed-form solution for the direct-effect network, facilitating its important applications to distinguish direct and indirect effects. Despite its application to undirected graphs, it is not well known why the method works, leaving it with skepticism. We first clarify the implicit linear model assumption underlying ND, then derive a surprisingly simple result on the equivalence between ND and use of precision matrices, offering insightful justification and interpretation for the application of ND to undirected graphs. We also establish a formal result to characterize the effect of scaling a total-effect graph. Finally, leveraging large-scale genome-wide association study data, we show a novel application of ND to contrast marginal versus conditional genetic correlations between body height and risk of coronary artery disease; the results align with an inferred causal directed graph using ND. We conclude that ND is a promising approach with its easy and wide applicability to both directed and undirected graphs.