This paper studies causal inference with observational network data. A challenging aspect of this setting is the possibility of interference in both potential outcomes and selection into treatment, for example due to peer effects in either stage. We therefore consider a nonparametric setup in which both stages are reduced forms of simultaneous-equations models. This results in high-dimensional network confounding, where the network and covariates of all units constitute sources of selection bias. The literature predominantly assumes that confounding can be summarized by a known, low-dimensional function of these objects, and it is unclear what selection models justify common choices of functions. We show that graph neural networks (GNNs) are well suited to adjust for high-dimensional network confounding. We establish a network analog of approximate sparsity under primitive conditions on interference. This demonstrates that the model has low-dimensional structure that makes estimation feasible and justifies the use of shallow GNN architectures.

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