  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.
