  Causal inference for network data is an area of active interest in the social
sciences. Unfortunately, the complicated dependence structure of network data
presents an obstacle to many causal inference procedures. We consider the task
of mediation analysis for network data, and present a model in which mediation
occurs in a latent embedding space. Under this model, node-level interventions
have causal effects on nodal outcomes, and these effects can be partitioned
into a direct effect independent of the network, and an indirect effect induced
by homophily. To estimate network-mediated effects, we embed nodes into a
low-dimensional space and fit two regression models: (1) an outcome model
describing how nodal outcomes vary with treatment, controls, and position in
latent space; and (2) a mediator model describing how latent positions vary
with treatment and controls. We prove that the estimated coefficients are
asymptotically normal about the true coefficients under a sub-gamma
generalization of the random dot product graph, a widely-used latent space
model. We show that these coefficients can be used in product-of-coefficients
estimators for causal inference. Our method is easy to implement, scales to
networks with millions of edges, and can be extended to accommodate a variety
of structured data.
