  Randomized experiments in which the treatment of a unit can affect the
outcomes of other units are becoming increasingly common in healthcare,
economics, and in the social and information sciences. From a causal inference
perspective, the typical assumption of no interference becomes untenable in
such experiments. In many problems, however, the patterns of interference may
be informed by the observation of network connections among the units of
analysis. Here, we develop elements of optimal estimation theory for causal
effects leveraging an observed network, by assuming that the potential outcomes
of an individual depend only on the individual's treatment and on the treatment
of the neighbors. We propose a collection of exclusion restrictions on the
potential outcomes, and show how subsets of these restrictions lead to various
parameterizations. Considering the class of linear unbiased estimators of the
average direct treatment effect, we derive conditions on the design that lead
to the existence of unbiased estimators, and offer analytical insights on the
weights that lead to minimum integrated variance estimators. We illustrate the
improved performance of these estimators when compared to more standard biased
and unbiased estimators, using simulations.
