  We consider the problem of how to assign treatment in a randomized
experiment, in which the correlation among the outcomes is informed by a
network available pre-intervention. Working within the potential outcome causal
framework, we develop a class of models that posit such a correlation structure
among the outcomes. Then we leverage these models to develop restricted
randomization strategies for allocating treatment optimally, by minimizing the
mean square error of the estimated average treatment effect. Analytical
decompositions of the mean square error, due both to the model and to the
randomization distribution, provide insights into aspects of the optimal
designs. In particular, the analysis suggests new notions of balance based on
specific network quantities, in addition to classical covariate balance. The
resulting balanced, optimal restricted randomization strategies are still
design unbiased, in situations where the model used to derive them does not
hold. We illustrate how the proposed treatment allocation strategies improve on
allocations that ignore the network structure, with extensive simulations.
