  We study the calculation of exact p-values for a large class of non-sharp
null hypotheses about treatment effects in a setting with data from experiments
involving members of a single connected network. The class includes null
hypotheses that limit the effect of one unit's treatment status on another
according to the distance between units; for example, the hypothesis might
specify that the treatment status of immediate neighbors has no effect, or that
units more than two edges away have no effect. We also consider hypotheses
concerning the validity of sparsification of a network (for example based on
the strength of ties) and hypotheses restricting heterogeneity in peer effects
(so that, for example, only the number or fraction treated among neighboring
units matters). Our general approach is to define an artificial experiment,
such that the null hypothesis that was not sharp for the original experiment is
sharp for the artificial experiment, and such that the randomization analysis
for the artificial experiment is validated by the design of the original
experiment.
