  Randomized experiments on a network often involve interference between
connected units; i.e., a situation in which an individual's treatment can
affect the response of another individual. Current approaches to deal with
interference, in theory and in practice, often make restrictive assumptions on
its structure---for instance, assuming that interference is local---even when
using otherwise nonparametric inference strategies. This reliance on explicit
restrictions on the interference mechanism suggests a shared intuition that
inference is impossible without any assumptions on the interference structure.
In this paper, we begin by formalizing this intuition in the context of a
classical nonparametric approach to inference, referred to as design-based
inference of causal effects. Next, we show how, always in the context of
design-based inference, even parametric structural assumptions that allow the
existence of unbiased estimators, cannot guarantee a decreasing variance even
in the large sample limit. This lack of concentration in large samples is often
observed empirically, in randomized experiments in which interference of some
form is expected to be present. This result has direct consequences for the
design and analysis of large experiments---for instance, in online social
platforms---where the belief is that large sample sizes automatically guarantee
small variance. More broadly, our results suggest that although strategies for
causal inference in the presence of interference borrow their formalism and
main concepts from the traditional causal inference literature, much of the
intuition from the no-interference case do not easily transfer to the
interference setting.
