  The possibility of nonclassicality in networks unrelated to Bell's original
eponymous theorem has recently attracted significant interest. Here, we
identify a sufficient condition for being "outside the shadow of Bell's
theorem" and introduce a testable criterion capable of certifying the novelty
of instances of network-nonclassicality which we call minimal network
nonclassicality. We provide examples of minimally network nonclassical
correlations realizable in quantum theory as well as examples coming from more
exotic operational probabilistic theories. In particular, we apply these
concepts to the simplest configuration of the 3-chain scenario (a.k.a. the
bilocality scenario) to prove that certain correlations have escaped the shadow
of Bell's theorem. While some of the examples herein are unprecedented, we also
revisit more familiar examples of network nonclassicality in order to highlight
the contrast between our approach versus prior approaches with respect to
assessing novelty.
