  Bell's theorem is typically understood as the proof that quantum theory is
incompatible with local-hidden-variable models. More generally, we can see the
violation of a Bell inequality as witnessing the impossibility of explaining
quantum correlations with classical causal models. The violation of a Bell
inequality, however, does not exclude classical models where some level of
measurement dependence is allowed, that is, the choice made by observers can be
correlated with the source generating the systems to be measured. Here, we show
that the level of measurement dependence can be quantitatively upper bounded if
we arrange the Bell test within a network. Furthermore, we also prove that
these results can be adapted in order to derive nonlinear Bell inequalities for
a large class of causal networks and to identify quantumly realizable
correlations that violate them.
