  The noise sensitivity of a Boolean function describes its likelihood to flip
under small perturbations of its input. Introduced in the seminal work of
Benjamini, Kalai and Schramm [Inst. Hautes \'{E}tudes Sci. Publ. Math. 90
(1999) 5-43], it was there shown to be governed by the first level of Fourier
coefficients in the central case of monotone functions at a constant critical
probability $p_c$. Here we study noise sensitivity and a natural stronger
version of it, addressing the effect of noise given a specific witness in the
original input. Our main context is the Erd\H{o}s-R\'{e}nyi random graph, where
already the property of containing a given graph is sufficiently rich to
separate these notions. In particular, our analysis implies (strong) noise
sensitivity in settings where the BKS criterion involving the first Fourier
level does not apply, for example, when $p_c\to0$ polynomially fast in the
number of variables.
