  We show that depending on the disorder, a small noise added to the threshold
distribution of the fuse network may or may not completely change the
subsequent breakdown process. When the threshold distribution has a lower
cutoff at a finite value and a power law dependence towards large thresholds
with an exponent which is less than $0.16\pm0.03$, the network is not sensitive
to the added noise, otherwise it is. The transition between sensitivity or not
appears to be second order, and is related to a localization-delocalization
transition earlier observed in such systems.
