  We systematically study and compare damage spreading for random Boolean and
threshold networks under small external perturbations (damage), a problem which
is relevant to many biological networks. We identify a new characteristic
connectivity $K_s$, at which the average number of damaged nodes after a large
number of dynamical updates is independent of the total number of nodes $N$. We
estimate the critical connectivity for finite $N$ and show that it
systematically deviates from the annealed approximation. Extending the approach
followed in a previous study, we present new results indicating that internal
dynamical correlations tend to increase not only the probability for small, but
also for very large damage events, leading to a broad, fat-tailed distribution
of damage sizes. These findings indicate that the descriptive and predictive
value of averaged order parameters for finite size networks - even for
biologically highly relevant sizes up to several thousand nodes - is limited.
