  We systematically study and compare damage spreading at the sparse
percolation (SP) limit for random boolean and threshold networks with
perturbations that are independent of the network size $N$. This limit is
relevant to information and damage propagation in many technological and
natural networks. Using finite size scaling, we identify a new characteristic
connectivity $K_s$, at which the average number of damaged nodes $\bar d$,
after a large number of dynamical updates, is independent of $N$. Based on
marginal damage spreading, we determine the critical connectivity
$K_c^{sparse}(N)$ for finite $N$ at the SP limit and show that it
systematically deviates from $K_c$, established by the annealed approximation,
even for large system sizes. Our findings can potentially explain the results
recently obtained for gene regulatory networks and have important implications
for the evolution of dynamical networks that solve specific computational or
functional tasks.
