  Random Threshold Networks with sparse, asymmetric connections show complex
dynamical behavior similar to Random Boolean Networks, with a transition from
ordered to chaotic dynamics at a critical average connectivity $K_c$. In this
type of model - contrary to Boolean Networks - propagation of local
perturbations (damage) depends on the in-degree of the sites. $K_c$ is
determined analytically, using an annealed approximation, and the results are
confirmed by numerical simulations. It is shown that the statistical
distributions of damage spreading near the percolation transition obey
power-laws, and dynamical correlations between active network clusters become
maximal. We investigate the effect of local damage suppression at highly
connected nodes for networks with scale-free in-degree distributions. Possible
relations of our findings to properties of real-world networks, like robustness
and non-trivial degree-distributions, are discussed.
