  We calculate analytically the critical connectivity $K_c$ of Random Threshold
Networks (RTN) for homogeneous and inhomogeneous thresholds, and confirm the
results by numerical simulations. We find a super-linear increase of $K_c$ with
the (average) absolute threshold $|h|$, which approaches $K_c(|h|) \sim
h^2/(2\ln{|h|})$ for large $|h|$, and show that this asymptotic scaling is
universal for RTN with Poissonian distributed connectivity and threshold
distributions with a variance that grows slower than $h^2$. Interestingly, we
find that inhomogeneous distribution of thresholds leads to increased
propagation of perturbations for sparsely connected networks, while for densely
connected networks damage is reduced; the cross-over point yields a novel,
characteristic connectivity $K_d$, that has no counterpart in Boolean networks.
Last, local correlations between node thresholds and in-degree are introduced.
Here, numerical simulations show that even weak (anti-)correlations can lead to
a transition from ordered to chaotic dynamics, and vice versa. It is shown that
the naive mean-field assumption typical for the annealed approximation leads to
false predictions in this case, since correlations between thresholds and
out-degree that emerge as a side-effect strongly modify damage propagation
behavior.
