Misc,
The Critical Line in Random Threshold Networks with Inhomogeneous Thresholds
(July 2007)
Abstract
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|^\alpha\$ with \$2\$ 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|^\alpha\$. 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.
Further, damage propagation in RTN with in-degree distributions that exhibit a
scale-free tail \$k\_in^\gamma\$ is studied; we find that a decrease of
\$\gamma\$ can lead to a transition from supercritical (chaotic) to subcritical
(ordered) dynamics. 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. Interestingly, in this case the annealed approximation fails to
predict the dynamical behavior for sparse connectivities \$K\$, suggesting
that even weak topological correlations can strongly limit its applicability
for finite N.
