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 \alpha \approx }  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 , suggesting that even weak topological correlations can strongly limit its applicability for finite N .
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 with \alpha \approx } 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 ^{\gamma} is studied; we find that a decrease of \gamma can lead to a transition from supercritical (chaotic) to subcritical (ordered) dynamics} ; 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 .
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|h|^{\alpha
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h^2/(
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|h|^{\alpha
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. Further, damage propagation in RTN with in-degree distributions that exhibit a scale-free tail k_{in
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; the cross-over point yields a novel, characteristic connectivity K_d, that has no counterpart in Boolean networks
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Interestingly, in this case the annealed approximation fails to predict the dynamical behavior for sparse connectivities
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, suggesting that even weak topological correlations can strongly limit its applicability for finite N
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