The Critical Line in Random Threshold Networks with Inhomogeneous Thresholds

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 a power-law with an exponent at the order of two for large thresholds, and show that this asymptotic scaling is universal for RTN with Poissonian distributed connectivity and threshold distributions with a variance increasing (approximately) less than quadratically with $|h|$. 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 is studied; we find that a decrease of the scaling exponent 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 system sizes.