Criticality in random threshold networks: annealed approximation and beyond

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 Kc. In this type of model—contrary to Boolean Networks—propagation of local perturbations (damage) depends on the in-degree of the sites. Kc 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.