When correlations matter - response of dynamical networks to small perturbations

We systematically study and compare damage spreading for random Boolean and threshold networks under small external perturbations (damage), a problem which is relevant to many biological networks. We identify a new characteristic connectivityKs, at which the average number of damaged nodes after a large number of dynamical updates is independent of the total number of nodes N . We estimate the critical connectivity for finite N and show that it systematically deviates from the annealed approximation. Extending the approach followed in a previous study [11], we present new results indicating that internal dynamical correlations tend to increase not only the probability for small, but also for very large damage events, leading to a broad, fat-tailed distribution of damage sizes. These findings indicate that the descriptive and predictive value of averaged order parameters for finite size networks even for biologically highly relevant sizes up to several thousand nodes is limited.