Fig. 4.

Systematic evaluation of the correlations in the dynamics generated by different rules. We quantify the long-range correlations in the dynamics by means of the detrended fluctuation-analysis exponent α (5) systematically estimated for time scales 40 < n < 4,000. We show α for 3,721 pairs of values of k_e and the noise η in the communication between the units comprising the network. For all simulations, we follow the time evolution of systems comprising 4,096 units for a transient period lasting 8,192 time steps, and we then record the time evolution of the system for an additional 10,000 time steps. To avoid artifacts due to the fact that the units switch states with period 2 for some of the rules, we consider in our analysis the state of the systems at every other time step. (a) RBN as defined by Kauffman (10). Our results show that the dynamics generated by these systems are generally of the white-noise type, with a weak dependence on the noise intensity and no dependence on the number of long-distance links. (b) Rule 232, also known as the majority rule. This rule is representative of two other rules: rules 19 and 1. Rule 232 displays a very rich phase space with various dynamical behaviors all of the way from white noise (white and green) to Brownian noise (black). (c) Rule 50 is a threshold rule with refractory period. This rule is representative of eight other rules: rules 5, 36, 37, 73, 77, 94, 108, and 164. These rules display a relatively simple phase space with behaviors extending from white noise to 1/f noise. The 1/f behavior is restricted to very small noise intensities and there is a very weak dependence on k_e. (d) Rule 104. This rule is representative of 12 other rules (see Figs. 7, 8, 9, 10, 11). Their phase space is extremely simple because it displays only white-noise behavior.