Relevant elements, magnetization and dynamical properties in Kauffman networks: A numerical study

This is the first of two papers about the structure of Kauffman networks. In this paper we define the relevant elements of random networks of automata, following previous work by Flyvbjerg [J. Phys. A 21 (1988) L955-L960] and Flyvbjerg and Kjaer [J. Phys. A 21 (1988) 1695-1718], and we study numerically their probability distributions in the chaotic phase and on the critical line of the model. A simple approximate argument predicts that their number scales as √N on the critical line, while it is linear with N in the chaotic phase and independent on system size in the frozen phase. This argument is confirmed by numerical results. The study of the relevant elements gives useful information about the properties of the attractors in critical networks, where the pictures coming from either approximate computation methods or from simulations are not very clear.