Random Networks of Automata: A Simple Annealed Approximation

Abstract

Kauffman's model is a random complex automata where nodes are randomly assembled. Each node σ_i receives K inputs from K randomly chosen nodes and the values of σ_i at time t + 1 is a random Boolean function of the K inputs at time t. Numerical simulations have shown that the behaviour of this model is very different for K > 2 and K ≤ 2. It is the purpose of this work to give a simple annealed approximation which predicts K = 2 as the critical value of K. This approximation gives also quantitative predictions for distances between iterated configurations. These predictions agree rather well with the numerical simulations. A possible way of improving this annealed approximation is proposed.