Exact solution of Kauffman's model with connectivity one

Abstract

Kauffman's model is a randomly assembled network of Boolean automata. Each automaton receives inputs from at most K other automata. Its state at discrete time t+1 is determined by a randomly chosen, but fixed, Boolean function of the K inputs at time t. The resulting quenched, random dynamics of the network demonstrates two phases: a frozen and a chaotic phase. The authors give an exact solution of the model for connectivity K=1, valid everywhere in the frozen phase and at a critical point, valid for finite as well as for infinite networks. They discuss the network's critical behaviour and finite-size effects. The results for the frozen phase presented complement recent exact results for the chaotic phase obtained for K= infinity .