Phase transition in NK-Kauffman networks and its correction for Boolean irreducibility

In a series of articles published in 1986, Derrida and his colleagues studied two mean field treatments (the quenched and the annealed) for NK-Kauffman networks. Their main results lead to a phase transition curve K c 2 p c ( 1 − p c ) = 1 ( 0 < p c < 1 ) for the critical average connectivity K c in terms of the bias p c of extracting a “ 1 ” for the output of the automata. Values of K bigger than K c correspond to the so-called chaotic phase, while K < K c , to an ordered phase. In Zertuche (2009), a new classification for the Boolean functions, called Boolean irreducibility, permitted the study of new phenomena of NK-Kauffman networks. In the present work we study once again the mean field treatment for NK-Kauffman networks, correcting it for Boolean irreducibility. A shifted phase transition curve is found. In particular, for p c = 1 / 2 the predicted value K c = 2 by Derrida et al. changes to K c = 2.62140224613 … . We support our results with numerical simulations.