Phase Transition in NK-Kauffman Networks and its Correction for Boolean Irreducibility

Original · EN

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 Kc 2 pc (1 - pc) = 1 (0 < pc < 1) for the critical average connectivity Kc in terms of the bias pc of extracting a "1" for the output of the automata. Values of K bigger than Kc correspond to the so-called chaotic phase; while K < Kc, to an ordered phase. In [F. Zertuche, On the robustness of NK-Kauffman networks against changes in their connections and Boolean functions. J. Math. Phys. 50 (2009) 043513], 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 pc = 1 / 2 the predicted value Kc = 2 by Derrida et al. changes to Kc = 2.62140224613 We support our results with numerical simulations.