The Role of Boolean Irreducibility in \textit{NK}-Kauffman Networks

Abstract

Boolean variables are such that they take only values on $ \mathbb{Z}_2 \cong \left\{0, 1 \right\} $. \textit{NK}-Kauffman networks are dynamical deterministic systems of $ N $ Boolean functions that depend only on $ K \leq N $ Boolean variables. They were proposed by Kauffman as a first step to understand cellular behaviour [Kauffman, S.A.; {\rm The Large Scale Structure and Dynamics of Gene Control Circuits: An Ensemble Approach}. {\it J. Theoret. Biol.} {\bf 44} (1974) 167.] with great success. Among the problems that still have been not well understood in Kauffman networks, is the mechanism that regulates the phase transition of the system from an ordered phase; where small changes of the initial state decay, to a chaotic, where they grow exponentially. We show, that this mechanism is regulated through \textsf{the irreducible decomposition} of Boolean functions proposed in [ Zertuche, F. {\rm On the robustness of NK-Kauffman networks against changes in their connections and Boolean functions}, {\it J. Math. Phys.} {\bf 50} (2009) 043513]. This is in contrast to previous knowledge that attributed it to \textsf{canalization}. We also review other statistical properties of Kauffman networks that have been shown that \textsf{Boolean irreducibility} explains.