On the Robustness of NK-Kauffman Networks Against Changes in their Connections and Boolean Functions

Abstract

NK-Kauffman networks {\cal L}^N_K are a subset of the Boolean functions on N Boolean variables to themselves, \Lambda_N = {\xi: \IZ_2^N \to \IZ_2^N}. To each NK-Kauffman network it is possible to assign a unique Boolean function on N variables through the function \Psi: {\cal L}^N_K \to \Lambda_N. The probability {\cal P}_K that \Psi (f) = \Psi (f'), when f' is obtained through f by a change of one of its K-Boolean functions (b_K: \IZ_2^K \to \IZ_2), and/or connections; is calculated. The leading term of the asymptotic expansion of {\cal P}_K, for N \gg 1, turns out to depend on: the probability to extract the tautology and contradiction Boolean functions, and in the average value of the distribution of probability of the Boolean functions; the other terms decay as {\cal O} (1 / N). In order to accomplish this, a classification of the Boolean functions in terms of what I have called their irreducible degree of connectivity is established. The mathematical findings are discussed in the biological context where, \Psi is used to model the genotype-phenotype map.