The Number of Different Binary Functions Generated by NK-Kauffman Networks and the Emergence of Genetic Robustness

We determine the average number $\vartheta (N,K)$, of $\mathit{\text{NK}}$-Kauffman networks that give rise to the same binary function. We show that, for $ N \gg 1 $,thereexistsaconnectivitycriticalvalue$ K_c $suchthat$ \vartheta(N,K) \approx e^{\phi N} $($ \phi > 0 $)for$ K < K_c $and$\vartheta(N,K) \approx 1 $for$ K > K_c $\mathrm{.}Wefindthat$ K_c $ is not a constant, but scales very slowly with $N$, as $ K_c \approx \log_2 \log_2 (2N / \ln 2) $. The problem of genetic robustness emerges as a statistical property of the ensemble of $\mathit{\text{NK}}$-Kauffman networks and impose tight constraints in the average number of epistatic interactions that the genotype-phenotype map can have.