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

Abstract

We determine the average number $\vartheta (N,K)$, of \textit{NK}-Kauffman networks that give rise to the same binary function. We show that, for $N\gg 1$, there exists a connectivity critical value $K_c$ such that $\vartheta (N,K)\approx e^{\varphi N}$ ($\varphi >0$) for $K<K_c$ and $\vartheta (N,K)\approx 1$ for $K>K_c$. We find that $K_c$ 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 \textit{NK}-Kauffman networks and impose tight constraints in the average number of epistatic interactions that the genotype-phenotype map can have.

4 figures 18 pages

References in corpus (1)

• Number and length of attractors in a critical Kauffman model with connectivity one