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

TL;DR

This paper analyzes how the number of binary functions generated by NK-Kauffman networks varies with connectivity, revealing a critical threshold that relates to genetic robustness and constrains epistatic interactions.

Contribution

It provides a detailed characterization of the average number of functions produced by NK-Kauffman networks and identifies a connectivity threshold scaling with network size.

Findings

Existence of a critical connectivity value K_c scaling with N

Exponential growth of function count for K < K_c

Constraints on epistatic interactions due to 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^{\phi N} $ ($ \phi > 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.