Analysis of random Boolean networks using the average sensitivity

Abstract

In this work we consider random Boolean networks that provide a general model for genetic regulatory networks. We extend the analysis of James Lynch who was able to proof Kauffman's conjecture that in the ordered phase of random networks, the number of ineffective and freezing gates is large, where as in the disordered phase their number is small. Lynch proved the conjecture only for networks with connectivity two and non-uniform probabilities for the Boolean functions. We show how to apply the proof to networks with arbitrary connectivity $K$ and to random networks with biased Boolean functions. It turns out that in these cases Lynch's parameter $λ$ is equivalent to the expectation of average sensitivity of the Boolean functions used to construct the network. Hence we can apply a known theorem for the expectation of the average sensitivity. In order to prove the results for networks with biased functions, we deduct the expectation of the average sensitivity when only functions with specific connectivity and specific bias are chosen at random.