The phase diagram of random Boolean networks with nested canalizing functions

TL;DR

This paper maps the phase diagram of random Boolean networks with nested canalizing functions, revealing complex behaviors including chaos, frozen states, and oscillations, contrasting previous findings and highlighting their stability.

Contribution

It provides the first detailed phase diagram for these networks, showing both chaotic and frozen phases, and clarifies previous misconceptions about their stability.

Findings

Networks exhibit rich phase behavior including chaos, frozen, and oscillatory regimes.

Contrary to prior reports, both chaotic and frozen phases coexist.

Nested canalizing functions confer greater stability compared to general Boolean functions.

Abstract

We obtain the phase diagram of random Boolean networks with nested canalizing functions. Using the annealed approximation, we obtain the evolution of the number $b_t$ of nodes with value one, and the network sensitivity $\lambda$, and we compare with numerical simulations of quenched networks. We find that, contrary to what was reported by Kauffman et al. [Proc. Natl. Acad. Sci. 2004 101 49 17102-7], these networks have a rich phase diagram, were both the "chaotic" and frozen phases are present, as well as an oscillatory regime of the value of $b_t$. We argue that the presence of only the frozen phase in the work of Kauffman et al. was due simply to the specific parametrization used, and is not an inherent feature of this class of functions. However, these networks are significantly more stable than the variants where all possible Boolean functions are allowed.