# The Dynamics of Canalizing Boolean Networks

## TL;DR

This paper explores how canalizing properties influence the dynamics of Boolean networks, revealing that higher canalizing depth generally leads to fewer and smaller attractors, with implications for biological modeling.

## Contribution

It provides analytical and simulation-based insights into the impact of canalizing depth on Boolean network dynamics, including explicit formulas for attractor counts.

## Key findings

- Higher canalizing depth results in fewer attractors.
- Attractors are smaller and basins are larger with increased canalizing depth.
- High canalizing depth has a modest impact on attractor structure.

## Abstract

Boolean networks are a popular modeling framework in computational biology to capture the dynamics of molecular networks, such as gene regulatory networks. It has been observed that many published models of such networks are defined by regulatory rules driving the dynamics that have certain so-called canalizing properties. In this paper, we investigate the dynamics of a random Boolean network with such properties using analytical methods and simulations. From our simulations, we observe that Boolean networks with higher canalizing depth have generally fewer attractors, the attractors are smaller, and the basins are larger, with implications for the stability and robustness of the models. These properties are relevant to many biological applications. Moreover, our results show that, from the standpoint of the attractor structure, high canalizing depth, compared to relatively small positive canalizing depth, has a very modest impact on dynamics. Motivated by these observations, we conduct mathematical study of the attractor structure of a random Boolean network of canalizing depth one (i.e., the smallest positive depth). For every positive integer $\ell$, we give an explicit formula for the limit of the expected number of attractors of length $\ell$ in an $n$-state random Boolean network as $n$ goes to infinity.