On the number of non-degenerate canalizing Boolean functions

TL;DR

This work addresses the problem of counting Boolean functions with prescribed numbers of essential variables and canalizing depth, specifically focusing on non-degenerate canalizing functions. The authors develop exact recursive formulas, notably for $N(n,m,k)$ and $N(n,n,k)$, building on the known $C(n,k)$ counts and the extended monomial (layered) representation to stratify function space. They provide explicit small-$n$ results and define prevalence measures $P_{\text{canalizing}}(n)$ and $P_{\text{NCF}}(n)$ to quantify how often canalization occurs among non-degenerate functions, revealing how degeneracy biases naive estimates. The findings yield a rigorous null model for canalization in biological networks and advance the combinatorial understanding of how hierarchical control and redundancy shape the space of Boolean update rules, with future directions toward closed-form expressions and multistate generalizations.

Abstract

Canalization is a key organizing principle in complex systems, particularly in gene regulatory networks. It describes how certain input variables exert dominant control over a function's output, thereby imposing hierarchical structure and conferring robustness to perturbations. Degeneracy, in contrast, captures redundancy among input variables and reflects the complete dominance of some variables by others. Both properties influence the stability and dynamics of discrete dynamical systems, yet their combinatorial underpinnings remain incompletely understood. Here, we derive recursive formulas for counting Boolean functions with prescribed numbers of essential variables and given canalizing properties. In particular, we determine the number of non-degenerate canalizing Boolean functions -- that is, functions for which all variables are essential and at least one variable is canalizing. Our approach extends earlier enumeration results on canalizing and nested canalizing functions. It provides a rigorous foundation for quantifying how frequently canalization occurs among random Boolean functions and for assessing its pronounced over-representation in biological network models, where it contributes to both robustness and to the emergence of distinct regulatory roles.

Figures (3)

• Figure 1: Proportion of $n$-input Boolean functions with a specific canalizing depth, computed using formulas for $C(n,k)$. Canalization becomes increasingly rare as $n$ increases, highlighting the need for exact formulas to study the prevalence of such functions.
• Figure 2: Proportion on a (A) linear and (B) log scale of $n$-input non-degenerate Boolean functions with a specific canalizing depth, computed using the formula for $N(n,m=n,k)$ from Theorem \ref{['thm:main']}. These exact formulas enable an exploration of the prevalence of canalization even in functions with higher inputs ($n\geq 5$) where it is very rare.
• Figure 3: Relative bias introduced by neglecting degeneracy when estimating the prevalence of canalizing and nested-canalizing Boolean functions. For each number of inputs $n$, the plotted values represent the $\log_2$-fold change $\Delta_{*}(n)=\log_{2}(\tilde{P}_{*}(n)/P_{*}(n))$, which quantifies how many powers of two the naïve estimate $\tilde{P}_{*}(n)$ (based on all Boolean functions) over- or under-represents the true prevalence among non-degenerate functions. Positive values correspond to an apparent enrichment of canalization when degeneracy is ignored.

Theorems & Definitions (18)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Theorem 2.5
• Remark 2.6
• Example 2.7
• Definition 3.1
• Remark 3.2
• Lemma 3.3