Multistate nested canalizing functions

TL;DR

This paper extends the concept of nested canalizing functions from Boolean to multi-state functions over finite fields, providing formulas for their enumeration and equivalence classes, with implications for biological network modeling.

Contribution

It generalizes nested canalizing functions to multi-state cases over finite fields and derives formulas for their counts and equivalence classes, advancing understanding of their mathematical and biological relevance.

Findings

Derived a recursive formula for multi-state nested canalizing functions.

Established a closed formula for counting these functions.

Provided a formula for the number of equivalence classes under variable permutation.

Abstract

The concept of a nested canalizing Boolean function has been studied over the course of the last decade in the context of understanding the regulatory logic of molecular interaction networks, such as gene regulatory networks. Such functions appear preferentially in published models of such networks. Recently, this concept has been generalized to include multi-state functions, and a recursive formula has been derived for their number, as a function of the number of variables. This paper carries out a detailed analysis of the class of nested canalizing functions over an arbitrary finite field. Furthermore, the paper generalizes the concept further, and derives a closed formula for the number of such generalized functions. The paper also derives a closed formula for the number of equivalence classes under permutation of variables. This is motivated by the fact that two nested canalizing functions that differ by a permutation of the variables share many important properties with each other. The paper contributes to the effort of identifying a class of functions over finite fields that are of interest in biology and also have interesting mathematical properties.