Nested Canalyzing Functions And Their Average Sensitivities

TL;DR

This paper characterizes nested canalyzing functions (NCFs) algebraically, introduces the concept of layer number, and derives formulas for key parameters including average sensitivity, showing NCFs are stable with bounded sensitivity.

Contribution

It provides a complete algebraic characterization of NCFs, introduces layer number, and derives explicit formulas for their properties and sensitivities, advancing understanding of their stability.

Findings

Average sensitivity of NCFs is between 0 and 2.

Minimal average sensitivity occurs with single-layer NCFs.

Conjecture: maximal average sensitivity is achieved with maximal layer number.

Abstract

In this paper, we obtain complete characterization for nested canalyzing functions (NCFs) by obtaining its unique algebraic normal form (polynomial form). We introduce a new concept, LAYER NUMBER for NCF. Based on this, we obtain explicit formulas for the the following important parameters: 1) Number of all the nested canalyzing functions, 2) Number of all the NCFs with given LAYER NUMBER, 3) Hamming weight of any NCF, 4) The activity number of any variable of any NCF, 5) The average sensitivity of any NCF. Based on these formulas, we show the activity number is greater for those variables in out layer and equal in the same layer. We show the average sensitivity attains minimal value when the NCF has only one layer. We also prove the average sensitivity for any NCF (No matter how many variables it has) is between 0 and 2. Hence, theoretically, we show why NCF is stable since a random Boolean function has average sensitivity $\frac{n}{2}$. Finally we conjecture that the NCF attain the maximal average sensitivity if it has the maximal LAYER NUMBER $n-1$. Hence, we guess the uniform upper bound for the average sensitivity of any NCF can be reduced to 4/3 which is tight.