The Number of Locally $p$-stable Functions on $Q_n$

TL;DR

This paper provides precise estimates for the number of locally $p$-stable Boolean functions on the hypercube, showing a double exponential lower bound, a matching upper bound, and stability of counts for certain parameters as dimension grows.

Contribution

It improves the lower bound to a double exponential and establishes an upper bound, also showing the count stabilizes for fixed parameters as the hypercube dimension increases.

Findings

Double exponential lower bound for the number of locally $p$-stable functions.

Matching upper bound for these functions.

Number of such functions stabilizes for fixed $k$ as $n$ increases.

Abstract

A Boolean function $f:V \to \{-1,1\}$ on the vertex set of a graph $G=(V,E)$ is locally $p$-stable if for every vertex $v$ the proportion of neighbours $w$ of $v$ with $f(v)=f(w)$ is exactly $p$. This notion was introduced by Gross and Grupel in [1] while studying the scenery reconstruction problem. They give an exponential type lower bound for the number of isomorphism classes of locally $p$-stable functions when $G=Q_n$ is the $n$-dimensional Boolean hypercube and ask for more precise estimates. In this paper we provide such estimates by improving the lower bound to a double exponential type lower bound and finding a matching upper bound. We also show that for a fixed $k$ and increasing $n$, the number of isomorphism classes of locally $(1-k/n)$-stable functions on $Q_n$ is eventually constant. The proofs use the Fourier decomposition of functions on the Boolean hypercube.