A Computation of the Expected Number of Posts in a Finite Random Graph Order

TL;DR

This paper refines the understanding of the expected number of posts in finite random graph orders, providing an asymptotic linear approximation with rapid error decay, and explores related bounds on the Euler function.

Contribution

It offers a new explicit expression for the expected number of posts in finite random graph orders, including edge effects, and establishes an asymptotic linearity result with a positive intercept.

Findings

Expected number of posts is asymptotically linear in n.

Error in approximation decreases rapidly with n.

Provides a bound on the difference between the Euler function and its partial products.

Abstract

A random graph order is a partial order achieved by independently sprinkling relations on a vertex set (each with probability $p$) and adding relations to satisfy the requirement of transitivity. A \textit{post} is an element in a partially ordered set which is related to every other element. Alon et al.\ \cite{Alon} proved a result for the average number of posts among the elements $\{1,2,...,n\}$ in a random graph order on $\mathbb{Z}$. We refine this result by providing an expression for the average number of posts in a random graph order on $\{1,2,...,n\}$, thereby quantifying the edge effects associated with the elements $\mathbb{Z}\backslash\{1,2,...,n\}$. Specifically, we prove that the expected number of posts in a random graph order of size $n$ is asymptotically linear in $n$ with a positive $y$-intercept. The error associated with this approximation decreases monotonically and rapidly in $n$, permitting accurate computation of the expected number of posts for any $n$ and $p$. We also prove, as a lemma, a bound on the difference between the Euler function and its partial products that may be of interest in its own right.