Barak-Erd\H{o}s graphs and the infinite-bin model

TL;DR

This paper investigates the growth rate of the longest path in Barak-Erdős graphs, using couplings with infinite-bin models and branching random walks to derive explicit estimates, analyticity, and asymptotic expansions of the growth rate.

Contribution

It introduces a novel coupling approach between Barak-Erdős graphs and infinite-bin models to analyze the linear growth rate of the longest path.

Findings

Proves the linear growth rate C(p) is analytic for p > 1/2.

Provides a power series expansion of C(p).

Derives the first two terms of C(p) as p approaches 0.

Abstract

A Barak-Erd\H{o}s graph is a directed acyclic version of the Erd\H{o}s-R\'enyi random graph. It is obtained by performing independent bond percolation with parameter $p$ on the complete graph with vertices $\{1,...,n\}$, in which the edge between two vertices $i<j$ is directed from $i$ to $j$. The length of the longest path in this graph grows linearly with the number of vertices, at rate $C(p)$. In this article, we use a coupling between Barak-Erd\H{o}s graphs and infinite-bin models to provide explicit estimates on $C(p)$. More precisely, we prove that the front of an infinite-bin model grows at linear speed, and that this speed can be obtained as the sum of a series. Using these results, we prove the analyticity of $C$ for $p >1/2$, and compute its power series expansion. We also obtain the first two terms of the asymptotic expansion of $C$ as $p \to 0$, using a coupling with branching random walks.