First critical probability for a problem on random orientations in $G(n,p)$

TL;DR

This paper investigates the correlation of directed paths between three vertices in a randomly oriented Erdős–Rényi graph, identifying a critical probability threshold where the correlation changes sign.

Contribution

It establishes the first critical probability for the sign change of correlation in randomly oriented $G(n,p)$ graphs, supported by theoretical analysis and computational evidence.

Findings

Correlation is negative for small $p$, specifically below $C_1/n$.

Correlation becomes positive for $p$ between $C_1/n$ and $2/n$.

Computations suggest a second critical point at approximately $7.5/n$, with conjectures on correlation behavior beyond.

Abstract

We study the random graph $G(n,p)$ with a random orientation. For three fixed vertices $s,a,b$ in $G(n,p)$ we study the correlation of the events $a \to s$ and $s\to b$. We prove that asymptotically the correlation is negative for small $p$, $p<\frac{C_1}n$, where $C_1\approx0.3617$, positive for $\frac{C_1}n<p<\frac2n$ and up to $p=p_2(n)$. Computer aided computations suggest that $p_2(n)=\frac{C_2}n$, with $C_2\approx7.5$. We conjecture that the correlation then stays negative for $p$ up to the previously known zero at $\frac12$; for larger $p$ it is positive.