Correlation of paths between distinct vertices in a randomly oriented graph

TL;DR

This paper proves positive correlation between certain path events in randomly oriented graphs, including tournaments and Erdős–Rényi models, with conjectures for broader cases and exact probability recursions.

Contribution

It establishes positive correlation of path events in various random graph models and provides an exact recursion for their joint probabilities.

Findings

Positive correlation in random tournaments.

Positive correlation in G(n,p) and G(n,m) models for large n.

Exact recursion for joint path event probabilities.

Abstract

We prove that in a random tournament the events $\{s\rightarrow a\}$ and $\{t\rightarrow b\}$ are positively correlated, for distinct vertices $a,s,b,t \in K_n.$ It is also proven that the correlation between the events $\{s\rightarrow a\}$ and $\{t\rightarrow b\}$ in the random graphs $G(n,p)$ and $G(n,m)$ with random orientation is positive for every fixed $p>0$ and sufficiently large $n$ (with $m=\left\lfloor p \binom{n}{2}\right\rfloor$). We conjecture it to be positive for all $p$ and all $n$. An exact recursion for $\P(\{s\rightarrow a\} \cap \{t\rightarrow b\})$ in $\gnp$ is given.