A counter-intuitive correlation in a random tournament

Abstract

Consider a randomly oriented graph $G=(V,E)$ and let $a$, $s$ and $b$ be three distinct vertices in $V$. We study the correlation between the events $\{a\to s\}$ and $\{s\to b\}$. We show that, when $G$ is the complete graph $K_n$, the correlation is negative for $n=3$, zero for $n=4$, and that, counter-intuitively, it is positive for $n\ge 5$. We also show that the correlation is always negative when $G$ is a cycle, $C_n$, and negative or zero when $G$ is a tree (or a forest).