A counter-intuitive correlation in a random tournament
by
Sven Erick Alm and Svante Linusson
Uppsala University and Royal Institute of Technology
arXiv:0906.0240
pdf
In Combinatorics, Probability and Computing (2011) 20, 1--9.
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 $n\ge5$.
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).