Path correlations in a randomly oriented complete bipartite graph

TL;DR

This paper investigates how the likelihood of directed paths between vertices in a randomly oriented complete bipartite graph correlates, revealing dependence on the sizes of the bipartite classes.

Contribution

It provides a classification of positive and negative correlations between path events based on the bipartite class sizes.

Findings

Correlation depends on class size proportions.

Positive correlation occurs under certain size ratios.

Negative correlation occurs under other size ratios.

Abstract

In a randomly oriented graph containing vertices $x$ and $y$, denote by $\{x\to y\}$ the event that there is a directed path from $x$ to $y$. We study the correlation between the events $\{x\to y\}$ and $\{y\to z\}$ for a (large) oriented complete bipartite graph with orientation chosen uniformly at random. We classify the cases of positive and negative correlation respectively in terms of the relative proportions of the sizes of the color classes of the graph.