Connections in randomly oriented graphs

TL;DR

This paper investigates the probabilities of connectivity in randomly oriented graphs, establishing a correlation inequality for the events of reaching different vertices from a common source.

Contribution

It proves a new inequality relating the probabilities of directed paths in randomly oriented graphs, extending understanding of connectivity properties in such probabilistic models.

Findings

Proves a correlation inequality for directed paths in random orientations

Establishes that the probability of reaching two vertices simultaneously from a source is at least the product of individual probabilities

Provides insights into the structure of connectivity in randomly oriented graphs

Abstract

Given an undirected graph $G$, let us randomly orient $G$ by tossing independent (possibly biased) coins, one for each edge of $G$. Writing $a\rightarrow b$ for the event that there exists a directed path from a vertex $a$ to a vertex $b$ in such a random orientation, we prove that $\mathbb{P}(s\rightarrow a \cap s\rightarrow b) \ge \mathbb{P}(s\rightarrow a) \mathbb{P}(s\rightarrow b)$ for any three vertices $s$, $a$ and $b$ of $G$.