Approximating inter-point distances in directed Bernoulli graphs

TL;DR

This paper develops an approximation for the joint distribution of directed distances between vertices in a directed Bernoulli random graph, extending classical models and providing error bounds.

Contribution

It introduces a novel approximation method for inter-point distances in directed Bernoulli graphs using a multivariate limiting distribution, extending prior undirected models.

Findings

Approximation error is typically of order O(n^{-1/2} log n)

The method involves two independent copies of a trivariate limiting random vector

The asymptotic order is likely optimal even for undirected Bernoulli graphs.

Abstract

In directed random graphs, in which edges can be assigned to have one of two directions, or perhaps both, the distance between two vertices $v$ and $v'$ can be computed along paths that are directed from $v$ to $v'$, or along paths that are directed from $v'$ to $v$. These two distances are in general dependent. Here, we approximate their joint distribution in the setting of the directed Bernoulli random graph $\mathcal{DG}(n,p,\theta)$, obtained as a natural extension of the Bernoulli random graph $\mathcal{G}(n,p)$ by assigning directions to the edges independently, bidirectional with probability $\theta$, and either of the two possible choices of single direction with probability $\frac12(1-\theta)$. The approximation involves two independent copies of a trivariate limiting random vector $(W^*_1,W^*_2,W_3)$ associated with a $3$-type Bienaym\'e--Galton--Watson process. The approximation error is shown to be typically of order $O(n^{-1/2}\log n)$; this asymptotic order is likely to be optimal, even for the corresponding approximation in the Bernoulli random graph $\mathcal{G}(n,p)$.