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Approximating inter-point distances in directed Bernoulli graphs

A. D. Barbour, Gesine Reinert

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 $\mathcal{DG}(n,p,θ)$, obtained as a natural extension of the Bernoulli random graph $\mathcal{G}(n,p)$ by assigning directions to the edges independently, bidirectional with probability $θ$, and either of the two possible choices of single direction with probability $\frac12(1-θ)$. The approximation involves two independent copies of a trivariate limiting random vector $(W^*_1,W^*_2,W_3)$ associated with a $3$-type Bienaymé--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 $\mathcal{G}(n,p)$.

Approximating inter-point distances in directed 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 and can be computed along paths that are directed from to , or along paths that are directed from to . These two distances are in general dependent. Here, we approximate their joint distribution in the setting of the directed Bernoulli random graph , obtained as a natural extension of the Bernoulli random graph by assigning directions to the edges independently, bidirectional with probability , and either of the two possible choices of single direction with probability . The approximation involves two independent copies of a trivariate limiting random vector associated with a -type Bienaymé--Galton--Watson process. The approximation error is shown to be typically of order ; this asymptotic order is likely to be optimal, even for the corresponding approximation in the Bernoulli random graph .
Paper Structure (3 sections, 11 theorems, 176 equations)

This paper contains 3 sections, 11 theorems, 176 equations.

Table of Contents

  1. Introduction
  2. The Bernoulli random graph
  3. The Bernoulli digraph

Key Result

Theorem 1.1

Let $V_n$ and $V'_n$ be a pair of vertices chosen independently and uniformly at random from the vertex set of a Bernoulli random graph $G \sim {\cal G}(n,m/n)$, with $m > 1$. Then, for any $n$ sufficiently large, and with $r_n$ and $\chi_n$ defined in (B-rn-def) and (B-chin-def), we have for any $u > -2r_n$, where $C(m)$ is uniformly bounded in any interval of the form $1+\delta \le m \le \Delta

Theorems & Definitions (25)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Remark 2.5
  • ...and 15 more