How to get the random graph with non-uniform probabilities?

TL;DR

This paper investigates how to generate the Rado (Random) Graph with non-uniform, pair-dependent probabilities, characterizing sequences that produce the almost sure emergence of the graph.

Contribution

It provides a characterization of probability sequences that, when used to generate edges with pair-specific probabilities, almost surely produce the Rado Graph.

Findings

Characterization of sequences $(p_n)$ for Rado Graph generation

Conditions for pair-dependent probabilities to yield the Rado Graph

Almost sure emergence of the Rado Graph with non-uniform probabilities

Abstract

The Rado Graph, sometimes also known as the (countable) Random Graph, can be generated almost surely by putting an edge between any pair of vertices with some fixed probability $p \in (0, 1)$, independently of other pairs. In this article, we study the influence of allowing different probabilities for each pair of vertices. More specifically, we characterize for which sequences $(p_n)_{n\in \mathbb{N}}$ of values in $[0, 1]$ there exists a bijection f from pairs of vertices in $\mathbb{N}$ to $\mathbb{N}$ such that if we put an edge between $v$ and $w$ with probability $p_{f(\{v,w\})}$, independently of other pairs, then the Random Graph arises almost surely.