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.
@article{10_37236_12960,
author = {Leonardo Coregliano and Jaros{\l}aw Swaczyna and Agnieszka Widz},
title = {How to get the random graph with non-uniform probabilities?},
journal = {The electronic journal of combinatorics},
year = {2025},
volume = {32},
number = {2},
doi = {10.37236/12960},
zbl = {1569.05318},
url = {http://geodesic.mathdoc.fr/articles/10.37236/12960/}
}
TY - JOUR
AU - Leonardo Coregliano
AU - Jarosław Swaczyna
AU - Agnieszka Widz
TI - How to get the random graph with non-uniform probabilities?
JO - The electronic journal of combinatorics
PY - 2025
VL - 32
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/12960/
DO - 10.37236/12960
ID - 10_37236_12960
ER -
%0 Journal Article
%A Leonardo Coregliano
%A Jarosław Swaczyna
%A Agnieszka Widz
%T How to get the random graph with non-uniform probabilities?
%J The electronic journal of combinatorics
%D 2025
%V 32
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/12960/
%R 10.37236/12960
%F 10_37236_12960
Leonardo Coregliano; Jarosław Swaczyna; Agnieszka Widz. How to get the random graph with non-uniform probabilities?. The electronic journal of combinatorics, Tome 32 (2025) no. 2. doi: 10.37236/12960
