Erd\"os-R\'enyi Poissonized

TL;DR

This paper introduces a Poissonized variant of the Erd"os--Rényi random graph, simplifying proofs of classical results like phase transition and connectedness through a Markov property and Poisson process techniques.

Contribution

It presents a new Poissonized model of Erd"os--Rényi graphs with a simple Markov property, enabling concise proofs of key graph properties.

Findings

Simplified proofs of phase transition for the giant component

Concise demonstration of connectedness threshold

Utilization of Poisson process for analysis

Abstract

We introduce a variant of the Erd\"os--R\'enyi random graph where the number of vertices is random and follows a Poisson law. A very simple Markov property of the model entails that the Lukasiewicz exploration is made of \textit{independent} Poisson increments. Using a vanilla Poisson counting process, this enables us to give very short proofs of classical results such as the phase transition for the giant component or the connectedness for the standard Erd\"os--R\'enyi model.