A Random Graph Growth Model

TL;DR

This paper introduces a new model for growing random graphs by successively sampling vertices and edges, analyzing the edge count distribution at different stages, and identifying Poisson limits and Gaussian fluctuations.

Contribution

It presents a novel random graph growth model with detailed asymptotic analysis of edge counts, including Poisson limits and Gaussian fluctuation descriptions.

Findings

Poisson limits identified for specific graph sizes

Edge count fluctuations follow a Gaussian bridge for large graphs

The model provides insights into the probabilistic structure of growing graphs

Abstract

A growing random graph is constructed by successively sampling without replacement an element from the pool of virtual vertices and edges. At start of the process the pool contains $N$ virtual vertices and no edges. Each time a vertex is sampled and occupied, the edges linking the vertex to previously occupied vertices are added to the pool of virtual elements. We focus on the edge-counting at times when the graph has $n\leq N$ occupied vertices. Two different Poisson limits are identified for $n\asymp N^{1/3}$ and $N-n\asymp 1$. For the bulk of the process, when $n\asymp N$, the scaled number of edges is shown to fluctuate about a deterministic curve, with fluctuations being of the order of $N^{3/2}$ and approximable by a Gaussian bridge.