A Preferential Attachment Process Approaching the Rado Graph

TL;DR

This paper analyzes a simple preferential attachment process starting from a finite graph and shows that its infinite limit almost surely resembles the Rado graph with some isolated or universal vertices.

Contribution

It demonstrates that a basic preferential attachment process converges to the Rado graph structure, extending understanding of graph limits in stochastic processes.

Findings

The process converges to the Rado graph with probability 1.

Initial graph conditions influence the final structure.

The limit includes a finite number of isolated or universal vertices.

Abstract

We consider a simple Preferential Attachment graph process, which begins with a finite graph, and in which a new $(t+1)$st vertex is added at each subsequent time step $t$, and connected to each previous vertex $u \leq t$ with probability $\frac{d_u(t)}{t}$ where $d_u(t)$ is the degree of $u$ at time $t$. We analyse the graph obtained as the infinite limit of this process, and show that so long as the initial finite graph is neither edgeless nor complete, with probability 1 the outcome will be a copy of the Rado graph augmented with a finite number of either isolated or universal vertices.