A Preferential Attachment Process Approaching the Rado Graph

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.