Preferential Attachment Processes Approaching The Rado Multigraph

We consider a preferential attachment process in which a multigraph is built one node at a time. The number of edges added at stage $t$, emanating from the new node, is given by some prescribed function $f(t)$, generalising a model considered by Kleinberg and Kleinberg in 2005 where $f$ was presumed constant. We show that if $f(t)$ is asymptotically bounded above and below by linear functions in $t$, then with probability $1$ the infinite limit of the process will be isomorphic to the \emph{Rado multigraph}. This structure is the natural multigraph analogue of the Rado graph, which we introduce here.