  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.
