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 Rado multigraph. This structure is the natural multigraph analogue of the Rado graph, which we introduce here.