A preferential attachment process approaching the Rado graph

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 that is connected to each previous vertex u <= t with probability 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 we show that, as 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.