  We consider an evolving preferential attachment random graph model where at
discrete times a new node is attached to an old node, selected with probability
proportional to a superlinear function of its degree. For such schemes, it is
known that the graph evolution condenses, that is a.s. in the limit graph there
will be a single random node with infinite degree, while all others have finite
degree.
  In this note, we establish a.s. law of large numbers type limits and
fluctuation results, as $n\uparrow\infty$, for the counts of the number of
nodes with degree $k\geq 1$ at time $n\geq 1$. These limits rigorously verify
and extend a physical picture of Krapivisky, Redner and Leyvraz (2000) on how
the condensation arises with respect to the degree distribution.
