  We study preferential attachment mechanisms in random graphs that are
parameterized by (i) a constant bias affecting the degree-biased distribution
on the vertex set and (ii) the distribution of times at which new vertices are
created by the model. The class of random graphs so defined admits a
representation theorem reminiscent of residual allocation, or "stick-breaking"
schemes. We characterize how the vertex arrival times affect the asymptotic
degree distribution, and relate the latter to neutral-to-the-left processes.
Our random graphs generate edges "one end at a time", which sets up a
one-to-one correspondence between random graphs and random partitions of
natural numbers; via this map, our representation induces a result on (not
necessarily exchangeable) random partitions that generalizes a theorem of
Griffiths and Span\'o. A number of examples clarify how the class intersects
with several known random graph models.
