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.