Preferential Attachment and Vertex Arrival Times

We study preferential attachment mechanisms in random graphs that are parameterized by (i) a constant bias affecting the degreebiased 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-toone 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 Spanó. A number of examples clarify how the class intersects with several known random graph models.