On a random graph evolving by degrees

We consider a random (multi)graph growth process {Gm} on a vertex set [n], which is a special case of a more general process proposed by Laci Lovász in 2002. G0 is empty, and Gm+1 is obtained from Gm by inserting a new edge e at random. Specifically, the conditional probability that e joins two currently disjoint vertices, i and j, is proportional to (di+α)(dj+α), where di, dj are the degrees of i, j in Gm, and α>0 is a fixed parameter. The limiting case α=∞ is the Erdős–Rényi graph process. We show that whp Gm contains a unique giant component iff c:=2m/n>cα=α/(1+α), and the size of this giant is asymptotic to , where c∗1, mn is the threshold for connectedness of Gm itself.