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For 0 p q 0 let Gq(n,p) denote the random graph with vertex set [n]={1,…,n} such that, for each graph G on [n] with e(G) edges and c(G) components, the probability that Gq(n,p)=G is proportional to $p^{e(G)}(1-p)^{({n \atop 2})-e(G)}q^{c(G)}$. The first systematic study of Gq(n,p) was undertaken by [Bollobás, Grimmett, and Janson (Probab Theory Relat Fields 104 (1996), 283–317)], who analyzed the phase transition phenomenon corresponding to the emergence of the giant component. In this paper we describe the structure of Gq(n,p) very close the critical threshold. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006