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On a Randomly Evolving Graph : Emergence of theGiant

A randomly evolving graph, with vertices immigrating at rate n and each possible edge appearing at rate 1=n, is studied. The detailed picture of emergence of giant components with O(n 2=3) vertices is shown to be the same as in the Erd} os-R enyi graph process with the number of vertices xed at n at the start. A major diierence is that now the transition occurs about a time t = =2, rather than t = 1. The proof has three ingredients. The size of the largest component in the subcritical phase is bounded by comparison with a certain multitype branching process. With this bound at hand, the growth of the sum-of-squares and sum-of-cubes of component sizes is shown, via martingale methods, to follow closely a solution of the Smoluchovsky-type equations. The approximation allows us to apply results of Aldous (1997) on emergence of giant components in the multiplicative coalescent, i.e. a non-uniform random graph process.