Component behavior near the critical point of the random graph process

We study the behavior of a random graph process (G(n, M)) 0 2 n for M(n) = n /2 + s and ∣ s ∣ 3 n −;2 → ∞. Among others we find the number of components in G(n, M) and estimate the number of vertices and edges in the k th largest component of G(n, M) , for any natural number k , Moreover, it is shown that, with probability 1 – o (1), when M(n) = n /2 + s , s 3 n −2 →−∞, then during a random graph process in some step M 1 > M a “new” largest component will emerge, whereas when s 3 n −2 →∞, the largest component of G(n, M) remains largest until the very end of the process.