Component behavior near the critical point of the random graph process

We study the behavior of a random graph process (G(n, M))₀²ⁿ for M(n) = n/2 + s and ∣s∣³n^?;2 → ∞. Among others we find the number of components in G(n, M) and estimate the number of vertices and edges in the kth 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, s³n^?2 →?∞, then during a random graph process in some step M¹ > M a “new” largest component will emerge, whereas when s³n^?2→∞, the largest component of G(n, M) remains largest until the very end of the process.