Random graph theory predicts that the giant component has of order theta(c)N vertices, where theta(c) is the percolation probability of a Poisson Galton-Watson process with mean c (meaning, the probability that such a process survives forever). In this simulation, we were lucky enough that 1 was belonging to the giant cluster as early as 1.05 (and maybe before as well). As c tends to 1, this probability tends to 0 since theta(c) tends to 0). However the slope of theta is infinite at c=1, (in this mean-field context, this is easy to show), so the probability that 1 belongs to the giant component is not that small.