Achlioptas processes are a class of modifications of the Erdős–Rényi
random graph. At each step of an Achlioptas process we add one of two
randomly selected edges to the graph according to a fixed rule. We
present new results on the phase transitions of a group of Achlioptas
processes called 'bounded-size rules' and show that qualitatively, these
processes belong to the same class as the Erdős–Rényi process. The
results include the size and structure of small components in the barely
sub- and supercritical time periods. We will also discuss another type
of Achlioptas process that seems to exhibit markedly different behavior
at the phase transition.