  We present a detailed study of the evolution of the number of connected
components in sub-critical multiplicative random graph processes. We consider a
model where edges appear independently after an exponential time at rate equal
to the product of the sizes of the vertices. We provide an explicit expression
for the fluid limit of the number of connected components normalized by its
initial value, when the time is smaller than the inverse of the sum of the
square of the initial vertex sizes. We also identify the diffusion limit of the
rescaled fluctuations around the fluid limit. This is applied to several
examples. In the particular setting of the Erd\H{o}s-R\'enyi graph process, we
explicit the fluid limit of the number of connected components normalized, and
the diffusion limit of the scaled fluctuations in the sub-critical regime,
where the mean degree is between zero and one.
