The number of connected components in sub-critical random graph processes

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.