  We construct a compound Poisson process conditioned on its random summation
that represents the sizes of the connected components in the sparse
Erd\H{o}s-R\'enyi random graph $G(n,c/n)$. This new representation depicts a
connection between the phase transition in the sparse random graph and the
condensation transition in the zero-range model. Under this framework, we can
derive moderate deviation principles for the maximun component, total number of
connected components and empirical measure of the sizes in the non-critical
regimes. Large deviation results are discussed.
