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

A conditional compound Poisson process approach to the sparse Erd\H{o}s-R\'enyi random graphs: moderate deviations