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We give an exact solution for the complete distribution of component sizes inrandom networks with arbitrary degree distributions. The solution tells us theprobability that a randomly chosen node belongs to a component of size s, forany s. We apply our results to networks with the three most commonly studieddegree distributions -- Poisson, exponential, and power-law -- as well as tothe calculation of cluster sizes for bond percolation on networks, whichcorrespond to the sizes of outbreaks of SIR epidemic processes on the samenetworks. For the particular case of the power-law degree distribution, we showthat the component size distribution itself follows a power law everywherebelow the phase transition at which a giant component forms, but takes anexponential form when a giant component is present.

Component sizes in networks with arbitrary degree distributions