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In 1998, Molloy and Reed showed that, under suitable conditions, if asequence of degree sequences converges to a probability distribution $D$, thenthe size of the largest component in corresponding $n$-vertex random graph isasymptotically $\rho(D)n$, where $\rho(D)$ is a constant defined by thesolution to certain equations that can be interpreted as the survivalprobability of a branching process associated to $D$. There have been a numberof papers strengthening this result in various ways; here we prove a strongform of the result (with exponential bounds on the probability of largedeviations) under minimal conditions.

An old approach to the giant component problem