An old approach to the giant component problem
In 1998, Molloy and Reed showed that, under suitable conditions, if a sequence dn of degree sequences converges to a probability distribution D, then the proportion of vertices in the largest component of the random graph associated to dn is asymptotically ρ(D), where ρ(D) is a constant defined by the solution to certain equations that can be interpreted as the survival probability of a branching process associated to D. There have been a number of papers strengthening this result in various ways; here we prove a strong form of the result (with exponential bounds on the probability of large deviations) under minimal conditions.
Journal of Combinatorial Theory. Series B
Bollobás, B., & Riordan, O. (2015). An old approach to the giant component problem. Journal of Combinatorial Theory. Series B, 113, 236-260. https://doi.org/10.1016/j.jctb.2015.03.002