An old approach to the giant component problem

Bela Bollobas; Oliver Riordan

doi:10.1016/j.jctb.2015.03.002

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arXiv · 1209.3691

An old approach to the giant component problem

Bela Bollobas·Oliver Riordan

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Abstract

In 1998, Molloy and Reed showed that, under suitable conditions, if a sequence of degree sequences converges to a probability distribution $D$, then the size of the largest component in corresponding $n$-vertex random graph is asymptotically $\rho(D)n$, where $\rho(D)$ 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.

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Bela Bollobas, Oliver Riordan. 2012-09-17. An old approach to the giant component problem. https://doi.org/10.1016/j.jctb.2015.03.002

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