An old approach to the giant component problem

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.

24 pages; only minor changes

References in corpus (2)

• The phase transition in the configuration model
• Edge percolation on a random regular graph of low degree

Cited by in corpus (5)

• On the method of typical bounded differences
• How to determine if a random graph with a fixed degree sequence has a giant component
• Mesoscopic scales in hierarchical configuration models
• Percolation on random graphs with a fixed degree sequence
• Connectivity of random graphs after centrality-based vertex removal