A critical point for random graphs with a given degree sequence
Random Graphs 93 Proceedings of the sixth international seminar on Random graphs and probabilistic methods in combinatorics and computer science
On the Fluctuations of the Giant Component
Combinatorics, Probability and Computing
The Size of the Giant Component of a Random Graph with a Given Degree Sequence
Combinatorics, Probability and Computing
The phase transition in the cluster-scaled model of a random graph
Random Structures & Algorithms
A simple solution to the k-core problem
Random Structures & Algorithms - Proceedings from the 12th International Conference “Random Structures and Algorithms”, August1-5, 2005, Poznan, Poland
The phase transition in inhomogeneous random graphs
Random Structures & Algorithms
The critical phase for random graphs with a given degree sequence
Combinatorics, Probability and Computing
Random graphs with forbidden vertex degrees
Random Structures & Algorithms
The scaling window for a random graph with a given degree sequence
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
The scaling window for a random graph with a given degree sequence
Random Structures & Algorithms
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We study the largest component of a random (multi)graph on n vertices with a given degree sequence. We let n → ∞. Then, under some regularity conditions on the degree sequences, we give conditions on the asymptotic shape of the degree sequence that imply that with high probability all the components are small, and other conditions that imply that with high probability there is a giant component and the sizes of its vertex and edge sets satisfy a law of large numbers; under suitable assumptions these are the only two possibilities. In particular, we recover the results by Molloy and Reed on the size of the largest component in a random graph with a given degree sequence. We further obtain a new sharp result for the giant component just above the threshold, generalizing the case of G(n,p) with np = 1 + ω(n)n-1-3, where ω(n) → ∞ arbitrarily slowly. Our method is based on the properties of empirical distributions of independent random variables, and leads to simple proofs. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009