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Connected Components in Random Graphs Graphs with Given Expected Degree Sequences
Fan Chung and Linyuan Lu
Department of Mathematics, University of California, San Diego, La Jolla, CA 92093-0112, USA
{fan, llu}@euclid.ucsd.edu
Annals of Combinatorics 6 (2) p.125-145 June, 2002
AMS Subject Classification: 05C80
Abstract:
We consider a family of random graphs with a given expected degree sequence. Each edge is chosen independently with probability proportional to the product of the expected degrees of its endpoints. We examine the distribution of the sizes/volumes of the connected components which turns out depending primarily on the average degree d and the second-order average degree Here denotes the weighted average of squares of the expected degrees. For example, we prove that the giant component exists if the expected average degree d is at least 1, and there is no giant component if the expected second-order average degree is at most 1. Examples are given to illustrate that both bounds are best possible.
Keywords: random graphs, connected components, expected degree sequence, power law, power law graphs

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