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 d~. Here d~ 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 d~ is at most 1. Examples are given to illustrate that both bounds are best possible.
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Received August 11, 2002
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ID="*"Research supported in part by NSF Grant DMS 0100472.
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Chung, F., Lu, L. Connected Components in Random Graphs with Given Expected Degree Sequences. Annals of Combinatorics 6, 125–145 (2002). https://doi.org/10.1007/PL00012580
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DOI: https://doi.org/10.1007/PL00012580