, Volume 6, Issue 2, pp 125-145

Connected Components in Random Graphs with Given Expected Degree Sequences

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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.

Received August 11, 2002
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ID="*"Research supported in part by NSF Grant DMS 0100472.