Annals of Combinatorics

, Volume 6, Issue 2, pp 125-145

Connected Components in Random Graphs with Given Expected Degree Sequences

  • Fan ChungAffiliated withDepartment of Mathematics, University of California, San Diego, La Jolla, CA 92093-0112, USA, e-mail: fan@euclid.ucsd.edu; llu@euclid.ucsd.edu
  • , Linyuan LuAffiliated withDepartment of Mathematics, University of California, San Diego, La Jolla, CA 92093-0112, USA, e-mail: fan@euclid.ucsd.edu; llu@euclid.ucsd.edu

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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.

Keywords: random graphs, connected components, expected degree sequence, power law, power law graphs