Connected components in random graphs with given expected degree sequences

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 secondorder average degree d̃. Here d̃ 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 d̃ is at most 1. Examples are given to illustrate that both bounds are best possible.