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There has been a lot of research on random graph models for large real-worldnetworks such as those formed by hyperlinks between web pages in the world wideweb. Though largely successful qualitatively in capturing their key properties,such models may lack important quantitative characteristics of Internet graphs.While preferential attachment random graph models were shown to be capable ofreflecting the degree distribution of the webgraph, their ability to reflectcertain aspects of the edge distribution was not yet well studied.  In this paper, we consider the Buckley--Osthus implementation of preferentialattachment and its ability to model the web host graph in two aspects. One isthe degree distribution that we observe to follow the power law, as often beingthe case for real-world graphs. Another one is the two-dimensional edgedistribution, the number of edges between vertices of given degrees. We fit asingle "initial attractiveness" parameter $a$ of the model, first with respectto the degree distribution of the web host graph, and then, absolutelyindependently, with respect to the edge distribution. Surprisingly, the valuesof $a$ we obtain turn out to be nearly the same. Therefore the same model withthe same value of the parameter $a$ fits very well the two independent andbasic aspects of the web host graph. In addition, we demonstrate that othermodels completely lack the asymptotic behavior of the edge distribution of theweb host graph, even when accurately capturing the degree distribution.  To the best of our knowledge, this is the first attempt for a real graph ofInternet to describe the distribution of edges between vertices with respect totheir degrees.

Empirical Validation of the Buckley--Osthus Model for the Web Host Graph: Degree and Edge Distributions