  There has been a lot of research on random graph models for large real-world
networks such as those formed by hyperlinks between web pages in the world wide
web. 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 of
reflecting the degree distribution of the webgraph, their ability to reflect
certain aspects of the edge distribution was not yet well studied.
  In this paper, we consider the Buckley--Osthus implementation of preferential
attachment and its ability to model the web host graph in two aspects. One is
the degree distribution that we observe to follow the power law, as often being
the case for real-world graphs. Another one is the two-dimensional edge
distribution, the number of edges between vertices of given degrees. We fit a
single "initial attractiveness" parameter $a$ of the model, first with respect
to the degree distribution of the web host graph, and then, absolutely
independently, with respect to the edge distribution. Surprisingly, the values
of $a$ we obtain turn out to be nearly the same. Therefore the same model with
the same value of the parameter $a$ fits very well the two independent and
basic aspects of the web host graph. In addition, we demonstrate that other
models completely lack the asymptotic behavior of the edge distribution of the
web host graph, even when accurately capturing the degree distribution.
  To the best of our knowledge, this is the first attempt for a real graph of
Internet to describe the distribution of edges between vertices with respect to
their degrees.
