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

Abstract: 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.