The Distribution of Second Degrees in the Buckley–Osthus Random Graph Model

In this article we consider a well-known generalization of the Barabási and Albert preferential attachment model—the Buckley–Osthus model. Buckley and Osthus proved that in this model, the degree sequence has a power law distribution. As a natural (and arguably more interesting) next step, we study the second degrees of vertices. Roughly speaking, the second degree of a vertex is the number of vertices at distance two from the given vertex. The distribution of second degrees is of interest because it is a good approximation of PageRank, where the importance of a vertex is measured by taking into account the popularity of its neighbors. We prove that the second degrees also obey a power law. More precisely, we estimate the expectation of the number of vertices with the second degree greater than or equal to k and prove the concentration of this random variable around its expectation using the now-famous Talagrand's concentration inequality over product spaces. As far as we know, this is the only application of Talagrand's inequality to random web graphs where the (preferential attachment) edges are not defined over a product distribution, making the application nontrivial and requiring a certain degree of novelty.