  In this paper we consider a well-known generalization of the Barab\'asi 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 this 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 certain novelty.
