  Preferential attachment is widely used to model power-law behavior of degree
distributions in both directed and undirected networks. Practical analyses on
the tail exponent of the power-law degree distribution use the Hill estimator
as one of the key summary statistics, whose consistency is justified mostly for
iid data. The major goal in this paper is to answer the question whether the
Hill estimator is still consistent when applied to non-iid network data. To do
this, we first derive the asymptotic behavior of the degree sequence via
embedding the degree growth of a fixed node into a birth immigration process.
We also need to show the convergence of the tail empirical measure, from which
the consistency of Hill estimators is obtained. This step requires checking the
concentration of degree counts. We give a proof for a particular linear
preferential attachment model and use simulation results as an illustration in
other choices of models.
