  We consider a class of scale-free inhomogeneous random graphs, which includes
some long-range percolation models. We study the maximum degree in such graphs
in a growing observation window and show that its limiting distribution is
Frechet. We achieve this by proving convergence of the underlying point process
of the degrees to a certain Poisson process. Estimating the index of the
power-law tail for the typical degree distribution is an important question in
statistics. We prove consistency of the Hill estimator for the inverse of the
tail exponent of the typical degree distribution.
