  We consider the extremal values of the stationary distribution of sparse
directed random graphs with given degree sequences and their relation to the
extremal values of the in-degree sequence. The graphs are generated by the
directed configuration model. Under the assumption of bounded
$(2+\eta)$-moments on the in-degrees and of bounded out-degrees, we obtain
tight comparisons between the maximum value of the stationary distribution and
the maximum in-degree. Under the further assumption that the order statistics
of the in-degrees have a power-law behavior, we show that the extremal values
of the stationary distribution also have a power-law behavior with the same
index. In the same setting, we prove that these results extend to the PageRank
scores of the random digraph, thus confirming a version of the so-called
power-law hypothesis. Along the way, we establish several facts about the
model, including the mixing time cutoff and the characterization of the typical
values of the stationary distribution, which were previously obtained under the
assumption of bounded in-degrees.
