  PageRank is a well-known algorithm for measuring centrality in networks. It
was originally proposed by Google for ranking pages in the World-Wide Web. One
of the intriguing empirical properties of PageRank is the so-called `power-law
hypothesis': in a scale-free network the PageRank scores follow a power law
with the same exponent as the (in-)degrees. Up to date, this hypothesis has
been confirmed empirically and in several specific random graphs models. In
contrast, this paper does not focus on one random graph model but investigates
the existence of an asymptotic PageRank distribution, when the graph size goes
to infinity, using local weak convergence. This may help to identify general
network structures in which the power-law hypothesis holds. We start from the
definition of local weak convergence for sequences of (random) undirected
graphs, and extend this notion to directed graphs. To this end, we define an
exploration process in the directed setting that keeps track of in- and
out-degrees of vertices. Then we use this to prove the existence of an
asymptotic PageRank distribution. As a result, the limiting distribution of
PageRank can be computed directly as a function of the limiting object. We
apply our results to the directed configuration model and continuous-time
branching processes trees, as well as preferential attachment models.
