  Based on observations in the web-graph, the power-law hypothesis states that
PageRank has a power-law distribution with the same exponent as the in-degree.
While this hypothesis has been analytically verified for many random graph
models, such as directed configuration models and generalized random graphs,
surprisingly it has recently been disproven for the directed preferential
attachment model. In this paper, we prove that in undirected networks, the
graph-normalized PageRank is always upper bounded by the degree. Furthermore,
we prove that the corresponding (asymptotic) lower bound holds true under
reasonable assumptions on the local weak limit, but not in general, and we
provide a counterexample. Our result shows that PageRank always has a lighter
tail than the degree, which contrasts the case of the directed preferential
attachment model, where PageRank has a heavier tail instead. We end the paper
with a discussion, where we extend our results to directed networks with a
bounded ratio of in- and out-degrees, and reflect on our methods by contrasting
the undirected and directed preferential attachment model.
