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PageRank, the prestige measure for Web pages used by Google, is thestationary probability of a peculiar random walk on directed graphs, whichinterpolates between a pure random walk and a process where all nodes have thesame probability of being visited. We give some exact results on thedistribution of PageRank in the cases in which the damping factor q approachesthe two limit values 0 and 1. When q -> 0 and for several classes of graphs thedistribution is a power law with exponent 2, regardless of the in-degreedistribution. When q -> 1 it can always be derived from the in-degreedistribution of the underlying graph, if the out-degree is the same for allnodes.Comment: 15 pages, 10 figures. Minor modifications, references added, final  version to appear in the Special Issue "Complex Networks' Structure and  Dynamics'' of the International Journal of Bifurcation and Chao

RANDOM WALKS ON DIRECTED NETWORKS: THE CASE OF PAGERANK