Personalized PageRank has found many uses in not only the ranking of webpages, but also algorithmic design, due to its ability to capture certain geometric properties of networks. In this paper, we study the diffusion of PageRank: how varying the jumping (or teleportation) constant affects PageRank values. To this end, we prove a gradient estimate for PageRank, akin to the Li–Yau inequality for positive solutions to the heat equation (for manifolds, with later versions adapted to graphs).

PageRank with personalization is used in Web search as an importance measure for Web documents. The goal of this paper is to characterize the tail behavior of the PageRank distribution in the Web and other complex networks characterized by power laws. To this end, we model the PageRank as a solution of a stochastic equation where the R i s are distributed as R. This equation is inspired by the original definition of the PageRank. In particular, N models the number of incoming links to a page, and B stays for the user preference. Assuming that N or B are heavy tailed, we employ the theory of regular variation to obtain the asymptotic behavior of R under quite general assumptions on the involved random variables. Our theoretical predictions show good agreement with experimental data.

PageRank is a ranking method that assigns scores to web pages using the limitdistribution of a random walk on the web graph. A fibration of graphs is amorphism that is a local isomorphism of in-neighbourhoods, much in the same waya covering projection is a local isomorphism of neighbourhoods. We show that adeep connection relates fibrations and Markov chains with restart, aparticular kind of Markov chains that include the PageRank one as aspecial case. This fact provides constraints on the values that PageRank canassume. Using our results, we show that a recently defined class of graphs thatadmit a polynomial-time isomorphism algorithm based on the computation ofPageRank is really a subclass of fibration-prime graphs, which possesssimple, entirely discrete polynomial-time isomorphism algorithms based onclassical techniques for graph isomorphism. We discuss efficiency issues in theimplementation of such algorithms for the particular case of web graphs, in whichO(n) space occupancy (where n is the number of nodes) may be acceptable, butO(m) is not (where m is the number of arcs).