PageRank is one of the principle criteria according to which Google ranks Web pages. PageRank can be interpreted as a frequency of visiting a Web page by a random surfer and thus it reflects the popularity of a Web page. Google computes the PageRank using the power iteration method which requires about one week of intensive computations. In the present work we propose and analyze Monte Carlo type methods for the PageRank computation. There are several advantages of the probabilistic Monte Carlo methods over the deterministic power iteration method: Monte Carlo methods provide good estimation of the PageRank for relatively important pages already after one iteration; Monte Carlo methods have natural parallel implementation; and finally, Monte Carlo methods allow to perform continuous update of the PageRank as the structure of the Web changes.

This paper studies the distribution of a family of rankings, which includes Google's PageRank, on a directed configuration model. In particular, it is shown that the distribution of the rank of a randomly chosen node in the graph converges in distribution to a finite random variable scriptR* that can be written as a linear combination of i.i.d. copies of the attracting endogenous solution to a stochastic fixed‐point equation of the form R=scriptD∑i=1NscriptCiscriptRi+Q, where (Q,N,{scriptCi}) is a real‐valued vector with N∈{0,1,2,…}, P(|Q|>0)>0, and the {scriptRi} are i.i.d. copies of scriptR, independent of (Q,N,{scriptCi}). Moreover, we provide precise asymptotics for the limit scriptR*, which when the in‐degree distribution in the directed configuration model has a power law imply a power law distribution for scriptR* with the same exponent. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 237–274, 2017

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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 equationwhere 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.