Asymptotic analysis for personalized web search

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 $R \buildrel\rm D\over= \sum^N_{i=1} A_i R_i + B$, 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.