Asymptotic analysis for personalized Web search

Personalized PageRank 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 $R\stackrel{d}{=}\sum_{i=1}^NA_iR_i+B$, where $R_i$'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 of 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 a good agreement with experimental data

In-Degree and PageRank of web pages: why do they follow similar power laws?

PageRank is a popularity measure designed by Google to rank Web pages. Experiments confirm that PageRank values obey a power law with the same exponent as In-Degree values. This paper presents a novel mathematical model that explains this phenomenon. The relation between PageRank and In-Degree is modelled through a stochastic equation, which is inspired by the original definition of PageRank, and is analogous to the well-known distributional identity for the busy period in the $M/G/1$ queue. Further, we employ the theory of regular variation and Tauberian theorems to analytically prove that the tail distributions of PageRank and In-Degree differ only by a multiple factor, for which we derive a closed-form expression. Our analytical results are in good agreement with experimental data

A framework for evaluating statistical dependencies and rank correlations in power law graphs

We analyze dependencies in power law graph data (Web sample, Wikipedia sample and a preferential attachment graph) using statistical inference for multivariate regular variation. To the best of our knowledge, this is the first attempt to apply the well developed theory of regular variation to graph data. The new insights this yields are striking: the three above-mentioned data sets are shown to have a totally different dependence structure between different graph parameters, such as in-degree and PageRank. Based on the proposed methodology, we suggest a new measure for rank correlations. Unlike most known methods, this measure is especially sensitive to rank permutations for topranked nodes. Using this method, we demonstrate that the PageRank ranking is not sensitive to moderate changes in the damping factor

Probabilistic relation between In-Degree and PageRank

This paper presents a novel stochastic model that explains the relation between power laws of In-Degree and PageRank. PageRank is a popularity measure designed by Google to rank Web pages. We model the relation between PageRank and In-Degree through a stochastic equation, which is inspired by the original definition of PageRank. Using the theory of regular variation and Tauberian theorems, we prove that the tail distributions of PageRank and In-Degree differ only by a multiplicative constant, for which we derive a closed-form expression. Our analytical results are in good agreement with Web data

Stochastic models for web ranking

Web search engines need to deal with hundreds and thousands of pages which are relevant to a user's query. Listing them in the right order is an important and non-trivial task. Thus Google introduced PageRank [1] as a popularity measure for Web pages. Besides its primary application in search engines, PageRank also became a major method for evaluating importance of nodes in different informational networks and database systems