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Asymptotic analysis for personalized Web search. Memorandum 1884
, 2008
"... 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 s ..."
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Cited by 17 (2 self)
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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 d = ∑N i=1 AiRi + B, where Ri’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 heavytailed, 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.
Information ranking and power laws on trees
 ADVANCES IN APPLIED PROBABILITY, 42(4), 2010
, 2010
"... We consider the stochastic analysis of information ranking algorithms of large interconnected data sets, e.g. Google’s PageRank algorithm for ranking pages on the World Wide Web. The stochastic formulation of the problem results in an equation of the form R D N∑ = Q + CiRi, where N, Q, {Ri}i≥1, {C, ..."
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Cited by 15 (11 self)
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We consider the stochastic analysis of information ranking algorithms of large interconnected data sets, e.g. Google’s PageRank algorithm for ranking pages on the World Wide Web. The stochastic formulation of the problem results in an equation of the form R D N∑ = Q + CiRi, where N, Q, {Ri}i≥1, {C, Ci}i≥1 are independent nonnegative random variables, {C, Ci}i≥1 are identically distributed, and {Ri}i≥1 are independent copies of R; D = stands for equality in distribution. We study the asymptotic properties of the distribution of R that, in the context of PageRank, represents the frequencies of highly ranked pages. The preceding equation is interesting in its own right since it belongs to a more general class of weighted branching processes that have been found useful in the analysis of many other algorithms. Our first main result shows that if ENE[C α] = 1, α> 0 and Q, N satisfy additional moment conditions, then R has a power law distribution of index α. This result is obtained using a new approach based on an extension of Goldie’s (1991) implicit renewal theorem. Furthermore, when N is regularly varying of index α> 1, ENE[C α] < 1 and Q, C have higher moments than α, then the distributions of R and N are tail equivalent. The latter result is derived via a novel sample path large deviation method for recursive random sums. Similarly, we characterize the situation when the distribution of R is determined by the tail of Q. The preceding approaches may be of independent interest, as they can be used for analyzing other functionals on trees. We also briefly discuss the engineering implications of our results.
Tail behavior of solutions of linear recursions on trees. Stochastic Process
 Appl
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Ranking algorithms on directed configuration networks
, 2014
"... 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 R ∗ that can b ..."
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Cited by 2 (2 self)
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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 R ∗ that can be written as a linear combination of i.i.d. copies of the endogenous solution to a stochastic fixed point equation of the form R D= N∑ i=1
Multiple equilibria of nonhomogeneous Markov chains and selfvalidating Web rankings
 in "SIAM Journal on Matrix Analysis and Applications", Accepted for publication
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A framework for evaluating statistical dependencies and
"... rank correlations in power law graphs ..."
On Page Rank
"... In this paper the concept of page rank for the world wide web is discussed. The possibility of describing the distribution of page rank by an exponential law is considered. It is shown that the concept is essentially equal to that of status score, a centrality measure discussed already in 1953 by Ka ..."
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In this paper the concept of page rank for the world wide web is discussed. The possibility of describing the distribution of page rank by an exponential law is considered. It is shown that the concept is essentially equal to that of status score, a centrality measure discussed already in 1953 by Katz. A structural classification of users in the web is given in terms of graph theoretical concepts. Key words: page rank, status score, graph
Coupling on weighted branching trees
"... This paper considers linear functions constructed on two different weighted branching trees and provides explicit bounds for their KantorovichRubinstein distance in terms of couplings of their corresponding generic branching vectors. By applying these bounds to a sequence of weighted branching tree ..."
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This paper considers linear functions constructed on two different weighted branching trees and provides explicit bounds for their KantorovichRubinstein distance in terms of couplings of their corresponding generic branching vectors. By applying these bounds to a sequence of weighted branching trees, we derive the weak convergence of the corresponding linear processes. In the special case where sequence of trees converges to a weighted branching process, the limits can be represented as the endogenous solution to a stochastic fixedpoint equation of the form R
PageRank in scalefree random graphs?
"... Abstract. We analyze the distribution of PageRank on a directed configuration model and show that as the size of the graph grows to infinity it can be closely approximated by the PageRank of the root node of an appropriately constructed tree. This tree approximation is in turn related to the solut ..."
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Abstract. We analyze the distribution of PageRank on a directed configuration model and show that as the size of the graph grows to infinity it can be closely approximated by the PageRank of the root node of an appropriately constructed tree. This tree approximation is in turn related to the solution of a linear stochastic fixed point equation that has been thoroughly studied in the recent literature. 1