An Arnoldi-type algorithm for computing page rank

Golub, Gene H.; Greif, Chen

doi:10.1007/s10543-006-0091-y

Cited by 102 publications

(142 citation statements)

References 19 publications

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“…Due to the gap 1 − α ≈ 0.15 between the largest eigenvalue λ = 1 and other eigenvalues the PageRank algorithm permits an efficient and simple determination of the PageRank by the power iteration method [7]. It is also possible to use the powerful Arnoldi method [12][13][14] to compute efficiently the eigenspectrum λ i of the Google matrix: N N nA 2003 455 436 2 033 173 6000 2005 1 635 882 11 569 195 6000 2007 2 902 764 34 776 800 6000 2009 3 484 341 52 846 242 6000 200908 3 282 257 71 012 307 6000 2011 3 721 339 66 454 329 6000 The Arnoldi method allows to find a several thousands of eigenvalues λ i with maximal |λ| for a matrix size N as large as a few tens of millions [10,11,14,15]. Usually, at α = 1 the largest eigenvalue λ = 1 is highly degenerate [15] due to many invariant subspaces which define many independent Perron-Frobenius operators providing (at least) one eigenvalue λ = 1.…”

Section: Introductionmentioning

confidence: 99%

Time evolution of Wikipedia network ranking

et al. 2013

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Abstract.We study the time evolution of ranking and spectral properties of the Google matrix of English Wikipedia hyperlink network during years [2003][2004][2005][2006][2007][2008][2009][2010][2011]. The statistical properties of ranking of Wikipedia articles via PageRank and CheiRank probabilities, as well as the matrix spectrum, are shown to be stabilized for [2007][2008][2009][2010][2011]. A special emphasis is done on ranking of Wikipedia personalities and universities. We show that PageRank selection is dominated by politicians while 2DRank, which combines PageRank and CheiRank, gives more accent on personalities of arts. The Wikipedia PageRank of universities recovers 80% of top universities of Shanghai ranking during the considered time period.

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Section: Introductionmentioning

confidence: 99%

Time evolution of Wikipedia network ranking

et al. 2013

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“…In many applications, α is given the value 0.85, but care must be taken to ensure that P α sufficiently resembles P [11]. For the balancing algorithm we are unable to prove a result as strong as Theorem 5.1.…”

Section: Practicalitiesmentioning

confidence: 96%

The Sinkhorn–Knopp Algorithm: Convergence and Applications

Knight¹

2008

SIAM J. Matrix Anal. & Appl.

238

191

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Abstract. As long as a square nonnegative matrix A contains sufficient nonzero elements, then the Sinkhorn-Knopp algorithm can be used to balance the matrix, that is, to find a diagonal scaling of A that is doubly stochastic. It is known that the convergence is linear and an upper bound has been given for the rate of convergence for positive matrices. In this paper we give an explicit expression for the rate of convergence for fully indecomposable matrices.We describe how balancing algorithms can be used to give a measure of web page significance. We compare the measure with some well known alternatives, including PageRank. We show that with an appropriate modification, the Sinkhorn-Knopp algorithm is a natural candidate for computing the measure on enormous data sets.

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“…The effectiveness of this ranking procedure is witnessed by the success of the search engine Google. Since the first investigations, various procedures and algorithms have been presented to determine the PageRank and its dynamical evolution [10,12,13,14]. It is fair to say that the study of PageRank constitutes a field of research on its own.…”

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confidence: 99%

PageRank equation and localization in the WWW

et al. 2009

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The World Wide Web is one of the most important communication systems we use in our everyday life. Despite its central role, the growth and the development of the WWW is not controlled by any central authority. This situation has created a huge ensemble of connections whose complexity can be fruitfully described and quantified by network theory [1,2,3,4]. One important application to sort out the information present in these connections is given by the PageRank alghorithm [5]. Computation of this quantity is usually made iteratively with a large use of computational time. In this paper we show that the PageRank can be expressed in terms of a wave function obeying a Schrödinger-like equation. In particular the topological disorder given by the unbalance of outgoing and ingoing links between pages, induces wave function and potential structuring. This allows to directly localize the pages with the largest score. Through this new representation we can now compute the PageRank without iterative techniques. For most of the cases of interest our method is faster than the original one. Our results also clarify the role of topology in the diffusion of information within complex networks. The whole approach opens the possibility to novel techniques inspired by quantum physics for the analysis of the WWW properties.

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An Arnoldi-type algorithm for computing page rank

Cited by 102 publications

References 19 publications

Time evolution of Wikipedia network ranking

Time evolution of Wikipedia network ranking

The Sinkhorn–Knopp Algorithm: Convergence and Applications

PageRank equation and localization in the WWW

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