An Inner-Outer Iteration for Computing PageRank

Gleich, David F.; Gray, Andrew P.; Greif, Chen; Lau, Tracy

doi:10.1137/080727397

Cited by 80 publications

(74 citation statements)

References 27 publications

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“…1. a fixed-point iteration, as in the power method and Richardson method; 2. a shifted fixed-point iteration, as in SS-HOPM [Kolda and Mayo, 2011]; 3. a non-linear inner-outer iteration, akin to Gleich et al [2010]; 4. an inverse iteration, as in the inverse power method; and 5. a Newton iteration. We will show that the first four of them converge in the case that α < 1/(m − 1) for an order-m tensor.…”

Section: Algorithms For Multilinear Pagerankmentioning

confidence: 99%

Multilinear PageRank

Gleich¹,

Lim²,

Yu³

2015

SIAM J. Matrix Anal. & Appl.

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102

219

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In this paper, we first extend the celebrated PageRank modification to a higher-order Markov chain. Although this system has attractive theoretical properties, it is computationally intractable for many interesting problems. We next study a computationally tractable approximation to the higher-order PageRank vector that involves a system of polynomial equations called multilinear PageRank, which is a type of tensor PageRank vector. It is motivated by a novel "spacey random surfer" model, where the surfer remembers bits and pieces of history and is influenced by this information. The underlying stochastic process is an instance of a vertex-reinforced random walk. We develop convergence theory for a simple fixed-point method, a shifted fixed-point method, and a Newton iteration in a particular parameter regime. In marked contrast to the case of the PageRank vector of a Markov chain where the solution is always unique and easy to compute, there are parameter regimes of multilinear PageRank where solutions are not unique and simple algorithms do not converge. We provide a repository of these non-convergent cases that we encountered through exhaustive enumeration and randomly sampling that we believe is useful for future study of the problem

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Section: Algorithms For Multilinear Pagerankmentioning

confidence: 99%

Multilinear PageRank

Gleich¹,

Lim²,

Yu³

2015

SIAM J. Matrix Anal. & Appl.

Self Cite

102

219

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show abstract

“…We used the compressed graph data from the Laboratory of Web Algorithms (LAW) at the Università degli studi di Milano [30,31]. We used the bvgraph MATLAB package [32]. The stanford-cs database [33] is a 2001 crawl that includes all pages in the cs.stanford.edu domain.…”

Section: Web Samples and Social Networkmentioning

confidence: 99%

Uncovering disassortativity in large scale-free networks

2013

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Mixing patterns in large self-organizing networks, such as the Internet, the World Wide Web, social and biological networks are often characterized by degree-degree dependencies between neighbouring nodes. In this paper we propose a new way of measuring degree-degree dependencies. One of the problems with the commonly used assortativity coefficient is that in disassortative networks its magnitude decreases with the network size. We mathematically explain this phenomenon and validate the results on synthetic graphs and real-world network data. As an alternative, we suggest to use rank correlation measures such as Spearman's rho. Our experiments convincingly show that Spearman's rho produces consistent values in graphs of different sizes but similar structure, and it is able to reveal strong (positive or negative) dependencies in large graphs. In particular, we discover much stronger negative degree-degree dependencies in Web graphs than was previously thought. Rank correlations allow us to compare the assortativity of networks of different sizes, which is impossible with the assortativity coefficient due to its genuine dependence on the network size. We conclude that rank correlations provide a suitable and informative method for uncovering network mixing patterns.

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“…Therefore, the larger the β in (1.3), the faster the rate of convergence, which inosculates with the analysis in [17].…”

Section: Regular Splitting Iteration Methodsmentioning

confidence: 69%