A general multi-splitting iteration method for computing PageRank

Mao, Tian; Zhang, Yan; Wang, Yudong; Tian, Zhaolu

doi:10.1007/s40314-019-0830-8

Cited by 12 publications

(11 citation statements)

References 32 publications

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“…Thus, f (ψ) is monotonically decreasing, and f (ψ) is smaller for larger ψ. However, similar to the analysis in [5,24,34], the outer iterations (3.3) converge faster if ψ is close to 1, while the inner iterations (3.4) is faster if ψ tends to zero. In practice, the values of ψ ∈ [0.5, 1) are appropriate choices for the GMMS iteration method.…”

Section: The Choices Of the Parameters In The Gmms Iteration Methodssupporting

confidence: 57%

“…For the parameter m k , just as the analysis in [5,34], which is also true for m k . A larger m k may spend a long computational time performing inner iterations (3.4), just to implement a single outer iteration (3.3) and m iterations (3.2) at a time, then slows the overall convergence and takes more computational time.…”

Section: The Choices Of the Parameters In The Gmms Iteration Methodsmentioning

confidence: 93%

“…In this section, we firstly review the GIO iteration method [5,34], then propose the GMMS iteration method, and analyze its global convergence without imposing any restrictions on the damping factors and the stopping tolerances. Lemma 3.1 [27].…”

Section: The Gmms Iteration Methodsmentioning

confidence: 99%

“…Next, Wen et al [6] gave the MPIO iteration method, which is a variant of the PIO iteration. Inspired by these iteration frameworks in [6,11], based on multi-step matrix splitting iterations and GIO iteration [5,34], we propose the GMMS iteration method for solving the linear system (1.2). In addition, by choosing appropriate matrix splittings and parameters, the GMMS iteration method can reduce to the MPIO and PIO iteration methods, respectively.…”

Section: Introductionmentioning

confidence: 99%

“…

Based on the general inner-outer (GIO) iteration method [5,34] and the iteration framework[6], we present a general multi-step matrix splitting (GMMS) iteration method for computing PageRank, and analyze its overall convergence property. Moreover, the same idea can be used as a preconditioning technique for accelerating the Krylov subspace methods, such as GMRES method.

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mentioning

confidence: 99%

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A general multi-step matrix splitting iteration method for computing PageRank

Tian

Liu

2021

Filomat

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Based on the general inner-outer (GIO) iteration method [5,34] and the iteration framework [6], we present a general multi-step matrix splitting (GMMS) iteration method for computing PageRank, and analyze its overall convergence property. Moreover, the same idea can be used as a preconditioning technique for accelerating the Krylov subspace methods, such as GMRES method. Finally, several numerical examples are given to illustrate the effectiveness of the proposed algorithm.

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Section: The Choices Of the Parameters In The Gmms Iteration Methodssupporting

confidence: 57%

Section: The Choices Of the Parameters In The Gmms Iteration Methodsmentioning

confidence: 93%

Section: The Gmms Iteration Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…

…”

mentioning

confidence: 99%

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A general multi-step matrix splitting iteration method for computing PageRank

Tian

Liu

2021

Filomat

Self Cite

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Based on the general inner-outer (GIO) iteration method [5,34] and the iteration framework [6], we present a general multi-step matrix splitting (GMMS) iteration method for computing PageRank, and analyze its overall convergence property. Moreover, the same idea can be used as a preconditioning technique for accelerating the Krylov subspace methods, such as GMRES method. Finally, several numerical examples are given to illustrate the effectiveness of the proposed algorithm.

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Preconditioned weighted full orthogonalization method for solving singular linear systems from PageRank problems

Shen,

Carpentieri,

Wen

et al. 2023

Numerical Linear Algebra App

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The PageRank model, which was first proposed by Google for its web search engine application, has since become a popular computational tool in a wide range of scientific fields, including chemistry, bioinformatics, neuroscience, bibliometrics, social networks, and others. PageRank calculations necessitate the use of fast computational techniques with low algorithmic and memory complexity. In recent years, much attention has been paid to Krylov subspace algorithms for solving difficult PageRank linear systems, such as those with large damping parameters close to one. In this article, we examine the full orthogonalization method (FOM). We present a convergence study of the method that extends and clarifies part of the conclusions reached in Zhang et al. (J Comput Appl Math. 2016; 296:397–409.). Furthermore, we demonstrate that FOM is breakdown free when solving singular PageRank linear systems with index one and we investigate the effect of using weighted inner‐products instead of conventional inner‐products in the orthonormalization procedure on FOM convergence. Finally, we develop a shifted polynomial preconditioner that takes advantage of the special structure of the PageRank linear system and has a good ability to cluster most of the eigenvalues, making it a good choice for an iterative method like FOM or GMRES. Numerical experiments are presented to support the theoretical findings and to evaluate the performance of the new weighted preconditioned FOM PageRank solver in comparison to other established solvers for this class of problem, including conventional stationary methods, hybrid combinations of stationary and Krylov subspace methods, and multi‐step splitting strategies.

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