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A Hessenberg-type algorithm for computing PageRank Problems


PageRank is a widespread model for analysing the relative relevance of nodes within large graphs arising in several applications. In the current paper, we present a cost-effective Hessenberg-type method built upon the Hessenberg process for the solution of difficult PageRank problems. The new method is very competitive with other popular algorithms in this field, such as Arnoldi-type methods, especially when the damping factor is close to 1 and the dimension of the search subspace is large. The convergence and the complexity of the proposed algorithm are investigated. Numerical experiments are reported to show the efficiency of the new solver for practical PageRank computations.

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    Here, we give its executable codes in the website:

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    See the details from

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    Numerical results with the IDR(s)-based PageRank method are omitted due to its unsatisfactory performance for large values of s and m. However, the MATLAB code of the IDR(s)-based PageRank method is still included in our GitHub repository: for testing purposes.

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    In fact, the restart number is often chosen in the PageRank literature as m ≤ 10 [27, 30, 31].


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The authors would like to thank Prof. Zhongxiao Jia for his comments about the strategy used in the refined Arnoldi algorithm. Meanwhile, the authors are grateful to Dr. Reinaldo Astudillo (ASML Holding N.V.) for his kind suggestions about executing the IDR-based Hessenberg decompositions used in Section 2.2.


This research is supported by NSFC (11601323 and 11801463), the Applied Basic Research Program of Sichuan Province (2020YJ0007), and the research grants MYRG2018-00025-FST, MYRG2020-00208-FST from University of Macau. The last author is member of the Gruppo Nazionale per il Calcolo Scientifico (GNCS) of the Istituto Nazionale di Alta Matematica (INdAM) and his work was partially supported by INdAM-GNCS under Progetti di Ricerca 2020.

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Correspondence to Siu-Long Lei.

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Gu, XM., Lei, SL., Zhang, K. et al. A Hessenberg-type algorithm for computing PageRank Problems. Numer Algor (2021).

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  • PageRank vector
  • Hessenberg process
  • Arnoldi process
  • Krylov subspace method
  • Ritz values

Mathematics Subject Classification (2010)

  • 65F15
  • 65F10
  • 65Y20