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A Hessenberg-type algorithm for computing PageRank Problems

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Abstract

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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Notes

  1. Here, we give its executable codes in the website: https://github.com/Hsien-Ming-Ku/PageRank-Hessenberghttps://github.com/Hsien-Ming-Ku/PageRank-Hessenberg.

  2. See the details from https://developer.nvidia.com/blog/six-ways-saxpy/.

  3. 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: https://github.com/Hsien-Ming-Ku/PageRank-Hessenberg for testing purposes.

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

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Acknowledgements

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.

Funding

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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Authors and Affiliations

  1. School of Economic Mathematics/Institute of Mathematics, Southwestern University of Finance and Economics, Chengdu, Sichuan, 611130, People’s Republic of China

    Xian-Ming Gu

  2. Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, University of Groningen, Nijenborgh 9, P.O. Box 407, 9700 AK, Groningen, The Netherlands

    Xian-Ming Gu

  3. Department of Mathematics, Faculty of Science and Technology, University of Macau, Avenida da Universidade, Macao, People’s Republic of China

    Siu-Long Lei

  4. College of Arts and Sciences, Shanghai Maritime University, Shanghai, 201306, People’s Republic of China

    Ke Zhang

  5. Department of Applied Mathematics, College of Science, Sichuan Agricultural University, Yaan, Sichuan, 625014, People’s Republic of China

    Zhao-Li Shen

  6. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan, 611731, People’s Republic of China

    Chun Wen

  7. Facoltà di Scienze e Tecnologie Informatiche, Libera Università di Bolzano, Dominikanerplatz 3 - piazza Domenicani, 3 Italy, 39100, Bozen-Bolzano, Italy

    Bruno Carpentieri

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Gu, XM., Lei, SL., Zhang, K. et al. A Hessenberg-type algorithm for computing PageRank Problems. Numer Algor 89, 1845–1863 (2022). https://doi.org/10.1007/s11075-021-01175-w

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