IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v9y2021i13p1522-d584751.html
   My bibliography  Save this article

Iterative Methods for the Computation of the Perron Vector of Adjacency Matrices

Author

Listed:
  • Anna Concas

    (Department of Mathematics and Computer Science, University of Cagliari, Via Ospedale 72, 09124 Cagliari, Italy)

  • Lothar Reichel

    (Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA)

  • Giuseppe Rodriguez

    (Department of Mathematics and Computer Science, University of Cagliari, Via Ospedale 72, 09124 Cagliari, Italy)

  • Yunzi Zhang

    (Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA)

Abstract

The power method is commonly applied to compute the Perron vector of large adjacency matrices. Blondel et al. [SIAM Rev. 46, 2004] investigated its performance when the adjacency matrix has multiple eigenvalues of the same magnitude. It is well known that the Lanczos method typically requires fewer iterations than the power method to determine eigenvectors with the desired accuracy. However, the Lanczos method demands more computer storage, which may make it impractical to apply to very large problems. The present paper adapts the analysis by Blondel et al. to the Lanczos and restarted Lanczos methods. The restarted methods are found to yield fast convergence and to require less computer storage than the Lanczos method. Computed examples illustrate the theory presented. Applications of the Arnoldi method are also discussed.

Suggested Citation

  • Anna Concas & Lothar Reichel & Giuseppe Rodriguez & Yunzi Zhang, 2021. "Iterative Methods for the Computation of the Perron Vector of Adjacency Matrices," Mathematics, MDPI, vol. 9(13), pages 1-16, June.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:13:p:1522-:d:584751
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/9/13/1522/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/9/13/1522/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Duncan J. Watts & Steven H. Strogatz, 1998. "Collective dynamics of ‘small-world’ networks," Nature, Nature, vol. 393(6684), pages 440-442, June.
    2. BLONDEL, Vincent D. & GAJARDO, Anahi & HYEMANS, Maureen & SENELLART, Pierre, 2004. "A measure of similarity between graph vertices: Applications to synonym extraction and web searching," LIDAM Reprints CORE 1798, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Huan Wang & Chuang Ma & Han-Shuang Chen & Ying-Cheng Lai & Hai-Feng Zhang, 2022. "Full reconstruction of simplicial complexes from binary contagion and Ising data," Nature Communications, Nature, vol. 13(1), pages 1-10, December.
    2. Vinayak, & Raghuvanshi, Adarsh & kshitij, Avinash, 2023. "Signatures of capacity development through research collaborations in artificial intelligence and machine learning," Journal of Informetrics, Elsevier, vol. 17(1).
    3. Supriya Tiwari & Pallavi Basu, 2024. "Quasi-randomization tests for network interference," Papers 2403.16673, arXiv.org, revised Oct 2024.
    4. Anzhi Sheng & Qi Su & Aming Li & Long Wang & Joshua B. Plotkin, 2023. "Constructing temporal networks with bursty activity patterns," Nature Communications, Nature, vol. 14(1), pages 1-10, December.
    5. Samrachana Adhikari & Beau Dabbs, 2018. "Social Network Analysis in R: A Software Review," Journal of Educational and Behavioral Statistics, , vol. 43(2), pages 225-253, April.
    6. Wang, Xiaojie & Slamu, Wushour & Guo, Wenqiang & Wang, Sixiu & Ren, Yan, 2022. "A novel semi local measure of identifying influential nodes in complex networks," Chaos, Solitons & Fractals, Elsevier, vol. 158(C).
    7. Lin, Dan & Wu, Jiajing & Xuan, Qi & Tse, Chi K., 2022. "Ethereum transaction tracking: Inferring evolution of transaction networks via link prediction," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 600(C).
    8. Ferreira, D.S.R. & Ribeiro, J. & Oliveira, P.S.L. & Pimenta, A.R. & Freitas, R.P. & Dutra, R.S. & Papa, A.R.R. & Mendes, J.F.F., 2022. "Spatiotemporal analysis of earthquake occurrence in synthetic and worldwide data," Chaos, Solitons & Fractals, Elsevier, vol. 165(P2).
    9. Mario V. Tomasello & Mauro Napoletano & Antonios Garas & Frank Schweitzer, 2017. "The rise and fall of R&D networks," Industrial and Corporate Change, Oxford University Press and the Associazione ICC, vol. 26(4), pages 617-646.
    10. Qinghu Liao & Wenwen Dong & Boxin Zhao, 2023. "A New Strategy to Solve “the Tragedy of the Commons” in Sustainable Grassland Ecological Compensation: Experience from Inner Mongolia, China," Sustainability, MDPI, vol. 15(12), pages 1-24, June.
    11. Xinyu Huang & Dongming Chen & Dongqi Wang & Tao Ren, 2020. "MINE: Identifying Top- k Vital Nodes in Complex Networks via Maximum Influential Neighbors Expansion," Mathematics, MDPI, vol. 8(9), pages 1-25, August.
    12. Xueguo Xu & Chen Xu & Wenxin Zhang, 2022. "Research on the Destruction Resistance of Giant Urban Rail Transit Network from the Perspective of Vulnerability," Sustainability, MDPI, vol. 14(12), pages 1-26, June.
    13. Jianhong Chen & Hongcai Ma & Shan Yang, 2023. "SEIOR Rumor Propagation Model Considering Hesitating Mechanism and Different Rumor-Refuting Ways in Complex Networks," Mathematics, MDPI, vol. 11(2), pages 1-22, January.
    14. Xiaokun Su & Chenrouyu Zheng & Yefei Yang & Yafei Yang & Wen Zhao & Yue Yu, 2022. "Spatial Structure and Development Patterns of Urban Traffic Flow Network in Less Developed Areas: A Sustainable Development Perspective," Sustainability, MDPI, vol. 14(13), pages 1-18, July.
    15. Manisha Sawant & Rupali Patil & Tanmay Shikhare & Shreyas Nagle & Sakshi Chavan & Shivang Negi & Neeraj Dhanraj Bokde, 2022. "A Selective Review on Recent Advancements in Long, Short and Ultra-Short-Term Wind Power Prediction," Energies, MDPI, vol. 15(21), pages 1-24, October.
    16. Vincenza Carchiolo & Marco Grassia & Michele Malgeri & Giuseppe Mangioni, 2022. "Co-Authorship Networks Analysis to Discover Collaboration Patterns among Italian Researchers," Future Internet, MDPI, vol. 14(6), pages 1-15, June.
    17. Gregory Gutin & Tomohiro Hirano & Sung-Ha Hwang & Philip R. Neary & Alexis Akira Toda, 2021. "The effect of social distancing on the reach of an epidemic in social networks," Journal of Economic Interaction and Coordination, Springer;Society for Economic Science with Heterogeneous Interacting Agents, vol. 16(3), pages 629-647, July.
    18. Jie, Ke-Wei & Liu, San-Yang & Sun, Xiao-Jun & Xu, Yun-Cheng, 2023. "A dynamic ripple-spreading algorithm for solving mean–variance of shortest path model in uncertain random networks," Chaos, Solitons & Fractals, Elsevier, vol. 167(C).
    19. Michael D. König & Xiaodong Liu & Yves Zenou, 2019. "R&D Networks: Theory, Empirics, and Policy Implications," The Review of Economics and Statistics, MIT Press, vol. 101(3), pages 476-491, July.
    20. Yu Gong & Xiaojiang Xu & Changping Zhao & Tobias Schoenherr, 2024. "Multi-Tier Supply Chain Learning Networks: A Simulation Study Based on the Experience-Weighted Attraction (EWA) Model," Sustainability, MDPI, vol. 16(10), pages 1-25, May.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:9:y:2021:i:13:p:1522-:d:584751. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.