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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
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    References listed on IDEAS

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    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).
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