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The Minimum Spectral Radius of an Edge-Removed Network: A Hypercube Perspective

Author

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  • Yingbo Wu
  • Tianrui Zhang
  • Shan Chen
  • Tianhui Wang

Abstract

The spectral radius minimization problem (SRMP), which aims to minimize the spectral radius of a network by deleting a given number of edges, turns out to be crucial to containing the prevalence of an undesirable object on the network. As the SRMP is NP-hard, it is very unlikely that there is a polynomial-time algorithm for it. As a result, it is proper to focus on the development of effective and efficient heuristic algorithms for the SRMP. For that purpose, it is appropriate to gain insight into the pattern of an optimal solution to the SRMP by means of checking some regular networks. Hypercubes are a celebrated class of regular networks. This paper empirically studies the SRMP for hypercubes with two/three/four missing edges. First, for each of the three subproblems of the SRMP, a candidate for the optimal solution is presented. Second, it is shown that the candidate is optimal for small-sized hypercubes, and it is shown that the proposed candidate is likely to be optimal for medium-sized hypercubes. The edges in each candidate are evenly distributed over the network, which may be a common feature of all symmetric networks and hence is instructive in designing effective heuristic algorithms for the SRMP.

Suggested Citation

  • Yingbo Wu & Tianrui Zhang & Shan Chen & Tianhui Wang, 2017. "The Minimum Spectral Radius of an Edge-Removed Network: A Hypercube Perspective," Discrete Dynamics in Nature and Society, Hindawi, vol. 2017, pages 1-8, April.
    • Handle: RePEc:hin:jnddns:1382980
    DOI: 10.1155/2017/1382980
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    References listed on IDEAS

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    1. Tzu-Liang Kung & Cheng-Kuan Lin & Lih-Hsing Hsu, 2014. "On the maximum number of fault-free mutually independent Hamiltonian cycles in the faulty hypercube," Journal of Combinatorial Optimization, Springer, vol. 27(2), pages 328-344, February.
    2. Lu-Xing Yang & Xiaofan Yang, 2015. "A Novel Virus-Patch Dynamic Model," PLOS ONE, Public Library of Science, vol. 10(9), pages 1-16, September.
    3. Xiaofan Yang & Pengdeng Li & Lu-Xing Yang & Yingbo Wu, 2016. "Reducing the Spectral Radius of a Torus Network by Link Removal," PLOS ONE, Public Library of Science, vol. 11(5), pages 1-15, May.
    4. P. Van Mieghem & H. Wang & X. Ge & S. Tang & F. A. Kuipers, 2010. "Influence of assortativity and degree-preserving rewiring on the spectra of networks," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 76(4), pages 643-652, August.
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