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International Conference on Research in Networking

NETWORKING 2012: NETWORKING 2012 pp 149–160Cite as

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Degree and Principal Eigenvectors in Complex Networks

Degree and Principal Eigenvectors in Complex Networks

  • Cong Li20,
  • Huijuan Wang20 &
  • Piet Van Mieghem20 
  • Conference paper
  • 1714 Accesses

  • 5 Citations

Part of the Lecture Notes in Computer Science book series (LNCCN,volume 7289)

Abstract

The largest eigenvalue λ 1 of the adjacency matrix powerfully characterizes dynamic processes on networks, such as virus spread and synchronization. The minimization of the spectral radius by removing a set of links (or nodes) has been shown to be an NP-complete problem. So far, the best heuristic strategy is to remove links/nodes based on the principal eigenvector corresponding to the largest eigenvalue λ 1. This motivates us to investigate properties of the principal eigenvector x 1 and its relation with the degree vector. (a) We illustrate and explain why the average E[x 1] decreases with the linear degree correlation coefficient ρ D in a network with a given degree vector; (b) The difference between the principal eigenvector and the scaled degree vector is proved to be the smallest, when \(\lambda _{1}=\frac{N_{2}}{N_{1}}\), where N k is the total number walks in the network with k hops; (c) The correlation between the principal eigenvector and the degree vector decreases when the degree correlation ρ D is decreased.

Keywords

  • networks
  • spectral radius
  • principal eigenvector
  • degree
  • assortativity

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Author information

Authors and Affiliations

  1. Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, P.O. Box 5031, 2600 GA, Delft, The Netherlands

    Cong Li, Huijuan Wang & Piet Van Mieghem

Authors
  1. Cong Li
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  2. Huijuan Wang
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  3. Piet Van Mieghem
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Editor information

Editors and Affiliations

  1. Department of Telecommunications Engineering, Czech Technical University in Prague, Technicka 2, 166 27, Prague 6, Czech Republic

    Robert Bestak & Lukas Kencl & 

  2. Alcatel-Lucent, Bell Labs, 600 Mountain Avenue, 07974-0636, Murray Hill, NJ, USA

    Li Erran Li

  3. Instituto IMDEA Networks, Avenida del Mar Mediterraneo 22, Leganes, 28918, Madrid), Spain

    Joerg Widmer

  4. Tsinghua-ChinaCache Joint Laboratory, Tsinghua University, FIT 3-429, Haidian District, 100016, Beijing, China

    Hao Yin

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© 2012 IFIP International Federation for Information Processing

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Cite this paper

Li, C., Wang, H., Van Mieghem, P. (2012). Degree and Principal Eigenvectors in Complex Networks. In: Bestak, R., Kencl, L., Li, L.E., Widmer, J., Yin, H. (eds) NETWORKING 2012. NETWORKING 2012. Lecture Notes in Computer Science, vol 7289. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30045-5_12

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  • DOI: https://doi.org/10.1007/978-3-642-30045-5_12

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  • Print ISBN: 978-3-642-30044-8

  • Online ISBN: 978-3-642-30045-5

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