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.
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References
Barabasi, A.-L., Albert, R.: Emergency of Scaling in Random Networks. Science 286, 509–512 (1999)
Barthélemy, M., Barrat, A., Pastor-Satorras, R., Vespignani, A.: Velocity and Hierarchical Spread of Epidemic Outbreaks in Scale-Free Networks. Phys. Rev. Lett. 92, 178701 (2004)
Boguñá, M., Pastor-Satorras, R., Vespignani, A.: Absence of Epidemic Threshold in Scale-Free Networks with Degree Correlations. Phys. Rev. Lett. 90, 028701 (2003)
Erdös, P., Rényi, A.: On Random Graphs. I. Publicationes Mathematicae 6, 290–297 (1959)
Li, C., Wang, H., Van Mieghem, P.: Bounds for the spectral radius of a graph when nodes are removed, accepted, Linear Algebra and its Applications
May, R.M., Lloyd, A.L.: Infection dynamics on scale-free networks. Phys. Rev. E 64, 066112 (2001)
Milanese, A., Sun, J., Nishikawa, T.: Approximating spectral impact of structural perturbations in large networks. Physical Review E 81, 046112 (2010)
Newman, M.E.J.: Assortative Mixing in Networks. Physic Rev. Lett. 89, 208701 (2002)
Newman, M.E.J.: Mixing patterns in networks. Phys. Rev. E 67, 026126 (2003)
Pastor-Satorras, R., Vespignani, A.: Epidemic dynamics and endemic states in complex networks. Phys. Rev. E 63, 066117 (2001)
Restrepo, J.G., Ott, E., Hunt, B.R.: Onset of synchronization in large networks of coupled oscillators. Physical Review E 71, 036151, 1–12 (2005)
Van Mieghem, P., Omic, J., Kooij, R.E.: Virus spread in networks. IEEE/ACM Transactions on Networking 17(1), 1–14 (2009)
Van Mieghem, P., Wang, H., Ge, X., Tang, S., Kuipers, F.A.: Influence of assortativity and degree-presreving rewiring on the spectra of networks. The European Physical Journal B, 643–652 (2010)
Van Mieghem, P.: Graph Spectra for Complex Networks. Cambridge University Press, Cambridge (2011)
Van Mieghem, P., Stevanović, D., Kuipers, F., Li, C., van de Bovenkamp, R., Liu, D., Wang, H.: Optimally decreasing the spectral radius of a graph by link removals. Physical Review E 84, 016101 (2011)
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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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