# Degree and Principal Eigenvectors in Complex Networks

Abstract : The largest eigenvalue λ1 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 λ1. This motivates us to investigate properties of the principal eigenvector x1 and its relation with the degree vector. (a) We illustrate and explain why the average E[x1] decreases with the linear degree correlation coefficient ρD 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 Nk 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 ρD is decreased.
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https://hal.inria.fr/hal-01531136
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### Citation

Cong Li, Huijuan Wang, Piet Mieghem. Degree and Principal Eigenvectors in Complex Networks. 11th International Networking Conference (NETWORKING), May 2012, Prague, Czech Republic. pp.149-160, ⟨10.1007/978-3-642-30045-5_12⟩. ⟨hal-01531136⟩

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