# Degree and Principal Eigenvectors in Complex Networks

Degree and Principal Eigenvectors in Complex Networks

Author Faculty Department AbstractThe largest eigenvalue ? 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 ? 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 ? 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 ?1=N2N1 , 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 ? D is decreased.

Subjectnetworks

spectral radius

principal eigenvector

degree

as-sortativity

http://resolver.tudelft.nl/uuid:055f7afd-22bf-4bc5-b841-dddf31b66217

DOI PublisherSpringer

ISBN978-3-642-30045-5

SourceProceedings 11th International IFIP TC 6 Networking Conference, Part 1, Praag, 21-25 Mei 2012

Part of collectionInstitutional Repository

Document typeconference paper

Rights(c) 2012 Li, C.

Wang, H.

Van Mieghem, P.