Degree and Principal Eigenvectors in Complex Networks

Title

Author

Li, C.
Wang, H.
Van Mieghem, P.

Faculty

Electrical Engineering, Mathematics and Computer Science

Department

Network Architectures and Services Group (NAS)

Abstract

The 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.

Subject

networks
spectral radius
principal eigenvector
degree
as-sortativity

To reference this document use:

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

DOI

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

Publisher

Springer

ISBN

978-3-642-30045-5

Source

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