Degree and Principal Eigenvectors in Complex Networks

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.Network Architectures and Services Group (NAS)Electrical Engineering, Mathematics and Computer Scienc