On perturbations of principal eigenvectors of substochastic matrices

In this report we consider substochastic matrices with some zero rows and study the sensitivity of their eigenvectors to the modifications of zero entries. The analysis is prompted by and directly applied to the Google method of ranking the web sites, which substitutes the pages without outlinks (dangling nodes) in the original web link matrix by non-zero rows (dangling vectors). We present an analysis of the influence of artificial links attributed to the dangling nodes on the principal eigenvectors of the web matrix. We clarify when the choice of the dangling vector does not change the original eigenvectors and give an evaluation for perturbations of the principal eigenvectors when they are subject to modification.