Abstract. An important problem in web search is to determine the importance of each page. From the mathematical point of view, this problem consists in finding the nonnegative left eigenvector of a matrix corresponding to its dominant eigenvalue 1. Since this matrix is neither stochastic nor irreducible, the power method has convergence problems. So, the matrix is replaced by a convex combination, depending on a parameter c, with a rank one matrix. Its left principal eigenvector now depends on c, and it is the PageRank vector we are looking for. However, when c is close to 1, the problem is ill-conditioned, and the power method converges slowly. So, the idea developed in this paper consists in computing the PageRank vector for several values of c, andthen to extrapolate them, by a conveniently chosen rational function, at a point near 1. The choice of this extrapolating function is based on the mathematical expression of the PageRank vector as a function of c. Numerical experiments end the paper. 1. The problem

Abstract. We present a new iterative scheme for PageRank computation. The algorithm is applied to the linear system formulation of the problem, using inner-outer stationary iterations. It is simple, can be easily implemented and parallelized, and requires minimal storage overhead. Our convergence analysis shows that the algorithm is effective for a crude inner tolerance and is not sensitive to the choice of the parameters involved. The same idea can be used as a preconditioning technique for non-stationary schemes. Numerical examples featuring matrices of dimensions exceeding 100,000,000 in sequential and parallel environments demonstrate the merits of our technique. Our code is available online for viewing and testing, along with several large scale examples.

Sandia is a multiprogram laboratory operated by Sandia Corporation,

Abstract. The PageRank algorithm is used today within web information retrieval to provide a content-neutral ranking metric over web pages. It employs power method iterations to solve for the steady-state vector of a DTMC. The defining one-step probability transition matrix of this DTMC is derived from the hyperlink structure of the web and a model of web surfing behaviour which accounts for user bookmarks and memorised URLs. In this paper we look to provide a more accessible, more broadly applicable explanation than has been given in the literature of how to make PageRank calculation more tractable through removal of the danglingpage matrix. This allows web pages without outgoing links to be removed before we employ power method iterations. It also allows decomposition of the problem according to irreducible subcomponents of the original transition matrix. Our explanation also covers a PageRank extension to accommodate TrustRank. In setting out our alternative explanation, we introduce and apply a general linear algebraic theorem which allows us to map homogeneous singular linear systems of index one to inhomogeneous non-singular linear systems with a shared solution vector. As an aside, we show in this paper that irreducibility is not required for PageRank to be well-defined. 1