Lumping algorithms for computing Google’s PageRank and its derivative, with attention to unreferenced nodes

In this paper, we introduce five type nodes for lumping the Web matrix, and give a unified presentation of some popular lumping methods for PageRank. We show that the PageRank problem can be reduced to solving the PageRank corresponding to the strongly non-dangling and referenced nodes, and the full PageRank vector can be easily derived by some recursion formulations. Our new lumping strategy can reduce the original PageRank problem to a much smaller one, and it is much cheaper than the recursively reordering scheme. Furthermore, we discuss sensitivity of the PageRank vector, and present a lumping algorithm for computing its first order derivative. Numerical experiments show that the new algorithms are favorable when the matrix is large and the damping factor is high.