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16 2 ⋅ pagerank background Instead of replacing 0 columns of P ̄ with a distribution, sink preferential PageRank inserts a 1 into the diagonal for each of these columns, which corresponds to using the stochastic matrix Pd = P ̄ + Diag[d], (2.9) where Diag[d] is a diagonal matrix with the entries of d along the diagonal. nota bene Unless otherwise noted, we use the strongly preferential Page- Rank formulation of the problem when P ̄ is sub-stochastic. This choice is made in most of the literature. pseudorank Recall that we defined a PageRank vector with eTx = 1. Relaxing that requirement on the strongly preferential PageRank problem yields a vector called PseudoRank [Boldi et al., 2007]. Problem 2 (PseudoRank). A PseudoRank vector y satisfies (I − αP ̄)y = σv (2.10) for σ = n, 1, or (1 − α). PageRank and PseudoRank are related by x = y/eT y. Proving it requires simplesubstitution.Notethatσ=eTy−αeTP ̄y.Consider y ( I − α P ̄ ) y − α v d T y σ − α d T y (I−αP)eTy = eTy = eTy v (2.11) (eT y − αeT P ̄y) − α(eT y − eT P ̄y) = eTy v (2.12) = (1 − α)v. (2.13) Many authors define PageRank as PseudoRank [McSherry, 2005; Gyöngyi et al., 2004]. While they share some equivalence, there is an important distinc- tion with regard to the limit when α → 1, and that’s discussed in section 2.7. 2.2.2 PageRank on a graph Most derivations of PageRank begin with PageRank on a graph, and most often it is the web graph. For an arbitrary directed graph G with adjacency matrixA(Aij =1ifnodeihasadirectededgetonode j,andAij =0ifthere is no edge), the PageRank vector is commonly defined by applying one of the sub-stochastic algorithms to the matrix P ̄ =ATD+, (2.14)PDF Image | ALGORITHMS FOR PAGERANK SENSITIVITY DISSERTATION
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