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18 2 ⋅ pagerank background preferential PageRank and weakly preferential PageRank [Boldi et al., 2007]. We also formalize a sink preferential PageRank model below. Each formulation corresponds to a different way of converting P ̄ into a stochastic matrix and focuses on the columns of the matrix P ̄ that are completely 0. The dangling indicator vector d is 1 for such columns and 0 for columns where P ̄ is not completely zero. Formally, ⎧ ⎪1 Pij =0foralli dT =eT −eTP ̄, dj =⎨ (2.6) ⎪⎩0 Pij=/0forsomei. The strongly preferential PageRank problem uses the fully stochastic ma- trix Pv = P ̄ + vdT , (2.7) where v is the same vector as in the PageRank problem (2.3) or (2.4). To convince ourselves it is a column stochastic matrix, note that dT is positive only in the columns that caused P ̄ to fail to be stochastic, and so the matrix Pv will be the matrix P ̄ where each 0 column is replaced by v. Weakly preferential PageRank replaces each 0 column of P ̄ with a different distribution vector u, and uses the stochastic matrix Pu = P ̄ + udT , (2.8) whereuisanarbitrarydistributionvectorwithu ≥0,eTu=1,andu=/v. i 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].PDF Image | Instagram Cheat Sheet
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