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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) 2.2 ⋅ the pagerank problem 19 = 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) where D is a diagonal matrix with diagonal entries Dii = (Ae)i = outdegree of node i, and D+ is the pseudo-inverse [Golub and van Loan, 1996], another diagonal matrix with (D+) ii ⎧ ⎪ ⎪ ⎪ 1 / D i i = ⎨ ⎪ ⎪ ⎪⎩ 0 D i i =/ 0 D i i = 0 . (2.15) In the context of web search, each web page corresponds to a node in G, and nodes u and v are connected with a directed edge if the page corresponding to node u links to the page corresponding to node v.PDF Image | Instagram Cheat Sheet
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