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38 2 ⋅ pagerank background To begin, we require that eTw = 0 so that R−1 = I − e wT. Our expanded equation is (I−e wT)(αJ+(1−α)e vTV)=(αJ+(1−α)e eT)(I−e wT). (2.33) 11111 A few steps of algebra using eTw = 0 and Je = e yield the equivalent 111 expression (1−α)e vTV−αe wTJ=(1−α)e eT −e wT, (2.34) 11111 where everything shares the common e1. This expression encodes only a single vector semi-simple eigenvalue of S. Write J = [ 1 αJ+(1−α)e eT =⎢ ⎥. 10 Note that J1 is different from J2 in the previous section. This result is a corollary of the old theorem from Brauer about eigenvalues of combinations of matrices. Elden’s proof is specific to the PageRank case and more modern. 11 In the remainder of the document, x(1) means this limiting value. 11 (1−α)vTV−αwTJ=(1−α)eT −wT 1 or more elegantly the linear system wT(I−αJ)=(1−α)(eT −vTV). 1 (2.35) (2.36) This last step completes the derivation of the Jordan form. Note, however, that the eigenvalues of G, which are also the eigenvalues of M, are given by the diagonal of αJ + (1 − α)e eT . Recall that J = 1, which corresponds to a 11 11 ] so that10 ⎡⎤ J1 ⎢1 ⎥ 1 1 ⎢⎣ αJ1⎥⎦ This analysis confirms the following theorem. Theorem5(Eldén[2004];Brauer[1952]theorem29). Iftheeigenvaluesof Pare1,λ2,...,λn thentheeigenvaluesofM(α)are1,αλ2,...,αλn. finding the limit With the Jordan form from eqs. (2.27) to (2.30) we can work out the PageRank vector x(1) in the limit sense.11 For α < 1, the PageRank vector +(α) satisfies +(α)T =+(α)TG, (2.37) =+(α)TVR(α)(αJ+(1−α)e eT)R(α)−1V−1, (2.38) 11PDF Image | Instagram Cheat Sheet
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