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54 3 ⋅ the pagerank derivative To be explicit, in the strongly-preferential case the PageRank vector satisfies (I − αP ̄ − αvdT)x(α) = (1 − α)x(α) and its derivative satisfies14 1 α system for z(α) from (3.6) is ( I − α P ̄ − α v d T ) z ( α ) = ( 1 − α ) x ( α ) , which is a weakly preferential PageRank system. At this point, Boldi et al. [2007] provide a solution for a related problem They formalize that the strongly and weakly preferential PageRank systems are related by a rank-one change. Applying the Sherman-Morrison-Woodbury formula and an extra PageRank solve transitions between these formulations. For the derivative, applying this technique, however, then requires three Page- Rank solves. The extra solve is not necessary because a bit of algebra fixes the situation entirely, and there is no need for an explicit application of the Sherman-Morrison-Woodbury formula. Notice that (I−αP ̄)x′(α)= 1x(α)+(αdTx′(α)−1/α)v, α (I−αP ̄)x(α)=(1−α+αdTx(α))v, and (I − αP ̄)z(α) = (1 − α + αdT z(α))x(α). Consequently, x′(α) is still a linear combination of z(α) and x(α) where each is a strongly preferential PageRank vector. The coefficient for z(α) is available, so x′(α) = 1 z(α) + ηx(α). α(1 − α + αdT z(α)) We now exploit eT x′(α) = 0 to compute η and present algorithm 1 to compute the derivative. No algorithm in this thesis is missing Matlab code, and program 4 shows a simple implementation of this algorithm. 14 This derivative is just the same as (I−αP)x′(α)withP = P ̄ +vdT, which supports working with P for PageRank theory. (I − αP ̄ − αvdT )x′(α) = Simply using strongly preferential solves does not work because the PageRank (x(α) − v). δPDF Image | Instagram Cheat Sheet
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