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48 3 ⋅ the pagerank derivative of the theoretical discussion. A lackluster conclusion is that the derivative seems too correlated with PageRank, and does not appear to provide any suf- ficient guidance with regard to appropriate value of α; although, experiments with web spam in the next chapter (section 4.8.4) show that the derivative does have some useful properties. 3.1 formulations Given that the introduction to the chapter mentions that the derivative of the PageRank vector exists, we first address the burning question: what is it? Remember figure 2.2 and all the different ways of looking at the PageRank problem from the previous chapter? Graph or Web graph PseudoRank PageRank Eigensystems Linear systems Algorithms Substochastic matrix Strongly preferential PageRank Weakly preferential PageRank Sink preferential PageRank Theory Other transformations Must we compute a derivative for all of these formulations? As hinted by the top of the figure, only a few formulations are theoretically relevant. The difference between strongly, weakly, and sink preferential are irrelevant for the derivative: all that matters is P.3 With P from any of these variations, 3 The matrix P is the fully column the derivative vector satisfies the same formulation in terms of P.4 Thus, it suffices to look at the derivative of the core PageRank problem alone. The core problem is still either a linear system or an eigensystem, and thus the difference between that choice may matter. Though, as shown shortly, it does not. The core PageRank problem has enough structure to support the following lemma. It is important to have this lemma about the derivative, because it uses only properties of the PageRank problem—nothing else. It could tell us if a formulation were wrong, for instance. Lemma 6. Let x(α) be the solution of a PageRank problem (problem 1) for P, v, and α. Then the derivative of PageRank with respect to α, denoted x′(α), sums to 0. stochastic matrix in the definition of PageRank. 4 This statement should not be surprising. All the variations con- verted P ̄ to P and did not involve α. The conversion has no effect on differentiating with respect to α. Proof. By definition, x(α + ω) − x(α) x′(α) = lim . ω→0 ωPDF Image | Instagram Cheat Sheet
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