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where σ may or may not depend on α. Most often, it does not [Gleich and Zhukov, 2005; McSherry, 2005; Langville and Meyer, 2006a]. When Pseudo- Rank is constructed with σ independent of α, then its derivative satisfies (I − αP ̄)y′(α) = P ̄y(α). This system has two important properties. First, while the PageRank deriva- tive satisfies eT x′(α) = 0, this PseudoRank derivative has y′(α) ≥ 0, which follows because (I − αP ̄ ) is an M-matrix (with a non-negative inverse) and P ̄y(α) is positive. Second, PseudoRank has the property that the deriva- tive of PseudoRank is another PseudoRank system, albeit for a possibility different value of θ.9 This second property is exploitable and is a component in an algorithm to compute the PageRank derivative given in the next section. 3.2 algorithms So far, PageRank has been differentiated to construct the PageRank deriva- tive algebraically. The next step is to analyze these derivatives, but experimen- tation is a powerful technique to suggest analysis. To experiment with the derivative requires computing a derivative—that is, an algorithm. Although the derivative vector in (3.1) is the solution of a linear system, can we solve the system more efficiently than a standard problem? It seems likely. After all, PageRank solves (I − αP)x(α) = (1 − α)v, whereas the derivative solves (3.1) (I − αP)x′(α) = Px(α) − v. These systems differ only in the right-hand side. Surely something efficient must be possible. It is. As shown towards the end of this section, a strongly personalized PageRank solver suffices to compute the derivative. Despite the similarities of (3.1) to PageRank, an algorithm with this property is not entirely trivial, 9 If P ̄ is taken to be a stochastic matrix instead, then θ does not change. 3.2 ⋅ algorithms 51PDF Image | Instagram Cheat Sheet
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