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52 3 ⋅ the pagerank derivative unless the goal is merely to approximate the PageRank derivative, in which case consider the following. a trivial idea A first idea for computing the derivative vector is a central finite difference method, x′(α)≈ 1 (x(α+ε)−x(α−ε)) 2ε for small ε. While attractive for its simplicity, this method requires computing a PageRank vector for a value of α larger than the value of α at the derivative. Additionally, it yields only an approximation to the derivative vector. A first order backward finite difference formula x(α) − x(α − ε) ε avoids computing PageRank at a larger value of α, but is less accurate—and dangerously so—when using inexact solutions x(α). Rather simple analysis shows that using the first order difference is unwise unless the PageRank problems are solved accurately.10 a big linear system From equation (3.1) we can express both the PageRank vector and its derivative vector in the solution to a single linear system. By solving: x′(α) ≈ ⎡⎤⎡⎤⎡⎤ ⎢I − αP 0 ⎥⎢x(α)⎥ ⎢(1 − α)v⎥ ⎢ ⎥⎢⎥=⎢⎥ ⎢ −P I−αP⎥⎢x′(α)⎥ ⎢ −v ⎥ ⎣⎦⎣⎦⎣⎦ (3.3) we simultaneously compute the solution vector for both PageRank and its derivative. Boldi et al. [2005] proposed computing the derivative in this manner by solving this linear system with a hybrid-Jacobi/block Gauss-Seidel procedure. One issue with this approach is that it requires solving a linear system twice the size of the PageRank system. Additionally, the linear system does not correspond to a PageRank problem, and it requires a general linear system solver. two smaller systems We could also, of course, solve the block-tri- 11 Techniques for multiple right- angular linear system (3.3) by first solving (I − αP)x(α) = (1 − α)v and then hand sides do not help in this case. The systems are generally too big solving (I − αP)x′(α) = Px(α) − v. But that algorithm is just solving the for any factorizations, and other derivative linear system (3.1) directly.11 approaches are also inappropriate. 10 If the PageRank problems are solved to a tolerance of γ, then each computed vector is roughly x(α) + γe for an error vector e. Both the central and backwards differences yield an error of γ/ε, and this suggests using a large ε. To get an accurate solution with a large ε requires central differencing.PDF Image | Instagram Cheat Sheet
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