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bound. Consequently, our final code for the path damping formulation uses E [AN +2 ] to test convergence instead. Next, we examine the runtime for these methods in the hard case of the Beta(2, 16, [0, 1]) distribution.26 Figure 4.12 displays the values of figure 4.11c against the time they took to compute. Again, the standard deviation was not computed for the path damping algorithm. These timings include all computations of moments and eigenvalues for path damping and Gaussian quadrature. 0 10 10−5 −10 10 −15 10 −2 −1 0 1 2 3 4 10 10 10 10 10 10 10 Time (sec) Based on these experiments, we advise the following. Path damping is the algorithm of choice when r ≪ 1 or the standard deviation is not required. Otherwise, the best method for computing both the expectation and stan- dard deviation for reasonably accurate (≈ 10−4–10−8) solutions is Gaussian quadrature with about 33 points.27 26 The case when r = 1 has the slowest convergence for all the methods. Figure 4.12 – Timing for the RAPr algo- rithms. The time required to compute the results from figure 4.11c for E [x(A)], A ∼ Beta(2, 16, [0, 1]). 4.7 ⋅ algorithm analysis 85 Monte Carlo Path Damping Quadrature 27 Using 33 quadrature points may seem like a lot to those accustomed to integrating smooth functions. With PageRank, there is a singularity near the region of integration and we need to use many points.PDF Image | MODELS AND ALGORITHMS FOR PAGERANK SENSITIVITY
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