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94 4 ⋅ random alpha pagerank We now make a few additional observations: • the Monte Carlo method has similar convergence behavior for all dis- tributions and does not achieve better than typical accuracy for all tests; • theBeta(2,16,[0,1])problem(solidlightblueline)requiresthelargest N for all methods except Monte Carlo; • theaccuracyofthestandarddeviationislessthantheaccuracyofthe expectation; and • using stepwise convergence as a proxy for analytical convergence in path damping can produce significant errors. The last statement merits further comment. A simple calculation shows that stepwise convergence of the path damping expression is ∥x(N) − x(N+1)∥ = E[AN+2]∥PN+2v − PN+1v∥ , (4.43) PD PD which is how we compute the values for the figures. The theoretical bound is much weaker with ∥PN+2 − PN+1∥ replaced by the trivial value 2. When the vectors PN v reach a small value, stepwise convergence is no longer a good 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 −5 10 −10 10 −15 10 −2 −1 0 1 2 3 4 10 10 10 10 10 10 10 Time (sec) 26 Thecasewhenr = 1hasthe slowest convergence for all the methods. Figure 4.12 – Timing for the RAPr al- gorithms. The time required to com- pute the results from figure 4.11c for E [x(A)], A ∼ Beta(2, 16, [0, 1]). Monte Carlo Path Damping QuadraturePDF Image | Instagram Cheat Sheet
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