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98 4 ⋅ random alpha pagerank considerable error from the quadrature approximation. But for big problems, running hundreds of Gauss points is not feasible.30 While the Matlab codes given throughout this chapter handle this graph through the bvgraph package, working in Matlab is roughly half the speed of an optimized computation. Consequently, we used a C++ implementation of the inner-outer iteration to solve the PageRank systems and compute the aggregated solution using a bvgraph structure to hold the graph in mem- ory [Boldi and Vigna, 2004]. The time required for our deterministic solves (tolerance 10−12) was α = 0.85 204 minutes, α = 0.5 51 minutes. Computing the expectation and standard deviation in the RAPr model re- quired A1 6199 minutes, A2 1569 minutes. Our codes solved each PageRank vector to a weighted tolerance of 10−12. This accuracy is far more than required when given the intrinsic error in the quadrature approximation mentioned above. Nevertheless, we might as well get something accurate with these computations when we can. To analyze the output, we use two schemes. First, we apply the truncated τ measure to the expectation and standard deviation vectors (table 4.4). The comparison shows that E [x(A)] ≈ x(E [A]) in terms of ranking and that the standard deviation vectors behave differently under this measure. Interest- ingly, the standard deviation vector for A2 appears to invert the orderings of all other measures and the magnitude of its anti-correlation is much stronger than for A1. 30 In chapter 7, we discuss a few ideas to make the codes more scalable. y x(0.85) x(0.5) x(0.85) x(0.95) E[x(A1)] E[x(A2)] x(0.95) E [x(A1 )] E [x(A2 )] Std [x(A1 )] Std [x(A2 )] Table 4.4 – PageRank vs. random alpha PageRank sensitivity on a big graph. The truncated τ values (τε(y,z)withε = 10−10)again show that the standard deviation vectors produce different rankings from the expectation vectors for the graph uk-2006 with 77 million vertices and 2.2 billion edges. A1 is a Beta(2, 16, [0, 1]) random vari- able with statistics computed using a 25-point quadrature rule, and the parameter A2 is a Beta(1, 1, [0, 1]) random variable computed using a 10-point quadrature rule. The coloring is the same as in table 4.3. z 0.850 0.765 0.845 0.956 0.412 -0.538 0.910 0.967 0.891 0.294 -0.675 0.916 0.808 0.219 -0.706 0.892 0.287 -0.675 0.378 -0.577PDF Image | Instagram Cheat Sheet
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