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164 7 ⋅ conclusion are quite large. Distinguishing differences among the small values implies we should use a strict tolerance. Also, the real answer to the tolerance question is: because we can. The graphs studied in this thesis are small compared with industrial web graphs. In this sense, the difference in computation time for extra accuracy is mean- ingless. There are no application requirements we have to meet, so why not get extra accuracy? 7.1.4 What about ties in the PageRank vector? At various points in this thesis, we illustrate a PageRank vector with an ordered list. For the nodes with highest PageRank, showing an ordered list is okay because the top few PageRank values are clearly separated from each other. The remainder of the PageRank vector, however, is often riddled with tied values. These ties are identical floating-point numbers and not just values within the machine precision tolerance. Exact floating-point ties occur when two pages have identical in-links, the value of v is the same on both pages, and the PageRank solver is invariant to permutations.2 While we do not attempt to quantify the total number of ties—they do seem to be common. This affects the results here in two ways. First, the Kendall-τ computation requires the order of the nodes. We used a version of τ that incorporates tied values, however. Second, the intersection similarity measure also uses a ranked order. This computation may change in the presence of ties. We only expect a slight change as the measure itself is considerably less sensitive to tied values. This follows because of the smoothing effect in the running average nature of the metric. In short, ties are a problem with some PageRank computations, but we do not expect them to alter the results of this thesis in any meaningful way. 7.2 future work We are almost done. There are some small loose ends (dangling nodes?) to address (connect?). Each is an idea to improve a piece of the results. 7.2.1 RAPr speed Although Gauss quadrature is the fastest method to compute the expecta- tion and standard deviation in the RAPr model, it is slow. We need to solve many PageRank problems at values of α that are close to 1. These vectors take considerable computation time. 2 It is worth noting that Gauss- Seidel algorithms are not invariant to permutations. This may sug- gest that they are less reliable for ranking purposes.PDF Image | Instagram Cheat Sheet
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