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52 3 ⋅ the pagerank derivative 3.4.3 Is the derivative related to PageRank? In the last table, many of the pages with large derivatives also had large PageRank values. The final experiment compares PageRank and its derivative to address whether x′(α)i is proportional to x(α)i. The next two figures (figures 3.2 and 3.3) show the magnitude of the PageRank derivative as a function of the PageRank value. Figure 3.2 – PageRank derivative magni- tude. For both of these graphs, PageRank and the magnitude of its derivative are roughly proportional. The red line shows the equality relationship. Unfortunately, this scatter plot does not show the den- sity of points inside the inner area. 0 10 −2 10 −4 10 −6 10 −8 10 −10 10 −12 10 −8 −6 −4 −2 0 10 10 10 10 10 PageRank (a) cnr-2000 (b) wiki-200611-scc 0 10 −1 10 −2 10 −3 10 −4 10 −5 10 −6 10 −7 10 −8 10 −6 −4 −2 0 10 10 10 10 Magnitude of PageRank derivative relative to PageRank (a) cnr-2000 Figure3.3–RelativemagnitudeofthePageRankderivative. Thedotsarevaluesof∣x′(α)/x(α)∣ ii with height x(α)i , the red line is a histogram of the horizontal density, and the black line is the cumulative histogram. For this plot, the three lines share the same vertical axis, which is either a PageRank value, the fraction of total points in a histogram bin, or the total fraction of all points in a cumulative histogram. Note the log-scale on both axes. This figure implies that between 5 and 10% of the density of a linear fit is accounted for by a single relationship between PageRank and its derivative. Each vertical line is at the position 1/(1 − α). These figures strongly support the idea that PageRank and its derivative are nearly proportional. Such a relationship is not entirely surprising. Pages with large PageRank may be more sensitive to changes in α because they depend on the PageRank values of all incoming links. Also, the PageRank derivative (b) wiki-200611-scc Derivative MagnitudePDF Image | MODELS AND ALGORITHMS FOR PAGERANK SENSITIVITY
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