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2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 Figure 4.6 – Empirically measured teleportationcoef cients. Fortwo hours of toolbar logs, the black histogram represents the raw values of α recorded. The dashed line is a bounded kernel density fit to the data with a kernel width of 0.1. The blue line is a Beta density approximation to α. In the original data, rational values of α with small denominators are likely—and cause spikes in the histogram—because many individuals are only briefly observed. 4.5 ⋅ empirical distribution 79 Histogram Density Fit Beta(1.5,0.5) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 α Following our presentation on the initial RAPr model at the Workshop on Algorithms for the Web Graph, Abraham Flaxman and Asela Gunawardana provided a summary of these logs. They reported values of α from one mil- lion “users” on the web collected in a two hour window. From this data, the mean value of α = 0.375. The data shows a good fit to a Beta(1.5, 0.5, [0, 1]) distribution (figure 4.6). For the figure, the analysis used a kernel density estimator [Asmussen and Glynn, 2007] to generate an approximate probability distribution from the raw data. The density fit itself looks quite similar to a Beta distribution. A nonlinear least squares fit produces a Beta(1.52, 0.53, [0, 1]) distribution. Instead, a Beta(1.5, 0.5, [0, 1]) is more simple and matches the mean of the data. For the estimate displayed in the figure, we dropped all values of α mea- sured at 0 and 1. Both of these values are impossible and represent problems with the sampling procedure. In an ideal case, the data would be collected with pseudo-counts [Agresti, 2002], where we estimate number of pages viewed after clicking a link + 1 α≈. total number of pages viewed + 2 Pseudo-counts correct for the two known, but unobserved, future actions. That is, a person will always click another link, so we add 1 to both totals. densityPDF Image | Instagram Cheat Sheet
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