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66 4 ⋅ random alpha pagerank 3 25 4 16 (a) A small web graph x 1 x 2 x 3 x 4 x 5 x 6 0 0.5 Figure 4.2 – An example of the Random Alpha PageRank model. A simple web graph and approx- imate probability density functions of the corresponding PageRank random variables. This shows that pages 5 and 6 have the highest variance (widest density function). These pages are traps from which the random surfer cannot leave. In this plot, A ∼ Beta(2, 16, [0, 1]). In figure 4.2b, the circle stems show the PageRank value for α = E [A] = 0.85, whereas the star stems show the expectation according to the PageRank density. 4.2 vision The PageRank vector x(E [A]) does not incorporate the surfing behavior of all users; we propose to use E [x(A)] instead. Because the PageRank vector is a nonlinear function of α, we do not expect E[x(A)] = x(E[A]), and section 4.4 gives a formal counterexample. For reasonable distributions of A, however, we expect x(E [A]) ≈ E [x(A)] . (4.4) Despite this similarity, moving from the deterministic x(α) to the random x(A) yields more information. For a given page, its “PageRank” is now a random variable. Figure 4.2 shows the probability density functions for the PageRank random variables on a small graph. We can use the standard deviation of the random variables to help “quantify the uncertainty” in the PageRank value. The standard deviation is a measure (b) Density functionsPDF Image | Instagram Cheat Sheet
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