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Simply put, the idea is that α < 1 is introducing a distortion into the rankings. As we will see, this sentiment is incorrect for the web.6 To look at what happens with α in PageRank, we study the implicitly defined PageRank function of α, 6 See the last paragraph of this section for another viewpoint. Also, we do not mean to suggest that such ideas were misguided. Newer research just provides better guidance. (I − αP)x(α) = (1 − α)v, which people sometimes write as x(α) = (1 − α)(I − αP)−1v. (2.23) Of course, we do not mean to suggest actually computing such functions explicitly and analyzing their properties, although, we are going to do so for purely expository purposes. For instance, 2.6 ⋅ the pagerank function of the damping parameter 33 2 3 45 16 ⎡1 α(α+3) ⎤ ⎢ 6−3α3+6α2−36 ⎥ ⎢⎥ ⎢ 2 (α−1) (α+3) ⎥ ⎢ 3 α3+6 α2−36 ⎥ ⎢⎥ G = with v = e/6, yields x(α) = ⎢ ⎢ ⎥ . (α−1) (α+2) (α+3) ⎥ 3 α3+6 α2−36 ⎥ ⎢ ⎢ 1−(α+2)(α−4) ⎥ ⎢ 2 2 α3+4 α2−24 ⎥ ⎢⎥ ⎢ α3 7α2 5α ⎥ ⎢1 3+6+3 ⎥ ⎢6−(α+1)(α3+2α2−12)⎥ ⎢⎥ ⎢α32 ⎥ ⎢− 6+α+4α+4 −1⎥ ⎣ (α+1)(α3+2α2−12) 6⎦ While this expression looks challenging to interpret, figure 2.5 shows the PageRank function of a single node, x1(α) on the 335-node largest strong component of the harvard500 graph. The expression in that figure looks nearly impossible to understand, and thus we need a more rational (pun fully intended) approach to the problem. Both of these examples were computed by the Matlab symbolic toolbox. A few groups started studying the effect of α around the same time. One of the first was Boldi et al. [2005]. Among other observations, they noted that x(α) is a rational vector function of α. Rational functions are ratios of poly- nomials. In figure 2.5 the caption notes that x1(α) = (−23/6030)f(α)/д(α). Carefully examining the functions f (α) and д(α) shows that they are indeed polynomials. It is not difficult to see why PageRank is a rational function. Consider Cramer’s rule [Meyer, 2000, page 476] for the solution of Ax = b: det(Ai ) x=, i det(A)PDF Image | Instagram Cheat Sheet
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