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The Kendall’s τ coefficient is a way to measure the correlation between two list rankings. τ is defined as: τ= 1−2s (IV.2) 12 n(n − 1) Where s, Kendall’s distance, is the number of different elements in the lists of ordered pairs in two rankings of n items. When τ = 0, then the two lists are identical. There are issues with the different ranked approximations. Wills listed sev- eral problems, where correct ranking could occur in one iteration and be de- stroyed in the next, instances where small residual norm does not guarantee a correct ranking, instances where τ = 0 does not guarantee a correct ranking, and instances where the correct ranking occurs much earlier than the termi- nation criteria. Wills created their own criteria for the ranking of the elements, described by them in theorem 4.5 and 4.7 (2007). Wills used these theorems to produce a computationally efficient criterion for ranking PageRank. 4.2.2 CombatingManipulation Since the original PageRank algorithm, dealing with manipulation by web- pages has been a major issue. This has been a problem mostly through spam, or the insertion of large numbers of links to point to a page in an attempt to inflate their PageRank number artificially. In their 2003 paper, Haveliwala and Kamvar showed how the second eigen- value of the Google matrix could be used to detect spam, as well as speed up PageRank computation. They show that for the web hyperlink matrix that the second eigenvalue λ2 = c, where c is the damping number used in the PageR- ank algorithm. This can be useful to make PageRank computations faster by not having to compute λ2. It can also help identify spam sites. The eigenvec- 20PDF Image | MATHEMATICS BEHIND GOOGLE PAGERANK ALGORITHM
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