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websites. Their linking schemes are indicated by arrows. The following examples and algorithm are taken from [3], and the figures are drawn using Geometer’s Sketchpad. Allow xk to equal the number of times page k is voted for by the other pages. Then, as can beseenbyfigure1,x1 =2, x2 =1, x3 =3andx4 =2. Inthiselementaryrankingsystem,x3 holds the majority vote. However, this elementary voting system fails to take into account that a link from a website of the highly frequented caliber of msn.com should have more weight than that of an obscure URL that has little actual clout in the way that the Internet is structured. For example, form the equation x1 = x3 +x4, the sum of the values of the two sites voting for page 1. This type of recursive approach is already begining to become incredibly ubiquitious, sincebythesametoken,wewouldhavex3 =x1+x2+x4,andsoon. Furthermore, in an ideal model, it is not desirable to have an obscure website artificially gain voting power by having a googol, or other large number, of links to other websites. The way this is rectified is by allowing each website to have exactly one vote, and then dividing that one vote into equal pieces among all of the websites that it links to. Stepping outside of the four-website model for a moment in order to form a general theory, suppose that the web has n pages, and fixing k, let page j contain a total of nj links, where one of these links belong to k. Then, add to k’s score a value of xj/nj. Using the convenient notation of [3], let Lk ⊂ {1,2,...,n}, where Lk is the set of all pages linking to k. Then: xk = xj (1) j∈L nj k ReturningtothemodelproposedinFigure1,x1 =x3+x4,x2 =x1 x3 =x1 +x2 +x4 and 23322 x4 = x1 + x2 . Placing these results into a matrix: 32 0 011/2xx 11 1/3 0 0 0 x2 x2 1/3 1/2 0 1/2x =x (2) 3 3 1/3 1/2 0 0 x4 x4 2PDF Image | Google PageRank algorithm powered by linear algebra
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