Shishigin M.I.
Attainment Of High Convergence Speed In The Iteration Method Of Computations Of The Page Rank (Part 1)

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    The article on the actual topic of site-promotion

ATTAINMENT OF HIGH CONVERGENCE SPEED IN THE ITERATION METHOD OF COMPUTATIONS OF THE PAGE RANK

FOR THE WEB-SITE PAGES

(The article on the actual topic of site-promotion)

  
   Let's examine the equation of the recursive computation scheme:
  
   PR (A) = (1-d) + d (Ci (T1) PR (T1)/Cr (T1) + …+Ci(Tn)PR(Tn)/Cr(Tn)),
   where
   PR (A) is Page Rank of Page A,
   PR (T1) is the Page Rank of the site (page) pointing to Page A,
   С1 (T1) is the number of the individual links off Page T1 to Page A,
   Сr (T1) is the general number of links off that Page T1,
   d is a dampening coefficient (usually 0.85)
   (1-d) is a normalization coefficient,
   n is the number of the Pages on the site.
  
  
  
   The convergence of the computation scheme is provided by a simple condition:

d<1

  
   The dampening coefficients for different site Pages can be different.
   It is important that the values of the normalization coefficients can be distributed depending on the Page preference. In acknowledged sense, to "encourage" this or that Page, the normalization coefficient value can be increased for the account of diminishing of the normalization coefficients of other Pages. This method of the Page "encouragement" one might call the normalization coefficients asymmetry.
   For example, it's already in the first sИance of the site indexation, that the search robot Googlebot can "encourage" the index Page of the site (home.index), having set the normalization coefficient value as (0.1*n + 0.9)*0.15 for it, and the normalization coefficient values as 0.9*0.15 for the other Pages.
  
   [ 0.1*n + 0.9+0.9* (n-1)=n, which n is the number of Pages in the site.]
  
  
   In the following sИances of the site indexation, the normalization coefficient values for the index Page (home.index) and the normalization coefficient values for the other Pages, can have the following values:
  
  

(0.2*n+0.8)*0.15

0.8*0.15

(0.3*n+0.7)*0.15

0.7*0.15

…

(0.9*n+0.1)*0.15

…

0.1*0.15

  
  
   This or that Page can receive the encouragement by the number of criteria (for example, thanks to the "great word mass"). In the computation algorithm PR the feedback links act, permitting to encourage the Pages operatively taking into account their dignity in different criteria.
   Thus, the Googlebot can encourage the site, which is in the Net, it is trusted and prestigious.
   The Search Robot Googlebot, seems trying to bring the Site owner to make his own decision about the optimal linking in favor of the Pages which describe the site topics more widely.
   Such Googlebot's approach explains why, as time went on, the PR value can be increased on the panel without any changes in the Site Pages structure.
  
   Let's explain the technique permitting the PR technology to attain the limit estimating values of the recursive procedure (equations of the recursive computation scheme) through the minimal number of iterations (the accelerated technology of computation of the Page Rank estimating values).
   In the event, when the site multigraph contains the directed cycle, in other terms, all graph tops can be passed by the directed elements, then the amount of the estimating values at any stage of iteration is equal to the number of Pages of the n Site.
   Suppose, PR i (A) is the estimating value for the Page A on i-iteration.
   The normalization coefficient for the PR1+1(A) computation (on i + 1-iteration) to be calculated by formula: (1-d)* PR i (A).
   The example designs in the Excel medium show that the iteration method with high convergence speed provides the reduction of the iteration number to two or three times in comparison with the constant values of the normalization coefficients.
  
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