The Google Matrix

To model the behavior of a random web surfer, Google forms the matrix: 
     G = αS + (1 - α)Iv,   where
0≤α<1
The damping factor, α, in the Google matrix indicates that random web surfers move to a different webpage by some means other than selecting a link with probability 1-α.

I, a column vector of ones
For example, I=(1 1 1 1)T.

v, a row probability distribution vector
The majority of experiments performed during the development of the PageRank algorithm used α=0.85 and v=(1/n 1/n ...1/n). Assigning the uniform vector for v suggests web surfers randomly choose new webpages to view when not selecting links.
Since each element Gij of G lies between 0 and 1 (0≤Gij≤1) and the sum of elements in each row of G is 1. The following gives an example of G calculation for the web graph on the right. 
 G = αS + (1 - α)Iv

            0.00  1.00  0.00  0.00                1
   = 0.85×[ 0.00  0.00  1.00  0.00 ] + (1-0.85)×[ 1 ]×[ 0.25 0.25 0.25 0.25 ]
            0.50  0.00  0.00  0.50                1
            0.25  0.25  0.25  0.25                1

       0.0000  0.8500  0.0000  0.0000       0.0375  0.0375  0.0375  0.0375
   = [ 0.0000  0.0000  0.8500  0.0000 ] + [ 0.0375  0.0375  0.0375  0.0375 ]
       0.4250  0.0000  0.0000  0.4250       0.0375  0.0375  0.0375  0.0375
       0.2125  0.2125  0.2125  0.2125       0.0375  0.0375  0.0375  0.0375

       0.0375  0.8875  0.0375  0.0375         3/80   71/80    3/80    3/80
   = [ 0.0375  0.0375  0.8875  0.0375 ] = [   3/80    3/80   71/80    3/80 ]
       0.4625  0.0375  0.0375  0.4625        37/80    3/80    3/80   37/80
       0.2500  0.2500  0.2500  0.2500         1/4     1/4     1/4     1/4


      Once in a blue moon (very rarely)    
      you see the Aurora here, but it is not like farther north.