Google's Page Rank Calculation Model
An example spreadsheet model for the - assumed - calculation of Google's Page Rank.
Referral Document: "Anatomy of a Large-Scale Hypertextual Web Search Engine"
Sergey Brin and Lawrence Page, Stanfort University, Stanford, CA 94305, USA
http://www-db.stanford.edu/~backrub/google.html
By: Whitelines Communications, The Hague, Netherlands
http://www.whitelines.net/
Initial content and structure of a Google Database
In this example we use an imaginary Google database containing only four (4) pages that all have an initial
PR value of 1. These four pages have the following linkstructure. In practice Google calculates the Page Rank
for billions of pages. Our model is simplified to show the creation of a Page Rank.
The left picture shows the interlink structure and the initial PR values. The right picture shows the final PR values
after 20 iterations and a damping factor of 0,85.
Calculation Basics
For the calculation of Google's Page Rank the following formula is used (Source: Referral Document):
Formula PR(A) = (1 - d) + d * SUM ((PR(I->A)/C(I))
where:
PR(A) Page Rank of page A
d Damping factor
PR(I->A) Page Rank of page I having a link to page A
C(I) Number of links to other pages from page I
PR(I->A)/C(I) Value from page I that is used for PR calculation of Page A
SUM ((PR(I->A)/C(I)) Sum of values of all pages (I) that point to page A
PR calculation using 20 iterations
In this calculation we use 20 iterations and a damping factor of 0,85. You can vary the damping factor
according to your wishes to see its effect on the results.
Damping factor 0.85
This calculation: PR(A) = 0.15 + 0.85 * SUM ((PR(I->A)/C(I))
Page A B C D Total
Initial PR 1.000 1.000 1.000 1.000 4.000
Final PR (20 iterations) 1.490 0.783 1.577 0.150 4.000
Iteration A B C D Sum
0 1.000 1.000 1.000 1.000 4.000
1 1.000 0.575 2.275 0.150 4.000
2 2.084 0.575 1.191 0.150 4.000
3 1.163 1.036 1.652 0.150 4.000
4 1.554 0.644 1.652 0.150 4.000
5 1.554 0.810 1.485 0.150 4.000
6 1.413 0.810 1.627 0.150 4.000
7 1.533 0.750 1.567 0.150 4.000
8 1.482 0.801 1.567 0.150 4.000
9 1.482 0.780 1.588 0.150 4.000
10 1.500 0.780 1.570 0.150 4.000
11 1.485 0.788 1.578 0.150 4.000
12 1.491 0.781 1.578 0.150 4.000
13 1.491 0.784 1.575 0.150 4.000
14 1.489 0.784 1.577 0.150 4.000
15 1.491 0.783 1.576 0.150 4.000
16 1.490 0.784 1.576 0.150 4.000
17 1.490 0.783 1.577 0.150 4.000
18 1.490 0.783 1.576 0.150 4.000
19 1.490 0.783 1.577 0.150 4.000
20 1.490 0.783 1.577 0.150 4.000
Page A
2.500
Google's PR Evaluation Page B
Page C
2.000
Page D
1.500
1.000
0.500
-
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Detailed Calculation:
Iteration A B C D Sum
0 1.000 1.000 1.000 1.000 4.00
from A to .. 0.150 0.425 0.425 - 1.00
from B to .. - 0.150 0.850 - 1.00
from C to .. 0.850 - 0.150 - 1.00
from D to .. - - 0.850 0.150 1.00
End PR value 1.000 0.575 2.275 0.150 4.00
1 1.000 0.575 2.275 0.150 4.00
from A to .. 0.150 0.425 0.425 - 1.00
from B to .. - 0.150 0.489 - 0.64
from C to .. 1.934 - 0.150 - 2.08
from D to .. - - 0.128 0.150 0.28
End PR value 2.084 0.575 1.191 0.150 4.00
2 2.084 0.575 1.191 0.150 4.00
from A to .. 0.150 0.886 0.886 - 1.92
from B to .. - 0.150 0.489 - 0.64
from C to .. 1.013 - 0.150 - 1.16
from D to .. - - 0.128 0.150 0.28
End PR value 1.163 1.036 1.652 0.150 4.00
3 1.163 1.036 1.652 0.150 4.00
from A to .. 0.150 0.494 0.494 - 1.14
from B to .. - 0.150 0.880 - 1.03
from C to .. 1.404 - 0.150 - 1.55
from D to .. - - 0.128 0.150 0.28
End PR value 1.554 0.644 1.652 0.150 4.00
4 1.554 0.644 1.652 0.150 4.00
from A to .. 0.150 0.660 0.660 - 1.47
from B to .. - 0.150 0.547 - 0.70
from C to .. 1.404 - 0.150 - 1.55
from D to .. - - 0.128 0.150 0.28
End PR value 1.554 0.810 1.485 0.150 4.00
5 1.554 0.810 1.485 0.150 4.00
from A to .. 0.150 0.660 0.660 - 1.47
from B to .. - 0.150 0.689 - 0.84
from C to .. 1.263 - 0.150 - 1.41
from D to .. - - 0.128 0.150 0.28
End PR value 1.413 0.810 1.627 0.150 4.00
6 1.413 0.810 1.627 0.150 4.00
from A to .. 0.150 0.600 0.600 - 1.35
from B to .. - 0.150 0.689 - 0.84
from C to .. 1.383 - 0.150 - 1.53
from D to .. - - 0.128 0.150 0.28
End PR value 1.533 0.750 1.567 0.150 4.00
7 1.533 0.750 1.567 0.150 4.00
from A to .. 0.150 0.651 0.651 - 1.45
from B to .. - 0.150 0.638 - 0.79
from C to .. 1.332 - 0.150 - 1.48
from D to .. - - 0.128 0.150 0.28
End PR value 1.482 0.801 1.567 0.150 4.00
Iteration A B C D Sum
8 1.482 0.801 1.567 0.150 4.00
from A to .. 0.150 0.630 0.630 - 1.41
from B to .. - 0.150 0.681 - 0.83
from C to .. 1.332 - 0.150 - 1.48
from D to .. - - 0.128 0.150 0.28
End PR value 1.482 0.780 1.588 0.150 4.00
9 1.482 0.780 1.588 0.150 4.00
from A to .. 0.150 0.630 0.630 - 1.41
from B to .. - 0.150 0.663 - 0.81
from C to .. 1.350 - 0.150 - 1.50
from D to .. - - 0.128 0.150 0.28
End PR value 1.500 0.780 1.570 0.150 4.00
10 1.500 0.780 1.570 0.150 4.00
from A to .. 0.150 0.638 0.638 - 1.43
from B to .. - 0.150 0.663 - 0.81
from C to .. 1.335 - 0.150 - 1.48
from D to .. - - 0.128 0.150 0.28
End PR value 1.485 0.788 1.578 0.150 4.00
11 1.485 0.788 1.578 0.150 4.00
from A to .. 0.150 0.631 0.631 - 1.41
from B to .. - 0.150 0.669 - 0.82
from C to .. 1.341 - 0.150 - 1.49
from D to .. - - 0.128 0.150 0.28
End PR value 1.491 0.781 1.578 0.150 4.00
12 1.491 0.781 1.578 0.150 4.00
from A to .. 0.150 0.634 0.634 - 1.42
from B to .. - 0.150 0.664 - 0.81
from C to .. 1.341 - 0.150 - 1.49
from D to .. - - 0.128 0.150 0.28
End PR value 1.491 0.784 1.575 0.150 4.00
13 1.491 0.784 1.575 0.150 4.00
from A to .. 0.150 0.634 0.634 - 1.42
from B to .. - 0.150 0.666 - 0.82
from C to .. 1.339 - 0.150 - 1.49
from D to .. - - 0.128 0.150 0.28
End PR value 1.489 0.784 1.577 0.150 4.00
14 1.489 0.784 1.577 0.150 4.00
from A to .. 0.150 0.633 0.633 - 1.42
from B to .. - 0.150 0.666 - 0.82
from C to .. 1.341 - 0.150 - 1.49
from D to .. - - 0.128 0.150 0.28
End PR value 1.491 0.783 1.576 0.150 4.00
15 1.491 0.783 1.576 0.150 4.00
from A to .. 0.150 0.634 0.634 - 1.42
from B to .. - 0.150 0.665 - 0.82
from C to .. 1.340 - 0.150 - 1.49
from D to .. - - 0.128 0.150 0.28
End PR value 1.490 0.784 1.576 0.150 4.00
16 1.490 0.784 1.576 0.150 4.00
from A to .. 0.150 0.633 0.633 - 1.42
from B to .. - 0.150 0.666 - 0.82
from C to .. 1.340 - 0.150 - 1.49
from D to .. - - 0.128 0.150 0.28
End PR value 1.490 0.783 1.577 0.150 4.00
Iteration A B C D Sum
17 1.490 0.783 1.577 0.150 4.00
from A to .. 0.150 0.633 0.633 - 1.42
from B to .. - 0.150 0.666 - 0.82
from C to .. 1.340 - 0.150 - 1.49
from D to .. - - 0.128 0.150 0.28
End PR value 1.490 0.783 1.576 0.150 4.00
18 1.490 0.783 1.576 0.150 4.00
from A to .. 0.150 0.633 0.633 - 1.42
from B to .. - 0.150 0.666 - 0.82
from C to .. 1.340 - 0.150 - 1.49
from D to .. - - 0.128 0.150 0.28
End PR value 1.490 0.783 1.577 0.150 4.00
19 1.490 0.783 1.577 0.150 4.00
from A to .. 0.150 0.633 0.633 - 1.42
from B to .. - 0.150 0.666 - 0.82
from C to .. 1.340 - 0.150 - 1.49
from D to .. - - 0.128 0.150 0.28
End PR value 1.490 0.783 1.577 0.150 4.00
20 1.490 0.783 1.577 0.150 4.00
from A to .. 0.150 0.633 0.633 - 1.42
from B to .. - 0.150 0.666 - 0.82
from C to .. 1.340 - 0.150 - 1.49
from D to .. - - 0.128 0.150 0.28
End PR value 1.490 0.783 1.577 0.150 4.00
Whitelines Communications 20-Jul-02
Iteration A B
0 1 1
1 0 0
2 0 0
3 0 0
4 0 0
5 0 0
6 0 0
7 0 0
8 0 0
9 0 0
10 0 0
11 0 0
12 0 0
13 0 0
14 0 0
15 0 0
16 0 0
17 0 0
18 0 0
19 0 0
20 0 0
C D
1 1
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0