The PR of each page depends on the PR of the pages pointing to it. But we won't know what PR those pages have until the pages pointing to them have their PR calculated and so on… And when you consider that page links can form circles it seems impossible to do this calculation!
But actually it's not that bad. Remember this bit of the Google paper:
PageRank or PR(A) can be calculated using a simple iterative algorithm, and corresponds to the principal eigenvector of the normalized link matrix of the web.
What that means to us is that we can just go ahead and calculate a page's PR without knowing the final value of the PR of the other pages . That seems strange but, basically, each time we run the calculation we're getting a closer estimate of the final value. So all we need to do is remember the each value we calculate and repeat the calculations lots of times until the numbers stop changing much.
Lets take the simplest example network: two pages, each pointing to the other:
Each page has one outgoing link (the outgoing count is 1, i.e. C(A) = 1 and C(B) = 1).
Guess 1
We don't know what their PR should be to begin with, so let's take a guess at 1.0 and do some calculations:
d |
= 0.85 |
PR(A) |
= (1 – d) + d(PR(B)/1) |
PR(B) |
= (1 – d) + d(PR(A)/1) |
i.e.
PR(A) |
= 0.15 + 0.85 * 1 = 1 |
PR(B) |
= 0.15 + 0.85 * 1 = 1 |
Hmm, the numbers aren't changing at all! So it looks like we started out with a lucky guess!!!
Guess 2
No, that's too easy, maybe I got it wrong (and it wouldn't be the first time). Ok, let's start the guess at 0 instead and re-calculate:
|
PR(A) |
= 0.15 + 0.85 * 0 = 0.15
|
|
PR(B) |
= 0.15 + 0.85 * 0.15 = 0.2775 |
NB. we've already calculated a “next best guess” at PR(A) so we use it here |
And again:
|
PR(A) |
= 0.15 + 0.85 * 0.2775 = 0.385875 |
|
PR(B) |
= 0.15 + 0.85 * 0.385875 = 0.47799375 |
And again
PR(A) |
= 0.15 + 0.85 * 0.47799375 = 0.5562946875 |
PR(B) |
= 0.15 + 0.85 * 0.5562946875 = 0.622850484375 |
and so on. The numbers just keep going up. But will the numbers stop increasing when they get to 1.0? What if a calculation over-shoots and goes above 1.0?
Guess 3
Well let's see. Let's start the guess at 40 each and do a few cycles:
PR(A) = 40
PR(B) = 40
First calculation
|
PR(A) |
= 0.15 + 0.85 * 40 = 34.25 |
|
PR(B) |
= 0.15 + 0.85 * 0.385875 = 29.1775 |
And again
|
PR(A) |
0.15 + 0.85 * 29.1775 = 24.950875 |
|
PR(B) |
= 0.15 + 0.85 * 24.950875 = 21.35824375 |
Yup, those numbers are heading down alright! It sure looks the numbers will get to 1.0 and stop
Here's the code used to calculate this example starting the guess at 0:
• Principle: it doesn't matter where you start your guess, once the PageRank calculations have settled down, the “ normalized probability distribution ” (the average PageRank for all pages) will be 1.0
The Iterative Computation of PageRank
Because of the size of the actual web, the Google search engine uses an approximative, iterative computation of PageRank values. This means that each page is assigned an initial starting value and the PageRanks of all pages are then calculated in several computation circles based on the equations determined by the PageRank algorithm. The iterative calculation shall again be illustrated by our three-page example, whereby each page is assigned a starting PageRank value of 1.
Iteration PR(A) PR(B) PR(C)
0 1 1 1
1 1 0.75 1.125
2 1.0625 0.765625 1.1484375
3 1.07421875 0.76855469 1.15283203
4 1.07641602 0.76910400 1.15365601
5 1.07682800 0.76920700 1.15381050
6 1.07690525 0.76922631 1.15383947
7 1.07691973 0.76922993 1.15384490
8 1.07692245 0.76923061 1.15384592
9 1.07692296 0.76923074 1.15384611
10 1.07692305 0.76923076 1.15384615
11 1.07692307 0.76923077 1.15384615
12 1.07692308 0.76923077 1.15384615
We see that we get a good approximation of the real PageRank values after only a few iterations. According to publications of Lawrence Page and Sergey Brin, about 100 iterations are necessary to get a good approximation of the PageRank values of the whole web.
Also, by means of the iterative calculation, the sum of all pages' PageRanks still converges to the total number of web pages. So the average PageRank of a web page is 1. The minimum PageRank of a page is given by (1-d). Therefore, there is a maximum PageRank for a page which is given by dN+(1-d), where N is total number of web pages. This maximum can theoretically occur, if all web pages solely link to one page, and this page also solely links to itself.
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