So what is PageRank?
In short PageRank is a “vote”, by all the other pages on the Web, about how important a page is. A link to a page counts as a vote of support. If there's no link there's no support (but it's an abstention from voting rather than a vote against the page).
Quoting from the original Google paper, PageRank is defined like this:
We assume page A has pages T1...Tn which point to it (i.e., are citations). The parameter d is a damping factor which can be set between 0 and 1. We usually set d to 0.85. There are more details about d in the next section. Also C(A) is defined as the number of links going out of page A. The PageRank of a page A is given as follows:
PR(A) = (1-d) + d (PR(T1)/C(T1) + ... + PR(Tn)/C(Tn))
Note that the PageRanks form a probability distribution over web pages, so the sum of all web pages' PageRanks will be one.
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.
but that's not too helpful so let's break it down into sections.
- PR(Tn) - Each page has a notion of its own self-importance. That's “PR(T1)” for the first page in the web all the way up to “PR(Tn)” for the last page
- C(Tn) - Each page spreads its vote out evenly amongst all of it's outgoing links. The count, or number, of outgoing links for page 1 is “C(T1)”, “C(Tn)” for page n, and so on for all pages.
- PR(Tn)/C(Tn) - so if our page (page A) has a backlink from page “n” the share of the vote page A will get is “PR(Tn)/C(Tn)”
- d(... - All these fractions of votes are added together but, to stop the other pages having too much influence, this total vote is “damped down” by multiplying it by 0.85 (the factor “d”)
- (1 - d) - The (1 – d) bit at the beginning is a bit of probability math magic so the “ sum of all web pages' PageRanks will be one ”: it adds in the bit lost by the d(... . It also means that if a page has no links to it (no backlinks) even then it will still get a small PR of 0.15 (i.e. 1 – 0.85). (Aside: the Google paper says “the sum of all pages” but they mean the “the normalised sum” – otherwise known as “the average” to you and me.
How is PageRank Calculated?
This is where it gets tricky. 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:
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:
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:
Show the code |
Run the program - 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
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Source..]
Linear Mode
