Page Rank is a topic much discussed by Search Engine Optimisation (SEO) experts. It's no secret anymore that Google ranks as the number one defacto-standard in the field of major search engines. In 2003, Google accounts for more than 85% of all Internet searches on a daily basis. Google now has many versions running in many different countries, including China, Japan, the U.K., Hong-Kong and many others.  At the heart of Page Rank is a mathematical formula that seems scary to look at but is actually fairly simple to understand

The Google Page Rank value relies on the uniquely democratic nature of the Internet by using its vast global link structure as a prime indicator of an individual page's value. In essence, Google interprets a link from page A to page B as a vote, by page A, for page B. But, Google looks at more than the sheer volume of votes, or links a page receives. It also analyzes the page that casts the vote. Votes cast by pages that are themselves important or are favorably viewed as"established firms" in the Web community weigh more heavily and help to make other pages look
established too.


Page Rank Terminologies

PR: Shorthand for Page Rank- the actual, real, page rank for each page as calculated by Google. As we’ll see later this can range from 0.15 to billions.

Toolbar PR: The Page Rank displayed in the Google toolbar in your browser. This ranges from 0 to 10.

Backlink: If page A links out to page B, then page B is said to have a “back link” from page A.


What is Page Rank?

In short Page Rank 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, Page Rank 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 Page Rank of a page A is given as follows:

PR(A) = (1-d) + d (PR(T1)/C(T1) + ... + PR(Tn)/C(Tn))

Note that the Page Ranks form a probability distribution over web pages, so the sum of all web pages' Page Ranks will be one.

Page Rank 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.

Breaking It Down Into Sections:

1. 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.
   
2. PR(Tn)/C(Tn) - so if our page (page A) has a back link from page “n” the share of the vote page A will get is “PR(Tn)/C(Tn)”
   
3. 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”)
   
4. (1 - d) - The (1 – d) bit at the beginning is a bit of probability math magic so the “sum of all web pages' Page Ranks 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 back links) 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 Page Rank Calculated?

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:

Page Rank 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

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