I had previously written an article on how page rank works, and how Google works on assigning the page rank to any web page, today’s article will be based on the mathematical approach to the Google page rank formula. If you are looking for a quick refresher on what is page rank, and how Google works on assigning the page rank, please read the previous blog entry titled “All about Page Rank”. For the purpose of this article, I shall consider a scenario where there are 4 web pages A, B, C and D. Based on these web pages I shall explain the impact various factors have on the page rank of a page.
There are a couple of basic theories that are circulating the internet, however the core of all theories on the internet is the same, any page when it receives an incoming link from another page, gets a certain percentage of the page rank from the page that is pointing to it. To put it in prospective if I were to say that page B has an outgoing link to page A, then B will contribute a certain percent of its page rank to A. But what percentage of B’s page rank actually reaches A depends on a very important factor, according to the page rank formula the number of outgoing links from B has an effect on the percentage of the page rank that A receives from B. So in effect if if B were contribution say .6 page rank value to A, if it were to have 2 outgoing links (other than A), then it will contribute only .2 outgoing value to A and .2 each to the remaining pages that B is linking to.
Now if we were to expand our scope to three pages that were giving an outbound link to A, then the page rank of A (if only B,C and D were pointing to it) can be written as:
PR(A) = PR(B) + PR(C)+PR(D)
The above formula holds good only when B,C and D have a single outgoing link to A. However if B had 3 outgoing links, C had 7 and D has 9 then the formula would become
PR(A) = PR(B)/3 + PR(C)/7 + PR(D)/9
This means that how much of a page’s page rank goes to another page is inverselyproportional to the number of outgoing links from the page. So the higher the number of links from a page, the lower the contribution the page will make to other pages per link. Hence the formula for page rank can be written as:
PR(A) = ∑ incoming links to A/ number of outgoing links from each page that points to A
What however is not clear is how Google assigns initial page ranks to new pages, it has been noted sometimes Google simply guesses the page rank of newer pages, till the page rank is calculated. Also it is not necessary if page B has page rank 5 and only one link to A that it will give all of its page rank to A, it could only contribute just 3 or 2 of its page rank 5 to A.
All said and done the formula is very important for search engine optimizers, as it clearly illustrates that even if a page has high page rank, and thousands of outgoing links, it might not contribute anything in terms of page rank to a new page, in spite of its high page rank.
A practical approach to page rank
Apart from the theoretical approach to page rank explained above, there is something more important to understand about page rank, that the number associated with page rank is actually exponential, this means that the number of links required to take a page from page rank 4 to 5 is not simply 10 links or so, it is more like a thousand more incoming links. As the page rank increases, the number of incoming links increases exponentially.
Let take an example, lets assume that a page rank of 1 represents 101 or to put it simply 10 incoming links, then a page rank of 2 would require 102 links or 100 links. Already you can see the difference between a page rank 1 and page rank 2 is 90 links. Now if we were to extrapolate this to the difference between a page rank 4 and a page rank 5, the difference in the links would be close to 90,000 links! Although there are certain factors that help ease the difference, for example; instead of 90,000 links one can get links from high page rank sites, and the search engine optimizer would then require only 100 high page rank links.
The above illustration is just an example of what I mean when I say that page rank is actually an exponential representation of the number of links. All in all the object of writing the article was to put into perspective mathematically how page rank works.