September 28, 2007

How to calculate Page Rank

Page Rank is a measure of the popularity of a web page. More pages link one page, this page will have a bigger Page Rank.

It is very useful for us, webmasters, to know the value of Page Rank that our page receives from pages that link it. To have this information, we need data for each page that links to our page: the real value of Page Rank and the total numbers of links on the page.

The value of Page Rank that we know is those that we can read from Google’s Toolbar. That value is a logarithmic representation of the real value. I will assume that the base of logarithm is 7 (the real value isn’t known).

The formula that is used to calculate the real value of Page Rank that one page (lets call it page A) passes to another page (lets call it page X) is:

PR[Xa] = d * ( PR[A] / C[A])

With words: the Page Rank that the page X receive from the page A (PR[Xa]) is obtained dividing the real Page Rank value of the page A (PR[A]) by total number of links on this page (C[A]) and multiplying by dumping factor (d). I will assume that the value of dumping factor is 0,75 (the real value isn’t known).

The total value of Page Rank of page X is the sum of all contributions that the page receives from the pages that link it. For example, if the page X is linked by pages A, B, C and D, the real value of Page Rank will be:

PR[X] = PR[Xa] + PR[Xb] + PR[Xc] + PR[Xd]

Using this information, we can produce a table with ranges of real values of Page Rank that one page (A), with a certain Page Rank, read from Google’s Toolbar, gives to an another page (X), supposing that on the page A there is only one link, to page X.

PR Minimum Maximum
toolbar Value Value
0 1 4
1 5 36
2 37 256
3 257 1.800
4 1.801 12.604
5 12.605 88.236
6 88.237 617.656

Attention: there are not real values of Page Rank, but real values passed from one page to another.

The minimum values are obtained raising number 7 (base of logarithm) to the PR power and multiplying by 0,75 (dumping factor), for example: the minimum value of Toolbar’s Page Rank 5 is (7^5)*0,75 = 12.605.

The maximum values are obtained simply by decrementing by one the minimum value of the superior Page Rank, for example: the maximum value of Toolbar’s Page Rank 5 is 88.237-1 = 88.236.

Now, to have real value that one page pass to our page, we must take the real value of Page Rank of that page from the table and divide it by total number of links, present on the same page. Here is the problem. What value to take from the table: the minimum or the maximum one, or an average?

It’s impossible to resolve this problem because we don’t know exact value of Page Rank, but only an interval of values. Personally, I always work with minimum value so, in this mode, I’m sure that my calculations are underestimated.

An another problem to resolve is how to find the number of links on the page that we examine? There are Internet sites that offer free tools with which we can obtain this information.

At the end of all calculations, summing Page Ranks that we receive from any page that link our page, we have a real value of Page Rank of our page. Lets suppose that this value is 780. To get an equivalent Toolbar’s Page Rank, we have to calculate logarithm to base 7 of this value. Normal calculators have not this function, so we can calculate it using logarithm on base 10, in this mode: log(780) / log(7) = 3,4. Toolbar’s Page Rank is always represented as an integer number, simply ignoring the decimal part of result, so Toolbar’s Page Rank will be 3.