It is based on the premise, prevalent in the world of academia, that the importance of a research paper can be judged by the number of citations the paper has from other research papers. Brin and Page have simply transferred this premise to its web equivalent: the importance of a web page can be judged by the number of hyperlinks pointing to it from other web pages

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

where PR(A) is the PageRank of a page A
PR(T₁) is the PageRank of a page T₁
C(T₁) is the number of outgoing links from the page T₁
d is a damping factor in the range 0 < d < 1, usually set to 0.85

The PageRank of a web page is therefore calculated as a sum of the PageRanks of all pages linking to it (its incoming links), divided by the number of links on each of those pages (its outgoing links).

• The number of incoming links. Obviously the more of these the better. But there is another thing the algorithm tells us: no incoming link can have a negative effect on the PageRank of the page it points at. At worst it can simply have no effect at all.
  
   
• The number of outgoing links on the page which points at your page. The fewer of these the better. This is interesting: it means given two pages of equal PageRank linking to you, one with 5 outgoing links and the other with 10, you will get twice the increase in PageRank from the page with only 5 outgoing links.

At this point we take a step back and ask ourselves just how important PageRank is to the position of your page in the Google search results.

The next thing we can observe about the PageRank algorithm is that it has nothing whatsoever to do with relevance to the search terms queried. It is simply one single (admittedly important) part of the entire Google relevance ranking algorithm.

Perhaps a good way to look at PageRank is as a multiplying factor, applied to the Google search results after all its other computations have been completed. The Google algorithm first calculates the relevance of pages in its index to the search terms, and then multiplies this relevance by the PageRank to produce a final list. The higher your PageRank therefore the higher up the results you will be, but there are still many other factors related to the positioning of words on the page which must be considered first

Note: There are scenarios where you may find that total PageRank is not conserved after a calculation. PageRank itself is supposed to represent a probability distribution, with the individual PageRank of a page representing the likelihood of a 'random surfer' chancing upon it.).

On a much larger scale, supposing Google's index contains a billion pages, each with a PageRank of 1, the total PageRank across all pages is equal to a billion. Moreover, each time we recalculate PageRank, no matter what changes in PageRank may occur between individual pages, the total PageRank across all one billion pages will still add up to a billion.

Firstly, this means that although we may not be able to change the total PageRank across all pages, by strategic linking of pages within our site, we can affect the distribution of PageRank between pages. For instance, we may want most of our visitors to come into the site through our home page. We would therefore want our home page to have a higher PageRank relative to other pages within the site. We should also recall that all of the PageRank of a page is passed on and divided equally between each of the outgoing links on a page. We would therefore want to keep as much combined PageRank as possible within our own site without passing it on to external sites and losing its benefit. This means we would want any page with lots of external links (ie. links to other people's web sites) to have a lower PageRank relative to other pages within the site to minimise the amount of PageRank which is 'leaked' to external sites. Bear in mind also our earlier statement, that PageRank is simply a multiplying factor applied once Google's other calculations regarding relevance have already been calculated. We would therefore want our more keyword-rich pages to also have a higher relative PageRank.

Secondly, if we assume that every new page in Google's index begins its life with a PageRank of 1, there is a way we can increase the combined PageRank of pages within our site - by increasing the number of pages! A site with 10 pages will start life with a combined PageRank of 10 which is then redistributed through its hyperlinks. A site with 12 pages will therefore start with a combined PageRank of 12. We can thus improve the PageRank of our site as a whole by creating new content (ie. more pages) and then control the distribution of that combined PageRank through strategic interlinking between the pages.

And this is the purpose of the PageRank Calculator - to create a model of the site on a small scale including the links between pages, and see what effect the model has on the distribution of PageRank.

Next, for ease of reference once the calculation has been performed, provide a label for each page (eg. 'Home Page', 'Links Page', 'Contact Us Page', etc) and again hit 'Submit'.

Finally, use the list boxes to select which pages each page links to. You can use CTRL and SHIFT to highlight multiple selections.

You can also use this screen to change the initial PageRanks of each page. For instance, if one of your pages is supposed to represent Yahoo, you may wish to raise its initial PageRank to, say, 3. However, in actual fact, starting PageRank is irrelevant to its final computed value. In other words, even if one page were to start with a PageRank of 100, after many iterations of the equation (see below), the final computed PageRank will converge to the same value as it would had it started with a PageRank of only 1!

Finally you can play around with the damping factor d, which defaults to 0.85 as this is the value quoted in Brin and Page's research paper.

One of the reasons for this apparent dancing of results is because Google does not simply calculate the PageRank once for each page. After it has calculated the PageRank for the first time it will then put the resulting PageRanks back into the PageRank algorithm and calculate again. Google will go through this process of iteration many times before the results settle down to their 'true' values. When it has been completed, the results will then appear on the 'official' www.google.com domain.

(NB: While Google is updating its index, the updates also occur at different times across its various data centres.

The PageRank Calculator defaults to 20 iterations, although you can increase this number should you choose. For a model of around 20 pages, 20 iterations is sufficient to see the PageRanks honing in on a single 'true' value. Google almost certainly performs many more.