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Calculating Page Ranks

It takes quite a bit of work to calculate stable PageRank figures even for simple examples. Rather than do this by hand there are a number of tools on the Web that enable us to observe the effects of simple site structures with inbound and outbound links. We have used Paul Ryan's page rank calculator which you can find at: <http://webworkshop.net/pagerank_calculator.php3>. It is very illustrative and the following simple examples will give you a feel for what happens with different linking scenarios.

Before we go further another interesting feature of the PageRank calculation is that the average of all the PageRanks in a closed system is 1.0. The entire Google index is just such a system. If you took all the pages in the index, added the page ranks together then divided this by the total number of pages the result would be 1.0. This means that there is finite amount of PageRank to share out, as people add pages and these get indexed, the PageRank of existing pages decreases.

This has a major implication: Content poor, small, badly linked sites are less interesting than large, content rich sites. We are, of course, defining content purely in terms of the number of pages in a site.

PageRank in an Hierarchical closed site
Figure 1: Hierarchical closed site

To illustrate this point look at the site in figure 1. It is a closed set of pages, essentially no different from the whole Google universe. Remember from the example above that an orphan page has an intrinsic page rank of 0.15. But what have we here? The page rank of the home page is 1.92 and the other sub-pages 0.69. How did that happen? Well the PageRank of the home page is equal to:

(1 - 0.15) + 0.85 x ((0.15/1) + (0.15/1) + (0.15/1)) = 0.5325

We feed this figure into the sub-pages to get their PageRank is:

(1 - 0.15) + 0.85 x (0.5325/3) = 0.300875

We then feed this figure back to the homepage. The PageRank calculator given above can show you all the intermediate calculations. The figures pretty much reach their final values after 18 iterations. If you add all the page ranks together and divide by the total number of pages, four, the result is 1.0. We have a fixed size pot which is distributed amongst the pages by the linking structure.

Figure 2: Page Rank Calculator for hierarchy with one inbound link

Calculating Page Ranks

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