Simple and Real modes
Check the Simple box (bottom right) to turn Simple mode on. Uncheck it to turn it off, and to turn Real mode on.
Simple mode causes the calculator to do the calculations, and return the results, as found in most "PageRank Explained" type documents. Simple mode calculations don't take account of orphan pages or dangling links.
Orphan pages are those that are not linked to by any other pages on the web - they are orphans. For a page to be indexed by Google it must have at least one page linking to it. If a page is not in the Google index, it and its links don't exist as far as the calculations are concerned. Therefore, it can't have a PR value and it can't share PR with other pages.
Dangling links are links that go to pages that don't have any links going from them. These links are dropped for the duration of the calculations [more info]. This action effectively turns the pages into orphans although many of them are in the index. It isn't clear if the pages that are in the index receive the nominal 0.15 PR or if they are treated as orphans. It doesn't make any difference, though, as 0.15 is as low as it gets. The calculator doesn't treat them as orphans.
Real mode calculates properly for orphan pages and dangling links. Note that if a page is linked to only from an orphan page then, since the orphan page doesn't exist, neither does the link and and so the linked-to page can't exist. This example demonstrates what happens:-
3 pages:- Page B links to Page A : Page C links to Page B (click here to set it up in the calculator)
Page A is linked to and is not an orphan
Page B is linked to and is not an orphan
Nothing links to Page C so it is an orphan
If Page C is an orphan, it doesn't exist in the index and can't link to Page B. Therefore, Page B turns out to be an orphan. Similarly, if Page B is an orphan it can't link to Page A, so it too is an orphan.
The result is that none of the pages end up with any Pagerank. You can check it on the calculator but remember to uncheck the Simple box.
If you add a link from Page A to Page C, suddenly none of them are orphans because the link has made Page C legitimate. And if C is legitimate, its link to Page B counts, which means that B is also legitimate and its link to Page A counts.
Tip: multi-site simulation and reciprocal links
It is possible to see how 2 small sites affect each other by creating a grid for the combined number of pages. E.g. for a 6 page site and a 4 page site, create a grid with 10 pages. Treat pages A to F as one site and pages G to J as the other. As long as there are no links between them, their results are totally isolated. Links between the 2 blocks of pages behave as outbounds and inbounds of their respective 'sites'.
This is particularly relevant with reciprocated links because the PR value of an outbound link is spread around the recipient site's pages, and a returned link from the same page will rarely return the same amount of PR.