How much additional PageRank do we need to move up the toolbar?

First, let me explain in more detail why the values shown in the Google toolbar are not the actual PageRank figures. According to the equation, and to the creators of Google, the billions of pages on the web average out to a PageRank of 1.0 per page. So the total PageRank on the web is equal to the number of pages on the web * 1, which equals a lot of PageRank spread around the web.
The Google toolbar range is from 1 to 10. (They sometimes show 0, but that figure isn't believed to be a PageRank calculation result). What Google does is divide the full range of actual PageRanks on the web into 10 parts - each part is represented by a value as shown in the toolbar. So the toolbar values only show what part of the overall range a page's PageRank is in, and not the actual PageRank itself. The numbers in the toolbar are just labels.

Whether or not the overall range is divided into 10 equal parts is a matter for debate - Google aren't saying. But because it is much harder to move up a toolbar point at the higher end than it is at the lower end, many people (including me) believe that the divisions are based on a logarithmic scale, or something very similar, rather than the equal divisions of a linear scale.

Let's assume that it is a logarithmic, base 10 scale, and that it takes 10 properly linked new pages to move a site's important page up 1 toolbar point. It will take 100 new pages to move it up another point, 1000 new pages to move it up one more, 10,000 to the next, and so on. That's why moving up at the lower end is much easier that at the higher end.

In reality, the base is unlikely to be 10. Some people think it is around the 5 or 6 mark, and maybe even less. Even so, it still gets progressively harder to move up a toolbar point at the higher end of the scale.

Note that as the number of pages on the web increases, so does the total PageRank on the web, and as the total PageRank increases, the positions of the divisions in the overall scale must change. As a result, some pages drop a toolbar point for no 'apparent' reason. If the page's actual PageRank was only just above a division in the scale, the addition of new pages to the web would cause the division to move up slightly and the page would end up just below the division. Google's index is always increasing and they re-evaluate each of the pages on more or less a monthly basis. It's known as the "Google dance". When the dance is over, some pages will have dropped a toolbar point. A number of new pages might be all that is needed to get the point back after the next dance.

The toolbar value is a good indicator of a page's PageRank but it only indicates that a page is in a certain range of the overall scale. One PR5 page could be just above the PR5 division and another PR5 page could be just below the PR6 division - almost a whole division (toolbar point) between them.

How the PageRank calculations are actually done?

Now we'll look at how the calculations are actually done.

For a page's calculation, its existing PageRank (if it has any) is abandoned completely and a fresh calculation is done where the page relies solely on the PageRank "voted" for it by its current inbound links, which may have changed since the last time the page's PageRank was calculated.

The equation shows clearly how a page's PageRank is arrived at. But what isn't immediately obvious is that it can't work if the calculation is done just once. Suppose we have 2 pages, A and B, which link to each other, and neither have any other links of any kind. This is what happens:-

Step 1: Calculate page A's PageRank from the value of its inbound links

Page A now has a new PageRank value. The calculation used the value of the inbound link from page B. But page B has an inbound link (from page A) and its new PageRank value hasn't been worked out yet, so page A's new PageRank value is based on inaccurate data and can't be accurate.

Step 2: Calculate page B's PageRank from the value of its inbound links

Page B now has a new PageRank value, but it can't be accurate because the calculation used the new PageRank value of the inbound link from page A, which is inaccurate.

It's a Catch 22 situation. We can't work out A's PageRank until we know B's PageRank, and we can't work out B's PageRank until we know A's PageRank.

Now that both pages have newly calculated PageRank values, can't we just run the calculations again to arrive at accurate values? No. We can run the calculations again using the new values and the results will be more accurate, but we will always be using inaccurate values for the calculations, so the results will always be inaccurate.

The problem is overcome by repeating the calculations many times. Each time produces slightly more accurate values. In fact, total accuracy can never be achieved because the calculations are always based on inaccurate values. 40 to 50 iterations are sufficient to reach a point where any further iterations wouldn't produce enough of a change to the values to matter. This is precisiely what Google does at each update, and it's the reason why the updates take so long.

One thing to bear in mind is that the results we get from the calculations are proportions. The figures must then be set against a scale (known only to Google) to arrive at each page's actual PageRank. Even so, we can use the calculations to channel the PageRank within a site around its pages so that certain pages receive a higher proportion of it than others.

NOTE:
You may come across explanations of PageRank where the same equation is stated but the result of each iteration of the calculation is added to the page's existing PageRank. The new value (result + existing PageRank) is then used when sharing PageRank with other pages. These explanations are wrong for the following reasons:-

1. They quote the same, published equation - but then change it

from PR(A) = (1-d) + d(......) to PR(A) = PR(A) + (1-d) + d(......)

It isn't correct, and it isn't necessary.

2. We will be looking at how to organize links so that certain pages end up with a larger proportion of the PageRank than others. Adding to the page's existing PageRank through the iterations produces different proportions than when the equation is used as published. Since the addition is not a part of the published equation, the results are wrong and the proportioning isn't accurate.

According to the published equation, the page being calculated starts from scratch at each iteration. It relies solely on its inbound links. The 'add to the existing PageRank' idea doesn't do that, so its results are necessarily wrong.