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# Toolbar PageRank Scale

#### tonyww

2:22 pm on Oct 27, 2004 (gmt 0)

#### Junior Member

joined:Jan 10, 2004
posts:56

Is it true that the gaps between Google PRs are logarithmic and it's base 8?

If that is true, does this mean that:

it takes 8 PR 1s to = a PR2
it takes 64 PR 2s to = a PR3
it takes 512 PR 3s to = a PR4
it takes 4096 PR 4s to = a PR5
it takes 32,768 PR 5s to = a PR6

Is that correct?

#### ogletree

2:54 pm on Oct 27, 2004 (gmt 0)

#### Senior Member from US

joined:Apr 14, 2003
posts:4249

It is much more complicated than that. If that were true it would only work if your were the only link on one of those pages. Number of links on a page is a factor. I'm sure there are even more factors than that.

#### diamondgrl

6:12 pm on Oct 27, 2004 (gmt 0)

#### Senior Member

joined:Jan 12, 2004
posts:961

many others claim that the formula is logarithmic and i think that's very misleading. your formulation is not correct. it is not nearly that hard to become a PR6.

the originally published formula relies is a series, with each element in the series representing the PR of the site linking to you divided by the number of outbound links on that page. then there is an iterative damping factor that is applied so that the top value will be 10 (you can't have PR 11).

i would not be surprised if the PR formula has not changed at all - although there are many things other than PR that determine your search engine ranking.

without knowing what every other link on the web is, you can't do a simple calculation as to what your PR will be based on the links to your site.

7:18 pm on Oct 27, 2004 (gmt 0)

#### New User

joined:Sept 29, 2004
posts:10

Yeah, I find it very hard to determine how PR is calc'd. My site is a PR5 site (used to be PR6), and it only has ~50 backlinks, and not all of them from high ranking sites. (Many of the links are PR0 sites.) And I'm ranking on the 2nd page of a very competitive keyword, as well.

DS

#### tonyww

8:05 pm on Oct 27, 2004 (gmt 0)

#### Junior Member

joined:Jan 10, 2004
posts:56

I probably mis-phrased the question. I wasn't asking "how does G calculate PR" - I heard there were more than 100 variables in the algorithm and I did a little bit of analysis of PR and backlinks and it is far from consistent.

What I am trying to get is a feel for how difficult it is to move from one PR to the next PR above.

If we kept the other variables more or less stable, e.g. link reputation, the average PR of the sites linking to us etc, is it harder to go from a PR4 to a PR5 than from a PR2 to a PR3. If 'yes' then why?

Also, can we say how many (for example) PR4s have the same value as a PR5?

#### tonyww

8:10 pm on Oct 27, 2004 (gmt 0)

#### Junior Member

joined:Jan 10, 2004
posts:56

darqShadow: I'm not expert at this and I don't want to teach granny to suck eggs, but maybe one of the sites linking to you has a high PR, or the links to you have very strong reputation with the right balance of keyword density etc. There is a software tool that shows you all this - I don't know if I'm allowed to mention the name of it (I'm not an affiliate btw!).

#### wellzy

8:14 pm on Oct 27, 2004 (gmt 0)

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joined:Dec 20, 2002
posts:352

That can't be a correct formula. I have a PR5 site with only 125 baclinks (400 page site)Each individual page has around 10 incoming links and points back to the index page. Icluding my internal pages I have about 500 PR4 pages pointing to my index page. I think you would have to know the exact PR of the page pointing to you. Toolbar PR is rounded off.

wellzy

#### diamondgrl

9:33 pm on Oct 27, 2004 (gmt 0)

#### Senior Member

joined:Jan 12, 2004
posts:961

I heard there were more than 100 variables in the algorithm and I did a little bit of analysis of PR and backlinks and it is far from consistent.

I doubt this is true. My hunch is the basic algorithm is the originally published one. I described it in words above, but here is the original formula.

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

PR(x) is the PageRank of x, C(x) is the number of outbound links on a page x, d is a damping factor set between 0 and 1 and is controlled by Google.

Note that it is an iterative process because in order to determine the PR of the other pages that link to you, you have to have know the PR of all pages that link to it. It's iterative because it's like a chicken and egg problem as to which to calculate first.

With that said, PR and backlinks are very inconsistent because of numerous factors, including that your toolbar value is not necessarily the actual value Google uses (both because it might be out of date or an approximation) and backlinks shown are notoriously incomplete.

The 100 different factors is probably more accurately attributed not to PR but how well you rank in the SERPs, with PR being one of those 100 factors.

If we kept the other variables more or less stable, e.g. link reputation, the average PR of the sites linking to us etc, is it harder to go from a PR4 to a PR5 than from a PR2 to a PR3.

Yes. If it was just as easy to get from one level to another, don't you think all PR 5s would simply spend twice as much time and money to get to twice as high, or PR10? I sure would.

Also, can we say how many (for example) PR4s have the same value as a PR5?

Good question. Wish I knew something more definitive. But it changes over time. As we all build more and more links, it takes more and more links to get to a certain PR value since the competition has gotten better. You can simply slip in PR if you sit on your duff and don't constantly strive for more links (or naturally get people to link to you).

And note that assuming the above formula is correct, suggests that you will be a lot better of being linked from a PR5 page with 2 links on it than a PR7 page with 10 links, since PR5/2 = 2.5 whereas PR7/10 is only .7.

#### ciml

12:42 pm on Oct 29, 2004 (gmt 0)

#### Senior Member

joined:June 22, 2001
posts:3805

diamondgrl, your last paragraph shows why a log scale would be used.

tonyww:

it takes 8 PR 1s to = a PR2
it takes 64 PR 2s to = a PR3
it takes 512 PR 3s to = a PR4
it takes 4096 PR 4s to = a PR5
it takes 32,768 PR 5s to = a PR6

Not quite, it would be more like these (assuming log 8):

it takes 8 PR 1s to = a PR2
it takes 64 PR 1s to = a PR3
it takes 512 PR 1s to = a PR4
it takes 4096 PR 1s to = a PR5
it takes 32,768 PR 1s to = a PR6

or
it takes 8 PR 1s to = a PR2
it takes 8 PR 2s to = a PR3
it takes 8 PR 3s to = a PR4
it takes 8 PR 4s to = a PR5
it takes 8 PR 5s to = a PR6

Of course this is answering 'how many PRx pages have the same PR as a PRy page', not 'how many links from PRx pages give PRy'.

For the latter you need to change the numbers by some amount, and take into account that the number of links on each page affects the answer.

#### tonyww

1:43 pm on Oct 29, 2004 (gmt 0)

#### Junior Member

joined:Jan 10, 2004
posts:56

Anyone any comment on this:

[seocompany.ca...]

#### asomani

5:45 am on Oct 30, 2004 (gmt 0)

#### Junior Member

joined:Sept 22, 2002
posts:80

My clients' websites homepage got a PR0, but tthe inside pages got PR3, how is it possible.

Initially the root url was holding a PR0, but the index.php under it was a pr4, but oday the root url along with the index.php are all PR0.

If so, that can i do to get the penalty lifted?

#### anallawalla

10:52 am on Nov 1, 2004 (gmt 0)

#### Moderator from AU

joined:Mar 3, 2003
posts:3701

My clients' websites homepage got a PR0, but tthe inside pages got PR3, how is it possible.

Initially the root url was holding a PR0, but the index.php under it was a pr4, but oday the root url along with the index.php are all PR0.

Search for earlier discussions on this topic - I didn't read them because I have long had deep pages with higher PR than the home page. In most cases the home page has no link to them, but the deep pages are good enough to attract numerous direct links.

I have also had the home page drop to PR <1, i.e. white bar, but it was explained away as a technical glitch. Sure enough, the page showed normal PR some months later.

#### brixton

12:29 pm on Nov 1, 2004 (gmt 0)

#### Junior Member

joined:Oct 10, 2004
posts:50

My index PR5 page droped to a few BW links BUT my internal pages got over hundr...s of BWL's ,strange, but i love that :)

#### the_nerd

1:02 pm on Nov 1, 2004 (gmt 0)

#### Senior Member

joined:June 29, 2003
posts:790

diamondgrl

And note that assuming the above formula is correct, suggests that you will be a lot better of being linked from a PR5 page with 2 links on it than a PR7 page with 10 links, since PR5/2 = 2.5 whereas PR7/10 is only .7.

don't forget the log-scale. If we take base 8 as given then a pr 7 link is about 64 times more valuable than a PR 5 , so the link from the PR7 page a waaaayyyyy better. (more than 10 times as good)

#### diamondgrl

4:29 pm on Nov 1, 2004 (gmt 0)

#### Senior Member

joined:Jan 12, 2004
posts:961

thanx, the_nerd. i think you may be incorrect about the logarithmic base assuption, but if so, my post was probably even more inaccurate so i don't blame you. i have since i have since read a little more that helps me clarify a few points in my own mind.

actual pr is different from toolbar pr in that toolbar pr is limited from 0 to 10. actual pr has no such limit, and that is exactly what you would predict with the pr formula. so a "high" 7 as measured in toolbar terms would probably pass a heck of a lot more pr than a "low" 7 since the actual pr might be dramatically different.

however, it would also follow from this that it is not logarithmic in that the range of pr 7 values scales logarithmically over pr 6 value. otherwise, there should not be a limit of 10 since some site might, in theory anyway, qualify for 11 or 12. instead, my hunch is that the actual pr ranges that are associated with each toolbar ranking are picked so there is a logarithmically diminishing number of sites that qualify in each toolbar ranking.

does this make sense to anyone, or strike anyone as off-based?

#### the_nerd

6:55 pm on Nov 1, 2004 (gmt 0)

#### Senior Member

joined:June 29, 2003
posts:790

diamondgrl,

I guess scaling is one of the minor problems Google has, so they probably do just this:

show PR in 10 steps:

0.5 - 1.5 -> TB 1
1.5 - 2.5 -> TB 1
...
9.5 - 10.5 -> TB 10

All you have to do now is take the page with the highest actual pagerank and choose a log base so this highest PR maps to 10.5 (note that in the PR alg every single page has actual PR = 1 to spare, but every link-hop is subject to a dampening factor, otherwise everybody could create his own PR 10 site just with internal links)

Let's assume these links have an average of PR 5 and the log scale is 8. Then a PR 5 Page is worth 8^4 = 4096.

4096 * 2.6000.000 = around 10^10 (10 billion)

If we scale this with our log base of 8 we should get around 10.5 so it can be shown as a TB 10 and leave some space for "lesser" PR 10 pages.

log (8) 10^10 = 17,8. That's far too high. So either the base is too low or the assumption about those links averaging PR 5 is to high.

Let's calculate with a log base of 9:

A PR5 page then is worth 9^4 = 6561

6561 * 2.6000.000 = 17 billion
log (9) 1.7 billion = 13
still too high, but closer to 10.5

let's try 10:

A PR5 page then is worth 10^4 = 10000
10000 * 2.6000.000 = 26 billion
log (10) 2.1 billion = 10.4 (BINGO!)

That means:

- if the average incoming link to Google.com has PR of around 5
- if the number of 2.600.000 incoming links is accurate
- if TB PR is indeed scaled linear on a log basis (no bumps, or emphasising special areas)

then right now the log base used to calculate Toolbar PR could be around 10.

Does that help me a lot? It tells me that on average a PR 5 link is about 10 times as "heavy" as a PR 4 and 100 times a PR 3.

A TB Pagerank of e.g. 5 reflects anything from 4.5 to 5.5 so a link with the same TB PR and the same number of outgoing links may still be 10 times more valuable than another one that "looks" identical.

The only thing that might be remarkable about this calculation: the higher the actual pagerank of the big guys becomes, the less accurate toolbar PR HAS to become. And: it's getting more difficult to increase or even keep your rank because the steps are getting steeper.

Hmmm. Do I help anybody by posting this junk? Took too long to write it, so here you go ;)

#### neuron

11:01 pm on Nov 1, 2004 (gmt 0)

#### Full Member

joined:Jan 28, 2004
posts:316

That's an interesting way to look at it, reverse engineer the exponential scale by roughly calculating the total of the what we assume to be the highest PR page.

- if the average incoming link to Google.com has PR of around 5
- if the number of 2.600.000 incoming links is accurate
- if TB PR is indeed scaled linear on a log basis (no bumps, or emphasising special areas)

alltheweb: 27.5M
altavista: 46.8M
msn: 9.67M
yahoo: 30.1M

and I'm sure a lot more than the 2.61M are included in the calculation, like 10 times that many.

there are 11 integer values between and including the end points of 0 (zero) and 10 (ten), and only 10 units of value. Thus the rescale might be 1/11 increments in the 10 units of value:
PR0= 0.00000 to 0.90909 (0 to 1/11)
PR1= 0.90909 to 1.81818 (1/11 to 2/11)
PR2= 1.81818 to 2.72727 (2/11 to 3/11)
...
PR8= 7.27272 to 8.18181 (8/11 to 9/11)
PR9= 8.18181 to 9.09090 (9/11 to 10/11)
PR10=9.09090 to 10.0000 (10/11 to 1)

And finally, the_nerd, why would you think that the average page linking to google would be a PR5? Wouldn't a PR of 1 (one) as an average PR for all pages be more reasonable? I have quite a few pages and I bet the average PR of my pages is well below PR5.

#### steve128

12:49 am on Nov 2, 2004 (gmt 0)

#### Full Member

joined:Jan 10, 2003
posts:300

>>Anyone any comment on this:<<

Yes, if the guy owning the webite had 1 pr8 link to the page you show he would be pr7, rather than pr2 -;

g-lag accepted

#### steve128

12:51 am on Nov 2, 2004 (gmt 0)

#### Full Member

joined:Jan 10, 2003
posts:300

It seems your post was removed -;

#### cabowabo

12:58 am on Nov 2, 2004 (gmt 0)

#### Senior Member

joined:June 12, 2003
posts:723

In short, PageRank is computing using an unknown numerial value for a page and then allocating a piece of that value based on the number of outgoing links. This is what allows your site to gain more reputation ... a link from a PR6 site with 4 outgoing links would be worth more, in essence, than a link from a PR8 site with 150 outgoing links.

That is a very basic concept, but describes how the process works in a relative sense.

CaboWabo

#### the_nerd

8:20 am on Nov 2, 2004 (gmt 0)

#### Senior Member

joined:June 29, 2003
posts:790

hey neuron,

And finally, the_nerd, why would you think that the average page linking to google would be a PR5? Wouldn't a PR of 1 (one) as an average PR for all pages be more reasonable? I have quite a few pages and I bet the average PR of my pages is well below PR5.

I'm sure that (if they bother reading this) they'll laugh at us for wasting our time calculating the length of the emperor's nose ....

You're right, the PR5 just is a wild guess. It's somewhere in the middle so I took it. But we KNOW they have lots of 9s coming in. And since a 9 is worth 10^8 = 100.000.000 1s I'm sure all your pages (and most of mine) are being accounted for ;)

Ah, yes: Google normally only shows the higher PR-Links with "link:......" So the other 30 million or so links can be offset with a couple of 8s.

I left out the "0" because I don't think it is a real value - just "not yet" or "no more".

Have a great week,

nerd.

#### phantombookman

9:04 am on Nov 2, 2004 (gmt 0)

#### Preferred Member

joined:Dec 30, 2003
posts:625

There is, in my experience, an important and basic factor.
The PR on a page is one less than the page that gave it the PR.

I got a new site of mine PR4 with one link from one of my own sites which was PR5.

I wonder if there is a level at which the algo kicks in and becomes more important?
The reason I ask is because the numbers of links some people are talking about per site massively exceed the total number of links I have over 15 sites. Granted I cannot get further than PR5 but all sites are #1 or top 5 for the KW searches

In my area backlinks are difficult and there is little SEO so basic trends are easy to identify.
I did see a PR6 site with only 2 incoming links a DMOZ (pr4) backlink but one link from a PR7 site.

#### Macro

12:10 pm on Nov 2, 2004 (gmt 0)

#### Senior Member

joined:July 31, 2003
posts:2280

the_nerd, excellent post! (msg 15)

>> The only thing that might be remarkable about this calculation: the higher the actual pagerank of the big guys becomes, the less accurate toolbar PR HAS to become. And: it's getting more difficult to increase or even keep your rank because the steps are getting steeper.

There have been convincing arguments here before for a log of 5 (with accompanying calculations)... but I can't seem to find them.

one small flaw:

I believe it is no longer the case that it's only the higher PR links. In fact, I suspect, that it shows randomly picked out lower PR links when using "link:".

#### the_nerd

5:26 pm on Nov 2, 2004 (gmt 0)

#### Senior Member

joined:June 29, 2003
posts:790

Thanks, Macro!

I we want to learn more about the scale, we could try this: get a random sample of maybe 1000 links pointing to google.com and look at the Toolbar PR to get an idea about the distribution. Maybe somebody did this before and wants to share her findings?

I wouldn't bet the farm on a base of 10, but I don't think it will be much lower than 9. Think of all the macromedia links, for example. They push up the average a lot.