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All the pr is never transferred, only a tiny percentage.

IMHO (assuming not many links out per page).....

I don't think anyone knows all the factors but a pr7 homepage linking to a new site will probably give it a PR3? A few more pr5 links in (maybe 4 or 5 ) will get it to PR5. Achieving PR5 is not too difficult, but from there onwards it is a big job. PR5 to PR6 probably needs about 30+ decent links in (PR4+). PR6 to PR7 needs at least 4 or 5 PR7 sites + another 40 decent links in.

If you get a PR8 site to link to you then that couts a lot. Links from homepages seem to count for a lot as well.

Big sites with internal pages pointing to home seemed to generate a certain amount of pr but this may no longer be the case.

That's my understanding.... I may well be very wrong! However, on page seo work seems to be just as, if not more important these days, once you have got to pr5.

But there is some drop in PR from this link? So site a is slightly higher than site b or would they hold identicle PR or am I (as I often do with pr issues) missing the point here?

two extra post nipped in while I was typing -

I see it's not only me with PR confusion we have three different answers!

[**edited by**: Dino_M at 3:48 pm (utc) on Oct. 24, 2002]

It was only a theoretical example but no just the one external link and no internal outbound links.

I think e-speed has the correct anwser with the dampner of .9 .

Next question then - site a now put in another link, so it has two outbound links it is pr7 giving it a numerical value of 280 000 (just a quess) does it pass on page rank (280 000 *0.1) divided by 2, so 125 971.2 to each page of the two it's linking too?

> Next question then - site a now put in another link, so it has two

> outbound links it is pr7 giving it a numerical value of 280 000

> (just a quess) does it pass on page rank (280 000 *0.1) divided by

> 2, so 125 971.2 to each page of the two it's linking too?

No, it passes on (280,000 * .9) / 2 to each page. Actually it would pass on ((280,000 * .9) / 2) + (1 - .9).

[**edited by**: espeed at 4:05 pm (utc) on Oct. 24, 2002]

The

If page A has exactly PR7 and only 1 link, then the page it links to gets PR6.97 (6 on the Toolbar) if it has no other links to it.

If page A is known only to have a Toolbar PR7 then Page B could be somewhere between PR7 and (almost) PR8. The page it links to is most likely (97% chance) to get a Toolbar PR7 if it has no other links to it, but maybe (3% chance) PR6.

If Page A adds a link to another page, then Page B gets roughly PR6.75 if Page A is exactly PR 7. If Page A has some Toolbar PR7 value then Page B gets probably (roughly 75% chance) PR7 but maybe (roughly 25% chance) PR6.

> .9 or .85?

To relate Toobar PR decay to the real damping factor; you need to know the effective Toolbar PR graph log base.

Many thanks to the poster who pointed out the effect of the final normalisation constant on the effective log base. I can't find that thread so I don't know who it was.

"PageRank: Bringing Order to the Web", by Larry Page:

[www-pcd.stanford.edu...]

Noted slides: Actual PageRank Calculation, Actual PageRank Model

As published, the PR equation is a page-specific measure. It has no factor which addresses whether the domain names involved in each link are the same as the page they appear on.

So 20 in-site links and 1 off site link should net the same result as 1 in-site link and 20 off-site.

There "may" have been some changes in this area recently, but it's still essentially true as far as I know.

>>I am wondering if the internal pages that site A links to have a PR4 as well. I am guessing that most of them will be assigned a PR6 even though their actual PageRank is comparable to site B with a PR4.<<

Just so we're talking about the same thing,

My site A:

PR7

20 links to other pages WITHIN site A.

Only 1 link to my site B.

No other links external to site A.

and, to answer your question, most of the internal pages of site A are PR6, with a few PR5, and one new page at PR 3.

Site B:

PR4

with most internal pages at PR3.

Site B doesn't link back to site A.

Thanks jk3210,

Sorry that my comments are somewhat unrelated to this thread but your example got me thinking about a related PageRank issue.

You example helps me to see why there is so much discrepancy between what people think the PR log base is. I tend to look your site A (home page?) to site B (home page?) numbers and say the log base is some low number like 4 or something (i.e. PR7 page with 20 links with one link to a PR4 page) while others look at site A (home page) to site A (internal page) numbers and arrive at a significantly higher log base (i.e. PR7 page with 20 links with one link to a PR6 page).

Anyway, I am just making a point (somewhat unrelated to this thread) that using actual PR decay values to determine the log base is best done between home pages and not within sites. I feel the toolbar has some way of guessing the PR of internal pages so as not to deviate too much from the PR value of the internal page linking to it (i.e. Toolbar PR values of internal pages do not necessarily relate solely to their actual PageRank values.)

Interesting thread.

espeed said it best, that in order to prevent an almost race condition (called a *Rank Sink* in some articles), there has to be a dampening factor (not all the pr can transfer).

It's why a mess of lower inbound pr links can really help "top off" the tank so to speak.

Nice example JK.

All the pr is never transferred, only a tiny percentage.

Not true. I have a site (pr 5) that is only linked to only from one site that is a pr 6. Before the last update both sites were a pr 6.

Dino_M

It depends alot on the actual pr of the site linking to your pr 0 site. The Toolbar only works by integers so it could be a pr 7.0 or 7.9, unless they round up and then it could be anywhere from 6.51 to 7.5. Personally, I think you are headed for pr 6. Keep in mind with only one link pointing at you that only gives you one opportunity for anchor text so make it good.

<added>The pr 6 site that is the only site linking to the pr 5 site links to it from 7 pages. It was linked back from every (6) page but I've since rectified that (if all you guys can be trusted as to what Gbody can't see:)). Now there is only 1 navigation bar link and 2 natural in content links.<added>

[**edited by**: Powdork at 6:13 am (utc) on Oct. 25, 2002]

Page A current PR7

only links to Page B

Page B current PR0 but becomes (possibly high PR6)

Page B now links only to page A

Page A PR now a higher PR7

This increase is also transferred back to page B and,

pretty much eliminates the original damping factor --Page B now PR7.

(not quite but pretty much there)

fathom:

> This increase is also transferred back to page B and,

>

> pretty much eliminates the original damping factor --Page B now PR7.

If we assume, for a moment, a damping factor of 0.85 (raw) then PageA gets increased by a factor of 3.6, which is about 0.4 to 0.7 PR notches on the Toolbar with an effective log base between 6 and 20.

This is for the case where PageA has other links to it, but PageA and PageB each have only one link; more than that and the dilution is much higher.

To help me make a point about the PageRank log scale, I would like to challenge others to test their math skills and their knowledge of the PageRank papers by determining an upper bound on the PR log scale using jk3210’s example (i.e. home page A with a PR7 and 21 links off the page with one of the links pointing to home page B with a PR4). The only constraint I would add is that “d”, the damping factor, should be between 0.8 and 0.99.

Starting point: PageA has 21 links, of which one goes to PageB. PageB has no other links. To get the upper bound; PageA has exactly PR7 (the minumum), PageB has almost PR5 (the maximum) and the decay factor is exactly 0.8 (the minimum).

log_{5.12}(5.12^7 / 21 * 0.80) = 4.9992

So I'd say just over 5.

I tend to find the best fit at about an effective log base of about 23 and a damping factor of 0.91, but I seem to be alone with these figures. I hope to come up with something better soon, but the low Toolbar resolution makes that an interesting task.

Thanks for your input ciml. I wasn’t sure if you were going to contribute your knowledge to a mission that has the objective of trying to lower the readily accepted value of the PR log scale base. Fortunately for me, I arrived at the same number as you for our example but with a slightly different approach.

Upper bound on log scale = (PageRankA/PageRankBmin)^(1/(7-5)) = (1/(0.8/21))^(1/2) = 5.12

Anyway, I just want to point out that this is not a guess of the log scale but it is an analytically found maximum possible PR log scale based on jk3210’s example with the assumption that there are no special circumstances to jk3210’s example and that the toolbar PR is a log scale of the actual PageRank described in the PageRank papers.

It is possible to find examples where the upper bound on the log scale is higher than 5.12 but that does not change the results of this analysis that the maximum is 5.12. However, if an example is found that showes that the maximum is lower than 5.12 then that result will supercede this one. Actually, if jk3210 specified his link structure for site B then we could probably show that the upper bound on the log scale is even lower than 5.12. Before I get too carried away, it is probably a good idea to let jk3210’s PR results settle for a few updates before we draw too much from his nice example.

It is also possible to find examples that establish a lower bound on the log scale but these are much more difficult because you need to be sure that all links to the receiving page (i.e. like jk3210’s home page B) are accounted for. With the upper bound you only need to account for one link to get an upper bound but the more links you account for, the lower, and more accurate your upper bound will be. Anyway, if our PR log scale is theory is to be consistent then nobody should be able to find an example that shows that the lower bound of the log scale is greater than 5.12. If anybody finds examples of this from home page to home page (i.e. not home page to internal page) then I would be very interested to hear about them and I will be more than willing to offer my interpretation of the numbers.:)

The year 2002 is heading towards its end. And there are really people believing a formula published in 1998 (and perhaps "invented" in 1996) as a hypothesis in a research paper is still (if ever) used today in the biggest search engine on earth.

1998, you remember? Altavista was search engine number one. boo.com hasn't been founded yet. And Bill Gates just learned how to spell WWW.

What do I think about PageRank? First: Forget the formulas you can read in various papers. Second: Do your own experiments. The experiments I did told me one thing. The PR-1 rule of thumb gives me for about 90 per cent of my websites the correct PR value. Of course, this is not _the_ formula. Of course, Google works more complicated than that. But why should bother about the details?

My client had a new website. I linked from my page (with dozens of links on it) with a PR 6 to it and it got PR 5 after the last update. Another example: The private page of a friend got PR 6; it had just one inbound link from a another page with PR 7. During the last update its PR dropped to 4. But nothing was wrong with it: The linking page itself dropped from 7 to 5.

These are just two examples from at least a dozen where the PR-1 rule gives a pretty good approximation. And as a physicist all I want are good approximations. ;-)

PS: I hope you are not offended by my harsh introduction.

Fischerlaender,

The PR minus 1 is a pretty good rule of thumb but doesn’t mean that studying PageRank is a waste of time.

Here is a quote from Google

[google.com...]

”The heart of our software is PageRank^{TM}, a system for ranking web pages developed by our founders Larry Page and Sergey Brin at Stanford University. And while we have dozens of engineers working to improve every aspect of Google on a daily basis, PageRank continues to provide the basis for all of our web search tools.”

It would be interesting to see how many research papers use the word PageRank within them. It would also be interesting to see how many research papers reference the original PageRank papers. I wonder if the rate of new citations is growing as time passes or shrinking. Because I like wasting my time studying PageRank, I am actually reading a more recent paper that was referenced in this post [webmasterworld.com] titled *Using PageRank to Characterize Web Structure*. So far it doesn't seem like a waste of time to me.

Just because a mathematical formulation was developed in the distant 1998 past does not make it obsolete. As a physicist, you have undoubtedly worked many modern problems using mathematical formulations that were developed decades and even centuries ago. I think PageRank is going to be around for a while.

Here is something I think is interesting regarding the title of this thread “How much PageRank is transferred between links?”. The first part of this post is just expressing my agreement with what others on this post have said. However, the second part to this post has some pretty strange conclusions and I would like opinions to help me feel better about the analysis or to correct any mistakes I have made.

Take the case where page A links to page B which in turn has one link to page C which in turn has one link to page D etc.(i.e. A--> B--> C--> D-->. . .) and there are no other links to pages B, C, D, etc.

PART I

I agree with others on this thread that page B will have a PageRank of:

PageRankB = 1-d + PageRankA*d / #linksA

where “d”, the damping factor, is some number like 0.85 or something

and “PageRankA” is the actual (final) PageRank of page A (i.e. not toolbar PR)

and “#linksA” is the number of links out of page A

PART II

However, it is interesting to look at how much PageRank is actually transferred through the link between page A and page B. As seen above, the actual amount of PageRank transferred from page A directly to page B is {PageRankA*d/#linksA}. However, page A also effectively gives page C a PageRank of {PageRankB*d} as well as giving page D a PageRank of {PageRankC*d} etc.

Effectively, page A channels a large amount of PageRank through its link to page B. The actual amount can be solved by:

Total PageRank transferred = (PagRankA/#linksA)*(d + d^2 + d^3 + d^4 + . . . )

After looking this series up in a math book I find that:

Total PageRank transferred = (PagRankA/#linksA)*d/(1-d)

Plugging in some numbers, for example d=0.85 and #linksA=2, yields a surprising result (for me anyway) that page A effectively lost 2.8 times its final PageRank through the link to page B.

This is not some special case that applies only to links strung together like in the example above. Take the case where page A links to page B which links only to page C which links only back to page B (i.e. A --> B <--> C). Plugging in the same numbers above (d=0.85 and #linksA=2) you get that page A effectively transfers 2.8 times its final PageRank through the link to page B. Thus page B (and page C) will have a higher PageRank than page A because of the link from page A to page B.

I know it all seems a bit odd but it appears that this is what math and the PageRank papers dictate. I guess the moral of this story is that the PageRank bleed rate through a link is not {(PageRank/#links)*d} but rather {(PageRank/#links)*d/(1-d)}. It is just nice to know how much PageRank you actually are giving away through a link according the original PageRank papers.

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