The Google Pagerank Algorithm
and How It Works

Introduction

Page Rank is a topic much discussed by Search Engine Optimisation (SEO) experts. At the heart of PageRank is a mathematical formula that seems scary to look at but is actually fairly simple to understand.

Despite this many people seem to get it wrong! In particular “Chris Ridings of www.searchenginesystems.net” has written a paper entitled “PageRank Explained: Everything you’ve always wanted to know about PageRank”, pointed to by many people, that contains a fundamental mistake early on in the explanation! Unfortunately this means some of the recommendations in the paper are not quite accurate.

By showing code to correctly calculate real PageRank I hope to achieve several things in this response:

1. Clearly explain how PageRank is calculated.
2. Go through every example in Chris’ paper, and add some more of my own, showing the correct PageRank for each diagram. By showing the code used to calculate each diagram I’ve opened myself up to peer review - mostly in an effort to make sure the examples are correct, but also because the code can help explain the PageRank calculations.
3. Describe some principles and observations on website design based on these correctly calculated examples.

Any good web designer should take the time to fully understand how PageRank really works - if you don’t then your site’s layout could be seriously hurting your Google listings!

[Note: I have nothing in particular against Chris. If I find any other papers on the subject I’ll try to comment evenly]

How is PageRank Used?

PageRank is one of the methods Google uses to determine a page’s relevance or importance. It is only one part of the story when it comes to the Google listing, but the other aspects are discussed elsewhere (and are ever changing) and PageRank is interesting enough to deserve a paper of its own.

PageRank is also displayed on the toolbar of your browser if you’ve installed the Google toolbar (http://toolbar.google.com/). But the Toolbar PageRank only goes from 0 – 10 and seems to be something like a logarithmic scale:

We can’t know the exact details of the scale because, as we’ll see later, the maximum PR of all pages on the web changes every month when Google does its re-indexing! If we presume the scale is logarithmic (although there is only anecdotal evidence for this at the time of writing) then Google could simply give the highest actual PR page a toolbar PR of 10 and scale the rest appropriately.

Also the toolbar sometimes guesses! The toolbar often shows me a Toolbar PR for pages I’ve only just uploaded and cannot possibly be in the index yet!

What seems to be happening is that the toolbar looks at the URL of the page the browser is displaying and strips off everything down the last “/” (i.e. it goes to the “parent” page in URL terms). If Google has a Toolbar PR for that parent then it subtracts 1 and shows that as the Toolbar PR for this page. If there’s no PR for the parent it goes to the parent’s parent’s page, but subtracting 2, and so on all the way up to the root of your site.  If it can’t find a Toolbar PR to display in this way, that is if it doesn’t find a page with a real calculated PR, then the bar is greyed out.

Note that if the Toolbar is guessing in this way, the Actual PR of the page is 0 - though its PR will be calculated shortly after the Google spider first sees it.

PageRank says nothing about the content or size of a page, the language it’s written in, or the text used in the anchor of a link!

Definitions

I’ve started to use some technical terms and shorthand in this paper. Now’s as good a time as any to define all the terms I’ll use:

That’s enough of that, let’s get back to the meat…

So what is PageRank?

In short PageRank is a “vote”, by all the other pages on the Web, about how important a page is. A link to a page counts as a vote of support. If there’s no link there’s no support (but it’s an abstention from voting rather than a vote against the page).

Quoting from the original Google paper, PageRank is defined like this:

We assume page A has pages T1...Tn which point to it (i.e., are citations). The parameter d is a damping factor which can be set between 0 and 1. We usually set d to 0.85. There are more details about d in the next section. Also C(A) is defined as the number of links going out of page A. The PageRank of a page A is given as follows:

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

Note that the PageRanks form a probability distribution over web pages, so the sum of all web pages' PageRanks will be one.

PageRank or PR(A) can be calculated using a simple iterative algorithm, and corresponds to the principal eigenvector of the normalized link matrix of the web.

but that’s not too helpful so let’s break it down into sections.

1. PR(Tn) - Each page has a notion of its own self-importance. That’s “PR(T1)” for the first page in the web all the way up to “PR(Tn)” for the last page
2. C(Tn) - Each page spreads its vote out evenly amongst all of it’s outgoing links. The count, or number, of outgoing links for page 1 is “C(T1)”, “C(Tn)” for page n, and so on for all pages.
3. PR(Tn)/C(Tn) - so if our page (page A) has a backlink from page “n” the share of the vote page A will get is “PR(Tn)/C(Tn)”
4. d(... - All these fractions of votes are added together but, to stop the other pages having too much influence, this total vote is “damped down” by multiplying it by 0.85 (the factor “d”)
5. (1 - d) - The (1 – d) bit at the beginning is a bit of probability math magic so the “sum of all web pages' PageRanks will be one”: it adds in the bit lost by the d(.... It also means that if a page has no links to it (no backlinks) even then it will still get a small PR of 0.15 (i.e. 1 – 0.85). (Aside: the Google paper says “the sum of all pages” but they mean the “the normalised sum” – otherwise known as “the average” to you and me.

How is PageRank Calculated?

This is where it gets tricky. The PR of each page depends on the PR of the pages pointing to it. But we won’t know what PR those pages have until the pages pointing to them have their PR calculated and so on… And when you consider that page links can form circles it seems impossible to do this calculation!

But actually it’s not that bad. Remember this bit of the Google paper:

PageRank or PR(A) can be calculated using a simple iterative algorithm, and corresponds to the principal eigenvector of the normalized link matrix of the web.

What that means to us is that we can just go ahead and calculate a page’s PR without knowing the final value of the PR of the other pages. That seems strange but, basically, each time we run the calculation we’re getting a closer estimate of the final value. So all we need to do is remember the each value we calculate and repeat the calculations lots of times until the numbers stop changing much.

Lets take the simplest example network: two pages, each pointing to the other:

Each page has one outgoing link (the outgoing count is 1, i.e. C(A) = 1 and C(B) = 1).

Guess 1

We don’t know what their PR should be to begin with, so let’s take a guess at 1.0 and do some calculations:

i.e.

Hmm, the numbers aren’t changing at all! So it looks like we started out with a lucky guess!!!

Guess 2

No, that’s too easy, maybe I got it wrong (and it wouldn’t be the first time). Ok, let’s start the guess at 0 instead and re-calculate:

And again:

And again

and so on. The numbers just keep going up. But will the numbers stop increasing when they get to 1.0? What if a calculation over-shoots and goes above 1.0?

Guess 3

Well let’s see. Let’s start the guess at 40 each and do a few cycles:

PR(A) = 40
PR(B) = 40

First calculation

And again

Yup, those numbers are heading down alright! It sure looks the numbers will get to 1.0 and stop

Here’s the code used to calculate this example starting the guess at 0: Show the code | Run the program

• Principle: it doesn’t matter where you start your guess, once the PageRank calculations have settled down, the “normalized probability distribution” (the average PageRank for all pages) will be 1.0

As you’d expect, the home page has the most PR – after all, it has the most incoming links! But what’s happened to the average? It’s only 0.378!!! That doesn’t tie up with what I said earlier so something is wrong somewhere!

Well no, everything is fine. But take a look at the “external site” pages – what’s happening to their PageRank? They’re not passing it on, they’re not voting for anyone, they’re wasting their PR like so much pregnant chad!!! (NB, a more accurate description of this issue can be found in this thread)

A Discussion on Averages

From the Brin and Page paper, the average Actual PR of all pages in the index is 1.0!

So if you add pages to a site you’re building the total PR will go up by 1.0 for each page (but only if you link the pages together so the equation can work), but the average will remain the same.

If you want to concentrate the PR into one, or a few, pages then hierarchical linking will do that. If you want to average out the PR amongst the pages then "fully meshing" the site (lots of evenly distributed links) will do that - examples 5, 6, and 7 in my above. (NB. this is where Ridings’ goes wrong, in his MiniRank model feedback loops will increase PR - indefinitely!)

Getting inbound links to your site is the only way to increase your site's average PR. How that PR is distributed amongst the pages on your site depends on the details of your internal linking and which of your pages are linked to.

If you give outbound links to other sites then your site's average PR will decrease (you're not keeping your vote "in house" as it were). Again the details of the decrease will depend on the details of the linking.

Given that the average of every page is 1.0 we can see that for every site that has an actual ranking in the millions (and there are some!) there must be lots and lots of sites who's Actual PR is below 1.0 (particularly because the absolute lowest Actual PR available is (1 - d)).

It may be that the Toolbar PR 1,2 correspond to Actual PR's lower than 1.0! E.g. the logbase for the Toolbar may be 10 but the Actual PR sequence could start quite low: 0.01, 0.1, 1, 10, 100, 1,000 etc...

Finally

PageRank is, in fact, very simple (apart from one scary looking formula). But when a simple calculation is applied hundreds (or billions) of times over the results can seem complicated.

PageRank is also only part of the story about what results get displayed high up in a Google listing. For example there’s some evidence to suggest that Google is paying a lot of attention these days to the text in a link’s anchor when deciding the relevance of a target page – perhaps more so than the page’s PR…

PageRank is still part of the listings story though, so it’s worth your while as a good designer to make sure you understand it correctly.

Links

• The original PageRank paper by Google’s founders Sergey Brin and Lawrence Page - http://www-db.stanford.edu/~backrub/google.html
• Phil Craven’s PageRank Calculator (fortunately his figures agree with mine)
• A detailed explanation of how easy it is to alter the PageRank equation by mistake
• An excellent discussion on chad-jams (including “pregnant chad”) by Douglas W. Jones - http://www.cs.uiowa.edu/~jones/cards/chad.html - I don’t think many people know the United States’ voting system is this flawed!!!
• Discussion forums on this topic:
• MarketPositionTalk - PageRank updates
• SearchEngineForums - PR documents and calculator
• WebmasterWorld - PR document and calculator

About the Author

Ian Rogers first used the Internet in 1986 sending email on a University VAX machine! He first installed a webserver in 1990, taught himself HTML and perl CGI scripting. Since then he has been a Senior Research Fellow in User Interface Design and a consultant in Network Security and Database Backed Websites. He has had an informal interest in topology and the mathematics and behaviour of networks for years and has also been known to do a little Jive dancing.

This paper was sponsored by IPR Computing Ltd – specialists in Secure Networks and Database Backed Websites