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	<p align="center"> </p>
	<h1 align="center">The Google Pagerank Algorithm<br>and How It Works</h1>
	<p align="center">Ian Rogers<br>IPR Computing Ltd.<br><a href="mailto:ian@iprcom.com">ian@iprcom.com</a></p>
	<h3>Introduction</h3>
	<p>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.</p>
	<p>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 <a href="http://www.iprcom.com/papers/pagerank/altered_equation.html">fundamental mistake</a> early on in the explanation! Unfortunately this means some of the recommendations in the paper are not quite accurate.
	</p>
	<p>By showing code to correctly calculate real PageRank I hope to achieve several things in this response:</p>
	<ol>
	<li>Clearly explain how PageRank is calculated.</li>
	<li>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.</li>
	<li>Describe some principles and observations on website design based on these correctly calculated examples.</li>
	</ol>
	<p>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!</p>
	<p>[Note: I have nothing in particular against Chris. If I find any other papers on the subject I’ll try to comment evenly]</p>
	<h3>How is PageRank Used?</h3>
	<p>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.</p>
	<p>PageRank is also displayed on the toolbar of your browser if you’ve installed the Google toolbar (<a href="http://toolbar.google.com/">http://toolbar.google.com/</a>). But the Toolbar PageRank only goes from 0 – 10
	and seems to be something like a logarithmic scale:</p>
	<p align="left">
	<table nof="TE" border="0" cellpadding="0" cellspacing="0" width="100%">
	<tbody><tr>
	<td align="center">
	<table id="Table1" border="0" cellpadding="1" cellspacing="3" width="68%">
	<tbody><tr valign="top">
	<td align="center" width="49%">
	<p align="center"><b>Toolbar PageRank<br>(log base 10)</b></p>
	</td>
	<td align="center" width="50%">
	<p align="center"><b>Real PageRank</b></p>
	</td>
	</tr>
	<tr>
	<td align="center" width="49%">
	<p align="center">0</p>
	</td>
	<td align="center" width="50%">
	<p align="center">0 - 10</p>
	</td>
	</tr>
	<tr>
	<td align="center" width="49%">
	<p align="center">1</p>
	</td>
	<td align="center" width="50%">
	<p align="center">100 - 1,000</p>
	</td>
	</tr>
	<tr>
	<td align="center" width="49%">
	<p align="center">2</p>
	</td>
	<td align="center" width="50%">
	<p align="center">1,000 - 10,000</p>
	</td>
	</tr>
	<tr>
	<td align="center" width="49%">
	<p align="center">3</p>
	</td>
	<td align="center" width="50%">
	<p align="center">10,000 - 100,000</p>
	</td>
	</tr>
	<tr>
	<td align="center" width="49%">
	<p align="center">4</p>
	</td>
	<td align="center" width="50%">
	<p align="center">and so on...</p>
	</td>
	</tr>
	</tbody></table>
	</td>
	</tr>
	</tbody></table>
	</p><p>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. </p>
	<p>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! </p>
	<p>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.</p>
	<p>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.</p>
	<p>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!</p>
	<h3>Definitions</h3>
	<p>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:</p>
	<p>
	<table nof="TE" border="0" cellpadding="0" cellspacing="0" width="100%">
	<tbody><tr>
	<td align="center">
	<table id="Table2" border="0" cellpadding="1" cellspacing="3" width="77%">
	<tbody><tr>
	<td valign="top" width="21%">
	<p><b>PR: </b></p>
	</td>
	<td width="78%">
	<p>Shorthand
	for PageRank: the actual, real, page rank for each page as calculated
	by Google. As we’ll see later this can range from 0.15 to billions.</p>
	</td>
	</tr>
	<tr>
	<td valign="top" width="21%">
	<p><b>Toolbar PR:</b></p>
	</td>
	<td width="78%">
	<p>The PageRank displayed in the Google toolbar in your browser. This ranges from 0 to 10.</p>
	</td>
	</tr>
	<tr>
	<td valign="top" width="21%">
	<p><b>Backlink:</b></p>
	</td>
	<td width="78%">
	<p>If page A links out to page B, then page B is said to have a “backlink” from page A.</p>
	</td>
	</tr>
	</tbody></table>
	</td>
	</tr>
	</tbody></table>
	</p><p>That’s enough of that, let’s get back to the meat…</p>
	<h3><a name="what"></a>So what is PageRank?</h3>
	<p>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). </p>
	<p>Quoting from the original Google paper, PageRank is defined like this:</p>
	<ul>
	<p align="left"><i>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:</i> </p>
	<p align="left"><i>PR(A) = (1-d) + d (PR(T1)/C(T1) + ... + PR(Tn)/C(Tn))</i> </p>
	<p align="left"><i>Note that the PageRanks form a probability distribution over web pages, so the sum of all web pages' PageRanks will be one.</i></p>
	<p><i>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.</i></p>
	</ul>
	<p>but that’s not too helpful so let’s break it down into sections.</p>
	<ol>
	<li><b>PR(Tn)</b>
	- 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</li>
	<li><b>C(Tn)</b> - 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. </li>
	<li><b>PR(Tn)/C(Tn)</b> - 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)”</li>
	<li><b>d(...</b>
	- 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”)</li>
	<li><b>(1 - d)</b> - The (1 – d) bit at the beginning is a bit of probability math magic so the “<i>sum of all web pages' PageRanks will be one</i>”: it adds in the bit lost by the <b>d(...</b>.
	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.</li>
	</ol>
	<h3>How is PageRank Calculated?</h3>
	<p>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 <b>them</b> have their PR calculated and so on…
	And when you consider that page links can form circles it seems
	impossible to do this calculation!</p>
	<p>But actually it’s not that bad. Remember this bit of the Google paper:</p>
	<ul>
	<p align="left"><i>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.</i></p>
	</ul>
	<p>What that means to us is that we can just go ahead and calculate a page’s PR <b>without knowing the final value of the PR of the other pages</b>.
	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.</p>
	<p>Lets take the simplest example network: two pages, each pointing to the other:</p>
	<p align="left">
	<table nof="TE" border="0" cellpadding="0" cellspacing="0" width="100%">
	<tbody><tr>
	<td align="center"><img id="Picture1" src="pagerank_files/image001.gif" border="0" height="51" width="227"></td>
	</tr>
	</tbody></table>
	</p><p align="left">Each page has one outgoing link (the outgoing count is 1, i.e. C(A) = 1 and C(B) = 1).</p>
	<p align="left"> </p>
	<p align="left"><b><font face="Arial,Helvetica,Univers,Zurich BT,sans-serif" size="+1">Guess 1</font></b></p>
	<p align="left">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:</p>
	<p align="left">
	<table id="Table3" border="0" cellpadding="1" cellspacing="3" width="94%">
	<tbody><tr>
	<td align="right" width="24%">
	<p align="right">d</p>
	</td>
	<td width="75%">
	<p>= 0.85</p>
	</td>
	</tr>
	<tr>
	<td align="right" width="24%">
	<p align="right">PR(A)</p>
	</td>
	<td width="75%">
	<p align="left">= (1 – d) + d(PR(B)/1)</p>
	</td>
	</tr>
	<tr>
	<td align="right" width="24%">
	<p align="right">PR(B)</p>
	</td>
	<td width="75%">
	<p align="left">= (1 – d) + d(PR(A)/1)</p>
	</td>
	</tr>
	</tbody></table>
	</p><p align="left">i.e.</p>
	<p align="left">
	<table id="Table4" border="0" cellpadding="1" cellspacing="3" width="94%">
	<tbody><tr>
	<td align="right" valign="top" width="23%">
	<p align="right">PR(A)</p>
	</td>
	<td width="76%">
	<p align="left">= 0.15 + 0.85 * 1<br>= 1</p>
	</td>
	</tr>
	<tr>
	<td align="right" valign="top" width="23%">
	<p align="right">PR(B)</p>
	</td>
	<td width="76%">
	<p align="left">= 0.15 + 0.85 * 1<br>= 1</p>
	</td>
	</tr>
	</tbody></table>
	</p><p align="left">Hmm, the numbers aren’t changing at all! So it looks like we started out with a lucky guess!!! </p>
	<p align="left"><b><font face="Arial,Helvetica,Univers,Zurich BT,sans-serif" size="+1">Guess 2</font></b></p>
	<p align="left">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:</p>
	<p align="left">
	<table id="Table5" border="0" cellpadding="1" cellspacing="3" width="94%">
	<tbody><tr>
	<td align="right" valign="top" width="24%">
	<p align="right">PR(A)</p>
	</td>
	<td valign="top" width="33%">
	<p align="left">= 0.15 + 0.85 * 0<br>= 0.15</p>
	</td>
	<td width="41%">
	<p> </p>
	</td>
	</tr>
	<tr>
	<td align="right" valign="top" width="24%">
	<p align="right">PR(B)</p>
	</td>
	<td valign="top" width="33%">
	<p align="left">= 0.15 + 0.85 * 0.15<br>= 0.2775</p>
	</td>
	<td valign="top" width="41%">
	<p align="left">NB. we’ve already calculated a “next best guess” at PR(A) so we use it here</p>
	</td>
	</tr>
	</tbody></table>
	</p><p align="left">And again:</p>
	<p align="left">
	<table id="Table6" border="0" cellpadding="1" cellspacing="3" width="94%">
	<tbody><tr>
	<td align="right" valign="top" width="24%">
	<p align="right">PR(A)</p>
	</td>
	<td width="75%">
	<p align="left">= 0.15 + 0.85 * 0.2775<br>= 0.385875</p>
	</td>
	</tr>
	<tr>
	<td align="right" valign="top" width="24%">
	<p align="right">PR(B)</p>
	</td>
	<td width="75%">
	<p align="left">= 0.15 + 0.85 * 0.385875 <br>= 0.47799375</p>
	</td>
	</tr>
	</tbody></table>
	</p><p align="left">And again</p>
	<p align="left">
	<table id="Table7" border="0" cellpadding="1" cellspacing="3" width="94%">
	<tbody><tr>
	<td align="right" valign="top" width="24%">
	<p align="right">PR(A)</p>
	</td>
	<td width="75%">
	<p align="left">= 0.15 + 0.85 * 0.47799375<br>= 0.5562946875</p>
	</td>
	</tr>
	<tr>
	<td align="right" valign="top" width="24%">
	<p align="right">PR(B)</p>
	</td>
	<td width="75%">
	<p align="left">= 0.15 + 0.85 * 0.5562946875 <br>= 0.622850484375</p>
	</td>
	</tr>
	</tbody></table>
	</p><p align="left">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?</p>
	<p align="left"> </p>
	<p align="left"><b><font face="Arial,Helvetica,Univers,Zurich BT,sans-serif" size="+1">Guess 3</font></b></p>
	<p align="left">Well let’s see. Let’s start the guess at 40 each and do a few cycles:</p>
	<ul>
	<p align="left">PR(A) = 40<br>PR(B) = 40</p>
	</ul>
	<p align="left">First calculation</p>
	<p align="left">
	<table id="Table8" border="0" cellpadding="1" cellspacing="3" width="94%">
	<tbody><tr>
	<td align="right" valign="top" width="24%">
	<p align="right">PR(A)</p>
	</td>
	<td width="75%">
	<p align="left">= 0.15 + 0.85 * 40<br>= 34.25</p>
	</td>
	</tr>
	<tr>
	<td align="right" valign="top" width="24%">
	<p align="right">PR(B)</p>
	</td>
	<td width="75%">
	<p align="left">= 0.15 + 0.85 * 0.385875 <br>= 29.1775</p>
	</td>
	</tr>
	</tbody></table>
	</p><p align="left">And again</p>
	<p align="left">
	<table id="Table9" border="0" cellpadding="1" cellspacing="3" width="94%">
	<tbody><tr>
	<td align="right" valign="top" width="24%">
	<p align="right">PR(A)</p>
	</td>
	<td width="75%">
	<p align="left">= 0.15 + 0.85 * 29.1775<br>= 24.950875</p>
	</td>
	</tr>
	<tr>
	<td align="right" valign="top" width="24%">
	<p align="right">PR(B)</p>
	</td>
	<td width="75%">
	<p align="left">= 0.15 + 0.85 * 24.950875 <br>= 21.35824375</p>
	</td>
	</tr>
	</tbody></table>
	</p><p align="left">Yup, those numbers are heading down alright! It sure looks the numbers will get to 1.0 and stop</p>
	<p align="left"><a name="ex0"></a>Here’s the code used to calculate this example starting the guess at 0: <a href="http://www.iprcom.com/papers/pagerank/src_0">Show the code</a> \| <a href="http://www.iprcom.com/papers/pagerank/pr0.pl">Run the program</a></p>
	<ul>
	<li><b>Principle:</b> it doesn’t matter where you start your guess, once the PageRank calculations have settled down, the “<i>normalized probability distribution</i>” (the average PageRank for all pages) will be 1.0
	</li>
	</ul>
	<p align="left"> </p>
	<p align="left"><b><font face="Arial,Helvetica,Univers,Zurich BT,sans-serif" size="+1">Getting the answer quicker</font></b></p>
	<p align="left">How
	many times do we need to repeat the calculation for big networks?
	That’s a difficult question; for a network as large as the World Wide
	Web it can be many millions of iterations! The “damping factor” is
	quite subtle. If it’s too high then it takes ages for the numbers to
	settle, if it’s too low then you get repeated over-shoot, both above
	and below the average - the numbers just swing about the average like a
	pendulum and never settle down.</p>
	<p align="left">Also
	choosing the order of calculations can help. The answer will always
	come out the same no matter which order you choose, but some orders
	will get you there quicker than others.</p>
	<p align="left">I’m
	sure there’s been several Master’s Thesis on how to make this
	calculation as efficient as possible, but, in the examples below, I’ve
	used very simple code for clarity and roughly 20 to 40 iterations were
	needed!</p>
	<p align="left"> </p>
	<p align="left"><a name="ex1"></a><b><font face="Arial,Helvetica,Univers,Zurich BT,sans-serif" size="+1">Example 1</font></b></p>
	<p align="left">
	<table nof="TE" border="0" cellpadding="0" cellspacing="0" width="100%">
	<tbody><tr>
	<td align="center"><img id="Picture2" src="pagerank_files/image002.gif" border="0" height="229" width="319"></td>
	</tr>
	</tbody></table>
	</p><p align="left">I’m
	not going to repeat the calculations here, but you can see them by
	running the program (yes, if you click the link the program really is
	re-run to do the calculations for you)</p>
	<ul>
	<p align="left"><a href="http://www.iprcom.com/papers/pagerank/src_1">Show the code</a> \| <a href="http://www.iprcom.com/papers/pagerank/pr1.pl">Run the program</a></p>
	</ul>
	<p align="left">So the correct PR for the example is:</p>
	<p align="left">
	<table nof="TE" border="0" cellpadding="0" cellspacing="0" width="100%">
	<tbody><tr>
	<td align="center"><img id="Picture3" src="pagerank_files/image003.gif" border="0" height="229" width="428"></td>
	</tr>
	</tbody></table>
	</p><p align="left">You can see it took about 20 iterations before the network began to settle on these values!</p>
	<p align="left">Look at Page D though - it has a PR of 0.15 even though no-one is voting for it (i.e. it has no incoming links)! Is this right?</p>
	<p align="left">The first part, or "term" to be techinal, of the PR equation is doing this:</p>
	<ul>
	<p align="left"><i>PR(A) = (1-d) + d (PR(T1)/C(T1) + ... + PR(Tn)/C(Tn))</i></p>
	</ul>
	<p align="left">So, for Page D, no backlinks means the equation looks like this:</p>
	<p align="left">
	<table id="Table10" border="0" cellpadding="1" cellspacing="3" width="94%">
	<tbody><tr>
	<td align="right" valign="top" width="24%">
	<p align="right">PR(A)</p>
	</td>
	<td width="75%">
	<p align="left">= (1-d) + d * (0)<br>= 0.15</p>
	</td>
	</tr>
	</tbody></table>
	</p><p align="left">no matter what else is going on or how many times you do it.</p>
	<ul>
	<p align="left"><b>Observation: </b>every
	page has at least a PR of 0.15 to share out. But this may only be in
	theory - there are rumours that Google undergoes a post-spidering phase
	whereby any pages that have no incoming links at all are completely
	deleted from the index...</p>
	<p align="left"> </p>
	</ul>
	<p align="left"><a name="ex2"></a><b><font face="Arial,Helvetica,Univers,Zurich BT,sans-serif" size="+1">Example 2</font></b></p>
	<p align="left">A simple hierarchy with some outgoing links</p>
	<p align="left">
	<table nof="TE" border="0" cellpadding="0" cellspacing="0" width="100%">
	<tbody><tr>
	<td align="center"><img id="Picture4" src="pagerank_files/image004.gif" border="0" height="235" width="429"></td>
	</tr>
	</tbody></table>
	</p><ul>
	<p align="left"><a href="http://www.iprcom.com/papers/pagerank/src_2">Show the code</a> \| <a href="http://www.iprcom.com/papers/pagerank/pr2.pl">Run the program</a></p>
	</ul>
	<p align="left">
	<table nof="TE" border="0" cellpadding="0" cellspacing="0" width="100%">
	<tbody><tr>
	<td align="center"><img id="Picture5" src="pagerank_files/image005.gif" border="0" height="235" width="493"></td>
	</tr>
	</tbody></table>
	</p><p align="left">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!</p>
	<p align="left">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 <a href="http://www.marketpositiontalk.com/forums/Index.cfm?CFApp=11&Message_ID=37917" target="_blank">thread</a>)
	</p>
	<p align="left"> </p>
	<p align="left"><a name="ex3"></a><b><font face="Arial,Helvetica,Univers,Zurich BT,sans-serif" size="+1">Example 3</font></b></p>
	<p align="left">Let’s link those external sites back into our home page just so we can see what happens to the average…</p>
	<ul>
	<p align="left"><a href="http://www.iprcom.com/papers/pagerank/src_3">Show the code</a> \| <a href="http://www.iprcom.com/papers/pagerank/pr3.pl">Run the program</a></p>
	</ul>
	<p align="left">
	<table nof="TE" border="0" cellpadding="0" cellspacing="0" width="100%">
	<tbody><tr>
	<td align="center"><img id="Picture6" src="pagerank_files/image006.gif" border="0" height="312" width="527"></td>
	</tr>
	</tbody></table>
	</p><p align="left">That’s
	better - it does work after all! And look at the PR of our home page!
	All those incoming links sure make a difference – we’ll talk more about
	that later.</p>
	<p align="left"> </p>
	<p align="left"><a name="ex4"></a><b><font face="Arial,Helvetica,Univers,Zurich BT,sans-serif" size="+1">Example 4</font></b></p>
	<p align="left">What happens to PR if we follow a suggestion about writing page reviews?</p>
	<p align="left">
	<table nof="TE" border="0" cellpadding="0" cellspacing="0" width="100%">
	<tbody><tr>
	<td align="center"><img id="Picture8" src="pagerank_files/image007.gif" border="0" height="370" width="459"></td>
	</tr>
	</tbody></table>
	</p><ul>
	<p align="left"><a href="http://www.iprcom.com/papers/pagerank/src_4">Show the code</a> \| <a href="http://www.iprcom.com/papers/pagerank/pr4.pl">Run the program</a></p>
	</ul>
	<p align="left">
	<table nof="TE" border="0" cellpadding="0" cellspacing="0" width="100%">
	<tbody><tr>
	<td align="center"><img id="Picture7" src="pagerank_files/image008.gif" border="0" height="370" width="457"></td>
	</tr>
	</tbody></table>
	</p><p align="left"> </p>
	<p align="left"> </p>
	<p align="left"><a name="ex5"></a><b><font face="Arial,Helvetica,Univers,Zurich BT,sans-serif" size="+1">Example 5</font></b></p>
	<p align="left">A simple hierarchy</p>
	<ul>
	<p align="left"><a href="http://www.iprcom.com/papers/pagerank/src_5">Show the code</a> \| <a href="http://www.iprcom.com/papers/pagerank/pr5.pl">Run the program</a></p>
	</ul>
	<p align="left">
	<table nof="TE" border="0" cellpadding="0" cellspacing="0" width="100%">
	<tbody><tr>
	<td align="center"><img id="Picture9" src="pagerank_files/image009.gif" border="0" height="202" width="236"></td>
	</tr>
	</tbody></table>
	</p><p align="left">Our home page has 2 and a half times as much PR as the child pages! Excellent!</p>
	<ul>
	<li><b>Observation</b>: a hierarchy concentrates votes and PR into one page</li>
	</ul>
	<p align="left"> </p>
	<p align="left"><a name="ex6"></a><b><font face="Arial,Helvetica,Univers,Zurich BT,sans-serif" size="+1">Example 6</font></b></p>
	<p align="left">Looping</p>
	<ul>
	<p align="left"><a href="http://www.iprcom.com/papers/pagerank/src_6">Show the code</a> \| <a href="http://www.iprcom.com/papers/pagerank/pr6.pl">Run the program</a></p>
	</ul>
	<p align="left">
	<table nof="TE" border="0" cellpadding="0" cellspacing="0" width="100%">
	<tbody><tr>
	<td align="center"><img id="Picture10" src="pagerank_files/image010.gif" border="0" height="177" width="242"></td>
	</tr>
	</tbody></table>
	</p><p align="left">This
	is what we’d expect. All the pages have the same number of incoming
	links, all pages are of equal importance to each other, all pages get
	the same PR of 1.0 (i.e. the “average” probability).</p>
	<p align="left"> </p>
	<p align="left"><a name="ex7"></a><b><font face="Arial,Helvetica,Univers,Zurich BT,sans-serif" size="+1">Example 7</font></b></p>
	<p align="left">Extensive Interlinking – or Fully Meshed</p>
	<ul>
	<p align="left"><a href="http://www.iprcom.com/papers/pagerank/src_7">Show the code</a> \| <a href="http://www.iprcom.com/papers/pagerank/pr7.pl">Run the program</a></p>
	</ul>
	<p align="left">
	<table nof="TE" border="0" cellpadding="0" cellspacing="0" width="100%">
	<tbody><tr>
	<td align="center"><img id="Picture11" src="pagerank_files/image011.gif" border="0" height="177" width="241"></td>
	</tr>
	</tbody></table>
	</p><p align="left">Yes, the results are the same as the Looping example above and for the same reasons.</p>
	<p align="left"> </p>
	<p align="left"><a name="ex8"></a><b><font face="Arial,Helvetica,Univers,Zurich BT,sans-serif" size="+1">Example 8</font></b></p>
	<p align="left">Hierarchical – but with a link in and one out.</p>
	<p align="left">We’ll
	assume there’s an external site that has lots of pages and links with
	the result that one of the pages has the average PR of 1.0. We’ll also
	assume the webmaster really likes us – there’s just one link from that
	page and it’s pointing at our home page.</p>
	<ul>
	<p align="left"><a href="http://www.iprcom.com/papers/pagerank/src_8">Show the code</a> \| <a href="http://www.iprcom.com/papers/pagerank/pr8.pl">Run the program</a></p>
	</ul>
	<p align="left">
	<table nof="TE" border="0" cellpadding="0" cellspacing="0" width="100%">
	<tbody><tr>
	<td align="center"><img id="Picture12" src="pagerank_files/image012.gif" border="0" height="202" width="485"></td>
	</tr>
	</tbody></table>
	</p><p align="left">In
	example 5 the home page only had a PR of 1.92 but now it is 3.31!
	Excellent! Not only has site A contributed 0.85 PR to us, but the
	raised PR in the “About”, “Product” and “More” pages has had a lovely
	“feedback” effect, pushing up the home page’s PR even further!</p>
	<ul>
	<li>Priciple: a well structured site will amplify the effect of any contributed PR</li>
	</ul>
	<p align="left"> </p>
	<p align="left"><a name="ex9"></a><b><font face="Arial,Helvetica,Univers,Zurich BT,sans-serif" size="+1">Example 9</font></b></p>
	<p align="left">Looping – but with a link in and a link out</p>
	<ul>
	<p align="left"><a href="http://www.iprcom.com/papers/pagerank/src_9">Show the code</a> \| <a href="http://www.iprcom.com/papers/pagerank/pr9.pl">Run the program</a></p>
	</ul>
	<p align="left">
	<table nof="TE" border="0" cellpadding="0" cellspacing="0" width="100%">
	<tbody><tr>
	<td align="center"><img id="Picture13" src="pagerank_files/image013.gif" border="0" height="177" width="489"></td>
	</tr>
	</tbody></table>
	</p><p align="left">Well, the PR of our home page has gone up a little, but what’s happened to the “More” page? </p>
	<p align="left">The
	vote of the “Product” page has been split evenly between it and the
	external site. We now value the external Site B equally with our “More”
	page. The “More” page is getting only half the vote it had before –
	this is good for Site B but very bad for us!</p>
	<p align="left"> </p>
	<p align="left"><a name="ex10"></a><b><font face="Arial,Helvetica,Univers,Zurich BT,sans-serif" size="+1">Example 10</font></b></p>
	<p align="left">Fully meshed – but with one vote in and one vote out</p>
	<ul>
	<p align="left"><a href="http://www.iprcom.com/papers/pagerank/src_10">Show the code</a> \| <a href="http://www.iprcom.com/papers/pagerank/pr10.pl">Run the program</a></p>
	</ul>
	<p align="left">
	<table nof="TE" border="0" cellpadding="0" cellspacing="0" width="100%">
	<tbody><tr>
	<td align="center"><img id="Picture14" src="pagerank_files/image014.gif" border="0" height="177" width="489"></td>
	</tr>
	</tbody></table>
	</p><p align="left">That’s
	much better. The “More” page is still getting less share of the vote
	than in example 7 of course, but now the “Product” page has kept three
	quarters of its vote within our site - unlike example 10 where it was
	giving away fully half of it’s vote to the external site!</p>
	<p align="left">Keeping
	just this small extra fraction of the vote within our site has had a
	very nice effect on the Home Page too – PR of 2.28 compared with just
	1.66 in example 10.</p>
	<ul>
	<li><b>Observation:</b>
	increasing the internal links in your site can minimise the damage to
	your PR when you give away votes by linking to external sites.</li>
	<li><b>Principle:</b> </li>
	<ul type="1">
	<li>If a particular page is highly important – use a hierarchical structure with the important page at the “top”.</li>
	<li>Where a group of pages may contain outward links – increase the number of internal links to retain as much PR as possible.</li>
	<li>Where a group of pages do not contain outward links – the number of internal links in the site has <b>no</b> effect on the site’s average PR. You might as well use a link
	structure that gives the user the best navigational experience.</li>
	</ul>
	</ul>
	<p align="left"> </p>
	<p align="left"><b><font face="Arial,Helvetica,Univers,Zurich BT,sans-serif" size="+1">Site Maps</font></b></p>
	<p align="left">Site maps are useful in at least two ways:</p>
	<ul type="1">
	<li>If
	a user types in a bad URL most websites return a really unhelpful “404
	– page not found” error page. This can be discouraging. Why not
	configure your server to return a page that shows an error has been
	made, but also gives the site map? This can help the user enormously</li>
	<li>Linking
	to a site map on each page increases the number of internal links in
	the site, spreading the PR out and protecting you against your vote
	“donations”</li>
	</ul>
	<p align="left"> </p>
	<p align="left"><a name="ex11"></a><b><font face="Arial,Helvetica,Univers,Zurich BT,sans-serif" size="+1">Example 11</font></b></p>
	<p align="left">Lets try to fix our site to artificially concentrate the PR into the home page.</p>
	<p align="left">
	<table nof="TE" border="0" cellpadding="0" cellspacing="0" width="100%">
	<tbody><tr>
	<td align="center"><img id="Picture15" src="pagerank_files/image015.gif" border="0" height="178" width="302"></td>
	</tr>
	</tbody></table>
	</p><p align="left">That looks good, most of the links seem to be pointing up to page A so we should get a nice PR.</p>
	<p align="left"> </p>
	<p align="left">Try to guess what the PR of A will be before you scroll down or run the code.</p>
	<ul>
	<p align="left"><a href="http://www.iprcom.com/papers/pagerank/src_11">Show the code</a> \| <a href="http://www.iprcom.com/papers/pagerank/pr11.pl">Run the program</a></p>
	</ul>
	<p align="left"> </p>
	<p align="left"> </p>
	<p align="left">
	<table nof="TE" border="0" cellpadding="0" cellspacing="0" width="100%">
	<tbody><tr>
	<td align="center"><img id="Picture16" src="pagerank_files/image016.gif" border="0" height="178" width="302"></td>
	</tr>
	</tbody></table>
	</p><p align="left">Oh
	dear, that didn’t work at all well – it’s much worse than just an
	ordinary hierarchy! What’s going on is that pages C and D have such
	weak incoming links that they’re no help to page A at all!</p>
	<ul>
	<li><b>Principle</b>: trying to abuse the PR calculation is harder than you think.</li>
	</ul>
	<p align="left"> </p>
	<p align="left"><a name="ex12"></a><b><font face="Arial,Helvetica,Univers,Zurich BT,sans-serif" size="+1">Example 12</font></b></p>
	<p align="left">A
	common web layout for long documentation is to split the document into
	many pages with a “Previous” and “Next” link on each plus a link back
	to the home page. The home page then only needs to point to the first
	page of the document.</p>
	<ul>
	<p align="left"><a href="http://www.iprcom.com/papers/pagerank/src_12">Show the code</a> \| <a href="http://www.iprcom.com/papers/pagerank/pr12.pl">Run the program</a></p>
	</ul>
	<p align="left">
	<table nof="TE" border="0" cellpadding="0" cellspacing="0" width="100%">
	<tbody><tr>
	<td align="center"><img id="Picture17" src="pagerank_files/image017.gif" border="0" height="178" width="302"></td>
	</tr>
	</tbody></table>
	</p><p align="left">In
	this simple example, where there’s only one document, the first page of
	the document has a higher PR than the Home Page! This is because page B
	is getting all the vote from page A, but page A is only getting
	fractions of pages B, C and D.</p>
	<ul>
	<li><b>Principle</b>:
	in order to give users of your site a good experience, you may have to
	take a hit against your PR. There’s nothing you can do about this - and
	neither should you try to or worry about it! If your site is a pleasure
	to use lots of other webmasters will link to it and you’ll get back
	much more PR than you lost.</li>
	</ul>
	<p align="left">Can
	you also see the trend between this and the previous example? As you
	add more internal links to a site it gets closer to the Fully Meshed
	example where every page gets the average PR for the mesh.</p>
	<ul>
	<li><b>Observation</b>: as you add more internal links in your site, the PR will be spread out more evenly between the pages.</li>
	</ul>
	<p align="left"> </p>
	<p align="left"><a name="ex13"></a><b><font face="Arial,Helvetica,Univers,Zurich BT,sans-serif" size="+1">Example 13</font></b></p>
	<p align="left">Getting high PR the wrong way and the right way.</p>
	<p align="left">Just as an experiment, let’s see if we can get 1,000 pages pointing to our home page, but only have one link leaving it…</p>
	<ul>
	<p align="left"><a href="http://www.iprcom.com/papers/pagerank/src_13">Show the code</a> \| <a href="http://www.iprcom.com/papers/pagerank/pr13.pl">Run the program</a></p>
	</ul>
	<p align="left">
	<table nof="TE" border="0" cellpadding="0" cellspacing="0" width="100%">
	<tbody><tr>
	<td align="center"><img id="Picture18" src="pagerank_files/image018.gif" border="0" height="320" width="511"></td>
	</tr>
	</tbody></table>
	</p><p align="left">Yup, those spam pages are pretty worthless but they sure add up!</p>
	<ul>
	<li><b>Observation</b>:
	it doesn’t matter how many pages you have in your site, your average PR
	will always be 1.0 at best. But a hierarchical layout can strongly
	concentrate votes, and therefore the PR, into the home page!</li>
	</ul>
	<p align="left">This
	is a technique used by some disreputable sites (mostly adult content
	sites). But I can’t advise this - if Google’s robots decide you’re
	doing this there’s a good chance you’ll be banned from Google! Disaster!</p>
	<p align="left">On the other hand there are at least two right ways to do this:</p>
	<p align="left"><b><font face="Arial,Helvetica,Univers,Zurich BT,sans-serif" size="+1">1. Be a Mega-site</font></b></p>
	<p align="left">Mega-sites, like <a href="http://news.bbc.co.uk/">http://news.bbc.co.uk</a>
	have tens or hundreds of editors writing new content – i.e. new pages -
	all day long! Each one of those pages has rich, worthwile content of
	its own and a link back to its parent or the home page! That’s why the
	Home page Toolbar PR of these sites is 9/10 and the rest of us just get
	pushed lower and lower by comparison…</p>
	<ul>
	<li><b>Principle</b>: Content Is King! There really is no substitute for lots of good content…</li>
	</ul>
	<p align="left"> </p>
	<p align="left"><b><font face="Arial,Helvetica,Univers,Zurich BT,sans-serif" size="+1">2. Give away something useful</font></b></p>
	<p align="left"><a href="http://www.phpbb.com/">www.phpbb.com</a> has a Toolbar PR of 8/10 (at the time of writing) and it has no big money or marketing behind it! How can this be?</p>
	<p align="left">What
	the group has done is write a very useful bulletin board system that is
	becoming very popular on many websites. And at the bottom of every
	page, in <b>every</b> installation, is this HTML code:</p>
	<blockquote align="LEFT">Powered by <a href="http://www.phpbb.com/" target="_blank">phpBB</a></blockquote>
	<p align="left">The administrator of each installation can remove that link, but most don’t because they want to return the favour…</p>
	<p align="left">Can you imagine all those millions of pages giving a fraction of a vote to <a href="http://www.phpbb.com/">www.phpbb.com</a>? Wow!</p>
	<ul>
	<li><b>Principle</b>:
	Make it worth other people’s while to use your content or tools. If
	your give-away is good enough other site admins will gladly give you a
	link back.</li>
	<li><b>Principle</b>:
	it’s probably better to get lots (perhaps thousands) of links from
	sites with small PR than to spend any time or money desperately trying
	to get just the one link from a high PR page.</li>
	</ul>
	<p align="left"> </p>
	<p align="left"><b><font face="Arial,Helvetica,Univers,Zurich BT,sans-serif" size="+1">A Discussion on Averages</font></b></p>
	<p align="left">From the Brin and Page paper, the average Actual PR of all pages in the index is 1.0! </p>
	<p align="left">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. </p>
	<p align="left">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!)</p>
	<p align="left">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.</p>
	<p align="left">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.</p>
	<p align="left">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)).</p>
	<p align="left">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... </p>
	<p align="left"><b><font face="Arial,Helvetica,Univers,Zurich BT,sans-serif" size="+1">Finally</font></b></p>
	<p align="left">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 <b>seem</b> complicated.</p>
	<p align="left">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…</p>
	<p align="left">PageRank <b>is</b> still part of the listings story though, so it’s worth your while as a good designer to make sure you understand it correctly.</p>
	<p align="left"> </p>
	<p align="left"><b><font face="Arial,Helvetica,Univers,Zurich BT,sans-serif" size="+1">Links</font></b></p>
	<ul>
	<li>The original PageRank paper by Google’s founders Sergey Brin and Lawrence Page<b> </b>- <a href="http://www-db.stanford.edu/%7Ebackrub/google.html" target="_blank">http://www-db.stanford.edu/~backrub/google.html</a>
	</li>
	<li>Chris Ridings’ “PageRank Explained” paper which, as of April 2002 <a href="http://web.archive.org/web//http://www.goodlookingcooking.co.uk/PageRank.pdf">http://web.archive.org/web//http://www.goodlookingcooking.co.uk/PageRank.pdf</a> ,
	contains one major mistake/misunderstanding - <a href="http://www.goodlookingcooking.co.uk/PageRank.pdf" target="_blank">http://www.goodlookingcooking.co.uk/PageRank.pdf</a> </li>
	<li>Phil Craven’s <a href="http://webworkshop.net/pagerank_calculator.html" target="_blank">PageRank Calculator</a> (fortunately his figures agree with mine)</li>
	<li>A detailed explanation of how easy it is to <a href="http://www.iprcom.com/papers/pagerank/altered_equation.html">alter the PageRank equation by mistake</a></li>
	<li>An excellent discussion on chad-jams (including “pregnant chad”) by Douglas W. Jones - <a href="http://www.cs.uiowa.edu/%7Ejones/cards/chad.html">http://www.cs.uiowa.edu/~jones/cards/chad.html</a> - I don’t think many people know the
	United States’ voting system is this flawed!!!</li>
	<li>Discussion forums on this topic:</li>
	<ul>
	<li><a href="http://www.marketpositiontalk.com/forums/index.cfm?cfapp=11" target="_blank">MarketPositionTalk - PageRank updates</a></li>
	<li><a href="http://searchengineforums.com/Forum28/HTML/002922.html" target="_blank">SearchEngineForums - PR documents and calculator</a></li>
	<li><a href="http://www.webmasterworld.com/forum3/3199.htm" target="_blank">WebmasterWorld - PR document and calculator</a></li>
	</ul>
	</ul>
	<p align="left"><b><font face="Arial,Helvetica,Univers,Zurich BT,sans-serif" size="+1">About the Author</font></b></p>
	<p align="left"><a href="http://www.ianrogers.net/">Ian Rogers</a>
	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.</p>
	<p align="left">This paper was sponsored by <a href="http://www.iprcom.com/index.html">IPR Computing Ltd – specialists in Secure Networks and Database Backed Websites</a></p>
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