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Google PageRank Explained:-
**(C)2002
eFactory**

**A Survey of Google's
PageRank **

Within the past few years, Google has become the far most utilized
search engine worldwide. A decisive factor therefore was, besides
high performance and ease of use, the superior quality of search
results compared to other search engines. This quality of search
results is substantially based on PageRank, a sophisticated
method to rank web documents.

The aim of this page is to provide
a broad survey of all aspects of PageRank. The contents of these
pages primarily rest upon papers by Google founders Lawrence Page
and Sergey Brin from their time as graduate students at Stanford
University.

It is often argued that, especially
considering the dynamic of the internet, too much time has passed
since the scientific work on PageRank, as that it still could
be the basis for the ranking methods of the Google search engine.
There is no doubt that within the past years most likely many
changes, adjustments and modifications regarding the ranking methods
of Google have taken place, but PageRank was absolutely crucial
for Google's success, so that at least the fundamental concept
behind PageRank should still be constitutive.

**The PageRank Concept**

Since the early stages of the world wide web, search engines have
developed different methods to rank web pages. Until today, the
occurence of a search phrase within a document is one major factor
within ranking techniques of virtually any search engine. The
occurence of a search phrase can thereby be weighted by the length
of a document (ranking by keyword density) or by its accentuation
within a document by HTML tags.

For the purpose of better search
results and especially to make search engines resistant against
automatically generated web pages based upon the analysis of content
specific ranking criteria (doorway pages), the concept of link
popularity was developed. Following this concept, the number of
inbound links for a document measures its general importance.
Hence, a web page is generally more important, if many other web
pages link to it. The concept of link popularity often avoids
good rankings for pages which are only created to deceive search
engines and which don't have any significance within the web,
but numerous webmasters elude it by creating masses of inbound
links for doorway pages from just as insignificant other web pages.

Contrary to the concept of link
popularity, PageRank is not simply based upon the total number
of inbound links. The basic approach of PageRank is that a document
is in fact considered the more important the more other documents
link to it, but those inbound links do not count equally. First
of all, a document ranks high in terms of PageRank, if other high
ranking documents link to it.

So, within the PageRank concept,
the rank of a document is given by the rank of those documents
which link to it. Their rank again is given by the rank of documents
which link to them. Hence, the PageRank of a document is always
determined recursively by the PageRank of other documents. Since
- even if marginal and via many links - the rank of any document
influences the rank of any other, PageRank is, in the end, based
on the linking structure of the whole web. Although this approach
seems to be very broad and complex, Page and Brin were able to
put it into practice by a relatively trivial algorithm

**The PageRank Algorithm**

The original PageRank algorithm was described by Lawrence Page
and Sergey Brin in several publications. It is given by
PR(A) = (1-d) + d (PR(T1)/C(T1)
+ ... + PR(Tn)/C(Tn))

where

PR(A) is the PageRank of page
A,

PR(Ti) is the PageRank of pages Ti which link to page A,

C(Ti) is the number of outbound links on page Ti and

d is a damping factor which can be set between 0 and 1.

So, first of all, we see that PageRank does not rank web sites
as a whole, but is determined for each page individually. Further,
the PageRank of page A is recursively defined by the PageRanks
of those pages which link to page A.

The PageRank of pages Ti which
link to page A does not influence the PageRank of page A uniformly.
Within the PageRank algorithm, the PageRank of a page T is always
weighted by the number of outbound links C(T) on page T. This
means that the more outbound links a page T has, the less will
page A benefit from a link to it on page T.

The weighted PageRank of pages
Ti is then added up. The outcome of this is that an additional
inbound link for page A will always increase page A's PageRank.

Finally, the sum of the weighted
PageRanks of all pages Ti is multiplied with a damping factor
d which can be set between 0 and 1. Thereby, the extend of PageRank
benefit for a page by another page linking to it is reduced.

**The Random Surfer Model**

In their publications, Lawrence Page and Sergey Brin give a very
simple intuitive justification for the PageRank algorithm. They
consider PageRank as a model of user behaviour, where a surfer
clicks on links at random with no regard towards content.

The random surfer visits a web
page with a certain probability which derives from the page's
PageRank. The probability that the random surfer clicks on one
link is solely given by the number of links on that page. This
is why one page's PageRank is not completely passed on to a page
it links to, but is devided by the number of links on the page.

So, the probability for the random
surfer reaching one page is the sum of probabilities for the random
surfer following links to this page. Now, this probability is
reduced by the damping factor d. The justification within the
Random Surfer Model, therefore, is that the surfer does not click
on an infinite number of links, but gets bored sometimes and jumps
to another page at random.

The probability for the random
surfer not stopping to click on links is given by the damping
factor d, which is, depending on the degree of probability therefore,
set between 0 and 1. The higher d is, the more likely will the
random surfer keep clicking links. Since the surfer jumps to another
page at random after he stopped clicking links, the probability
therefore is implemented as a constant (1-d) into the algorithm.
Regardless of inbound links, the probability for the random surfer
jumping to a page is always (1-d), so a page has always a minimum
PageRank.

**A Different Notation of the PageRank
Algorithm**

Lawrence Page and Sergey Brin have published two different versions
of their PageRank algorithm in different papers. In the second
version of the algorithm, the PageRank of page A is given as

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

where N is the total number of
all pages on the web. The second version of the algorithm, indeed,
does not differ fundamentally from the first one. Regarding the
Random Surfer Model, the second version's PageRank of a page is
the actual probability for a surfer reaching that page after clicking
on many links. The PageRanks then form a probability distribution
over web pages, so the sum of all pages' PageRanks will be one.

Contrary, in the first version
of the algorithm the probability for the random surfer reaching
a page is weighted by the total number of web pages. So, in this
version PageRank is an expected value for the random surfer visiting
a page, when he restarts this procedure as often as the web has
pages. If the web had 100 pages and a page had a PageRank value
of 2, the random surfer would reach that page in an average twice
if he restarts 100 times.

As mentioned above, the two versions
of the algorithm do not differ fundamentally from each other.
A PageRank which has been calculated by using the second version
of the algorithm has to be multiplied by the total number of web
pages to get the according PageRank that would have been caculated
by using the first version. Even Page and Brin mixed up the two
algorithm versions in their most popular paper "The Anatomy
of a Large-Scale Hypertextual Web Search Engine", where they
claim the first version of the algorithm to form a probability
distribution over web pages with the sum of all pages' PageRanks
being one.

In the following, we will use
the first version of the algorithm. The reason is that PageRank
calculations by means of this algorithm are easier to compute,
because we can disregard the total number of web pages.

The Characteristics of PageRank

The characteristics of PageRank shall be illustrated by a small
example.

We regard a small web consisting
of three pages A, B and C, whereby page A links to the pages B
and C, page B links to page C and page C links to page A. According
to Page and Brin, the damping factor d is usually set to 0.85,
but to keep the calculation simple we set it to 0.5. The exact
value of the damping factor d admittedly has effects on PageRank,
but it does not influence the fundamental principles of PageRank.
So, we get the following equations for the PageRank calculation:

PR(A) = 0.5 + 0.5 PR(C)

PR(B) = 0.5 + 0.5 (PR(A) / 2)

PR(C) = 0.5 + 0.5 (PR(A) / 2 + PR(B))

These equations can easily be
solved. We get the following PageRank values for the single pages:

PR(A) = 14/13 = 1.07692308

PR(B) = 10/13 = 0.76923077

PR(C) = 15/13 = 1.15384615

It is obvious that the sum of
all pages' PageRanks is 3 and thus equals the total number of
web pages. As shown above this is not a specific result for our
simple example.

For our simple three-page example
it is easy to solve the according equation system to determine
PageRank values. In practice, the web consists of billions of
documents and it is not possible to find a solution by inspection.

The Iterative Computation of PageRank

Because of the size of the actual web, the Google search engine
uses an approximative, iterative computation of PageRank values.
This means that each page is assigned an initial starting value
and the PageRanks of all pages are then calculated in several
computation circles based on the equations determined by the PageRank
algorithm. The iterative calculation shall again be illustrated
by our three-page example, whereby each page is assigned a starting
PageRank value of 1.

Iteration PR(A) PR(B) PR(C)

0 1 1 1

1 1 0.75 1.125

2 1.0625 0.765625 1.1484375

3 1.07421875 0.76855469 1.15283203

4 1.07641602 0.76910400 1.15365601

5 1.07682800 0.76920700 1.15381050

6 1.07690525 0.76922631 1.15383947

7 1.07691973 0.76922993 1.15384490

8 1.07692245 0.76923061 1.15384592

9 1.07692296 0.76923074 1.15384611

10 1.07692305 0.76923076 1.15384615

11 1.07692307 0.76923077 1.15384615

12 1.07692308 0.76923077 1.15384615

We see that we get a good approximation
of the real PageRank values after only a few iterations. According
to publications of Lawrence Page and Sergey Brin, about 100 iterations
are necessary to get a good approximation of the PageRank values
of the whole web.

Also, by means of the iterative
calculation, the sum of all pages' PageRanks still converges to
the total number of web pages. So the average PageRank of a web
page is 1. The minimum PageRank of a page is given by (1-d). Therefore,
there is a maximum PageRank for a page which is given by dN+(1-d),
where N is total number of web pages. This maximum can theoretically
occur, if all web pages solely link to one page, and this page
also solely links to itself

The Implementation of PageRank
in the Google Search Engine

Regarding the implementation of PageRank, first of all, it is
important how PageRank is integrated into the general ranking
of web pages by the Google search engine. The proceedings have
been described by Lawrencec Page and Sergey Brin in several publications.
Initially, the ranking of web pages by the Google search engine
was determined by three factors:

Page specific factors

Anchor text of inbound links

PageRank

Page specific factors are, besides the body text, for instance
the content of the title tag or the URL of the document. It is
more than likely that since the publications of Page and Brin
more factors have joined the ranking methods of the Google search
engine. But this shall not be of interest here.

In order to provide search results,
Google computes an IR score out of page specific factors and the
anchor text of inbound links of a page, which is weighted by position
and accentuation of the search term within the document. This
way the relevance of a document for a query is determined. The
IR-score is then combined with PageRank as an indicator for the
general importance of the page. To combine the IR score with PageRank
the two values are multiplicated. It is obvious that they cannot
be added, since otherwise pages with a very high PageRank would
rank high in search results even if the page is not related to
the search query.

Especially for queries consisting
of two or more search terms, there is a far bigger influence of
the content related ranking criteria, whereas the impact of PageRank
is mainly visible for unspecific single word queries. If webmasters
target search phrases of two or more words it is possible for
them to achieve better rankings than pages with high PageRank
by means of classical search engine optimisation.

If pages are optimised for highly
competitive search terms, it is essential for good rankings to
have a high PageRank, even if a page is well optimised in terms
of classical search engine optimisation. The reason therefore
is that the increase of IR score deminishes the more often the
keyword occurs within the document or the anchor texts of inbound
links to avoid spam by extensive keyword repetition. Thereby,
the potentialities of classical search engine optimisation are
limited and PageRank becomes the decisive factor in highly competitive
areas.

The PageRank Display of the Google Toolbar

PageRank became widely known by the PageRank display of the Google
Toolbar. The Google Toolbar is a browser plug-in for Microsoft
Internet Explorer which can be downloaded from the Google web
site. The Google Toolbar provides some features for searching
Google more comfortably.

The Google Toolbar displays PageRank
on a scale from 0 to 10. First of all, the PageRank of an actually
visited page can be estimated by the width of the green bar within
the display. If the user holds his mouse over the display, the
Toolbar also shows the PageRank value.

Caution: The PageRank display
is one of the advanced features of the Google Toolbar. And if
those advanced features are enabled, Google collects usage data.
Additionally, the Toolbar is self-updating and the user is not
informed about updates. So, Google has access to the user's hard
drive.

If we take into account that PageRank
can theoretically have a maximum value of up to dN+(1-d), where
N is the total number of web pages and d is usually set to 0.85,
PageRank has to be scaled for the display on the Google Toolbar.
It is generally assumed that the scalation is not linearly but
logarithmically. At a damping factor of 0.85 and, therefore, a
minimum PageRank of 0.15 and at an assumed logaritmical basis
of 6 we get a scalation as follows:

Toolbar PageRank Real PageRank

0/10 0.15 - 0.9

1/10 0.9 - 5.4

2/10 5.4 - 32.4

3/10 32.4 - 194.4

4/10 194.4 - 1,166.4

5/10 1,166.4 - 6,998.4

6/10 6,998.4 - 41,990.4

7/10 41,990.4 - 251,942.4

8/10 251,942.4 - 1,511,654.4

9/10 1,511,654.4 - 9,069,926.4

10/10 9,069,926.4 - 0.85 × N + 0.15

It is uncertain if in fact a logarithmical scalation in a strictly
mathematical sense takes place. There is likely a manual scalation
which follows a logarithmical scheme, so that Google has control
over the number of pages within the single Toolbar PageRank ranges.
The logarithmical basis for this scheme should be between 6 and
7, which can for instance be rudimentary deduced from the number
of inbound links of pages with a high Toolbar PageRank from pages
with a Toolbar PageRank higher than 4, which are shown by Googe
using the link command.

The Toolbar's PageRank Files

Even webmasters who do not want to use the Google Toolbar or the
Internet Explorer permanently for security and privacy concerns
have the possibility to check the PageRank values of their pages.
Google submits PageRank values in simple text files to the Toolbar.
In former times, this happened via XML. The switch to text files
occured in August 2002.

The PageRank files can be requested
directly from the domain www.google.com. Basically, the URLs for
those files look like follows (without line breaks):

http://www.google.com/search?

client=navclient-auto&

ch=0123456789&

features=Rank&

q=info:http://www.domain.com/

There is only one line of text
in the PageRank files. The last cipher in this line is PageRank.

The parameters incorporated in
the above shown URL are inevitable for the display of the PageRank
files in a browser. The value "navclient-auto" for the
parameter "client" identifies the Toolbar. Via the parameter
"q" the URL is submitted. The value "Rank"
for the parameter "features" determines that the PageRank
files are requested. If it is omitted, Google's servers still
transmit XML files. The parameter "ch" transfers a checksum
for the URL to Google, whereby this checksum can only change when
the Toolbar version is updated by Google.

Thus, it is necessary to install
the Toolbar at least once to find out about the checksum of one's
URLs. To track the communication between the Toolbar and Google,
often the use of packet sniffers, local proxies an similar tools
is suggested. But this is not necessarily needed, since the PageRank
files are cached by the Internet Explorer. So, the checksums can
simply been found out by having a look at the folder Temporary
Internet Files. Knowing the checksums of your URLs, you can view
the PageRank files in your browser and you do not have to accept
Google's 36 years lasting cookies.

Since the PageRank files are kept
in the browser cache and, thus, are clearly visible, and as long
as requests are not automated, watching the PageRank files in
a browser should not be a violation of Google's Terms of Service.
However, you should be cautious. The Toolbar submits its own User-Agent
to Google. It is:

Mozilla/4.0 (compatible; GoogleToolbar
1.1.60-deleon; OS SE 4.10)

1.1.60-deleon is a Toolbar version
which may of course change. OS is the operating system that you
have installed. So, Google is able to identify requests by browsers,
if they do not go out via a proxy and if the User-Agent is not
modified accordingly.

Taking a look at IE's cache, one
will normally notice that the PageRank files are not requested
from the domain www.google.com but from IP addresses like 216.239.33.102.
Additionally, the PageRank files' URLs often contain a parameter
"failedip" that is set to values like "216.239.35.102;1111"
(Its function is not absolutely clear). The IP addresses are each
related to one of Google's seven data centers and the reason for
the Toolbar querying IP-addresses is most likely to control the
PageRank display in a better way, especially in times of the "Google
Dance".

The PageRank Display at the Google Directory

Webmasters who do not want to check the PageRank files that are
used by the toolbar have another possibility to receive information
about the PageRank of their sites by means of the Google Directory
(directory.google.com).

The Google Directory is a dump
of the Open Directory Project (dmoz.org), which shows the PageRank
for listed documents similarly to the Google Toolbar display scaled
and by means of a green bar. In contrast to the Toolbar, the scale
is from 1 to 7. The exact value is not displayed, but it can be
determined by the divided bar respectively the width of the single
graphics in the source code of the page if one is not sure by
looking at the bar.

By comparing the Toolbar PageRank
of a document with its Directory PageRank, a more exact estimation
of a pages PageRank can be deduced, if the page is listed with
the ODP. This connection was mentioned first by Chris Raimondi
(www.searchnerd.com/pagerank).

Especially for pages with a Toolbar
PageRank of 5 or 6, one can appraise if the page is on the upper
or the lower end of its Toolbar scale. It shall be noted that
for the comparison the Toolbar PageRank of 0 was not taken into
account. It can easily be verified that this is appropriate by
looking at pages with a Toolbar PageRank of 3. However, it has
to be considered that for a verification pages of the Google Directory
respectively the ODP with a Toolbar PageRank of 4 or lower have
to be chosen, since otherwise no pages linked from there with
a Toolbar PageRank of 3 will be found

The Effect of Inbound Links

It has already been shown that each additional inbound link for
a web page always increases that page's PageRank. Taking a look
at the PageRank algorithm, which is given by

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

one may assume that an additional
inbound link from page X increases the PageRank of page A by

d × PR(X) / C(X)

where PR(X) is the PageRank of
page X and C(X) is the total number of its outbound links. But
page A usually links to other pages itself. Thus, these pages
get a PageRank benefit also. If these pages link back to page
A, page A will have an even higher PageRank benefit from its additional
inbound link.

The single effects of additional
inbound links shall be illustrated by an example.

We regard a website consisting
of four pages A, B, C and D which are linked to each other in
circle. Without external inbound links to one of these pages,
each of them obviously has a PageRank of 1. We now add a page
X to our example, for which we presume a constant Pagerank PR(X)
of 10. Further, page X links to page A by its only outbound link.
Setting the damping factor d to 0.5, we get the following equations
for the PageRank values of the single pages of our site:

PR(A) = 0.5 + 0.5 (PR(X) + PR(D))
= 5.5 + 0.5 PR(D)

PR(B) = 0.5 + 0.5 PR(A)

PR(C) = 0.5 + 0.5 PR(B)

PR(D) = 0.5 + 0.5 PR(C)

Since the total number of outbound
links for each page is one, the outbound links do not need to
be considered in the equations. Solving them gives us the following
PageRank values:

PR(A) = 19/3 = 6.33

PR(B) = 11/3 = 3.67

PR(C) = 7/3 = 2.33

PR(D) = 5/3 = 1.67

We see that the initial effect
of the additional inbound link of page A, which was given by

d × PR(X) / C(X) = 0,5 ×
10 / 1 = 5

is passed on by the links on our
site.

The Influence of the Damping Factor

The degree of PageRank propagation from one page to another by
a link is primarily determined by the damping factor d. If we
set d to 0.75 we get the following equations for our above example:

PR(A) = 0.25 + 0.75 (PR(X) + PR(D))
= 7.75 + 0.75 PR(D)

PR(B) = 0.25 + 0.75 PR(A)

PR(C) = 0.25 + 0.75 PR(B)

PR(D) = 0.25 + 0.75 PR(C)

Solving these equations gives
us the following PageRank values:

PR(A) = 419/35 = 11.97

PR(B) = 323/35 = 9.23

PR(C) = 251/35 = 7.17

PR(D) = 197/35 = 5.63

First of all, we see that there
is a significantly higher initial effect of additional inbound
link for page A which is given by

d × PR(X) / C(X) = 0.75
× 10 / 1 = 7.5

This initial effect is then propagated
even stronger by the links on our site. In this way, the PageRank
of page A is almost twice as high at a damping factor of 0.75
than it is at a damping factor of 0.5. At a damping factor of
0.5 the PageRank of page A is almost four times superior to the
PageRank of page D, while at a damping factor of 0.75 it is only
a little more than twice as high. So, the higher the damping factor,
the larger is the effect of an additional inbound link for the
PageRank of the page that receives the link and the more evenly
distributes PageRank over the other pages of a site.

The Actual Effect of Additional Inbound Links

At a damping factor of 0.5, the accumulated PageRank of all pages
of our site is given by

PR(A) + PR(B) + PR(C) + PR(D)
= 14

Hence, by a page with a PageRank
of 10 linking to one page of our example site by its only outbound
link, the accumulated PageRank of all pages of the site is increased
by 10. (Before adding the link, each page has had a PageRank of
1.) At a damping factor of 0.75 the accumulated PageRank of all
pages of the site is given by

PR(A) + PR(B) + PR(C) + PR(D)
= 34

This time the accumulated PageRank
increases by 30. The accumulated PageRank of all pages of a site
always increases by

(d / (1-d)) × (PR(X) / C(X))

where X is a page additionally
linking to one page of the site, PR(X) is its PageRank and C(X)
its number of outbound links. The formula presented above is only
valid, if the additional link points to a page within a closed
system of pages, as, for instance, a website without outbound
links to other sites. As far as the website has links pointing
to external pages, the surplus for the site itself diminishes
accordingly, because a part of the additional PageRank is propagated
to external pages.

The justification of the above
formula is given by Raph Levien and it is based on the Random
Surfer Model. The walk length of the random surfer is an exponential
distribution with a mean of (d/(1-d)). When the random surfer
follows a link to a closed system of web pages, he visits on average
(d/(1-d)) pages within that closed system. So, this much more
PageRank of the linking page - weighted by the number of its outbound
links - is distributed to the closed system.

For the actual PageRank calculations
at Google, Lawrence Page und Sergey Brin claim to usually set
the damping factor d to 0.85. Thereby, the boost for a closed
system of web pages by an additional link from page X is given
by

(0.85 / 0.15) × (PR(X) /
C(X)) = 5.67 × (PR(X) / C(X))

So, inbound links have a far larger
effect than one may assume.

The PageRank-1 Rule

Users of the Google Toolbar often notice that pages with a certain
Toolbar PageRank have an inbound link from a page with a Toolbar
PageRank which is higher by one. Some take this observation to
doubt the validity of the PageRank algorithm presented here for
the actual ranking methods of the Google search engine. It shall
be shown, however, that the PageRank-1 rule complies with the
PageRank algorithm.

Basically, the PageRank-1 rule
proves the fundamental principle of PageRank. Web pages are important
themselves if other important web pages link to them. It is not
necessary for a page to have many inbound links to rank well.
A single link from a high ranking page is sufficient.

To show the actual consistance
of the PageRank-1 rule with the PageRank algorithm several factors
have to be taken into consideration. First of all, the toolbar
PageRank is a logarithmically scaled version of real PageRank
values. If the PageRank value of one page is one higher than the
PageRank value of another page in terms of Toolbar PageRank, than
its real PageRank can at least be higher by an amount which equals
the logarithmical basis for the scalation of Toolbar PageRank.
If the logarithmical basis for the scalation is 6 and the toolbar
PageRank of a linking Page is 5, then the real PageRank of the
page which receives the link can be at least 6 times smaller to
make that page still get a toolbar PageRank of 4.

However, the number of outbound
links on the linking page thwarts the effect of the logarithmical
basis, because the PageRank propagation from one page to another
is devided by the number of outbound links on the linking page.
But it has already been shown that the PageRank benefit by a link
is higher than PageRank algorithm's term d(PR(Ti)/C(Ti)) pretends.
The reason is that the PageRank benefit for one page is further
distributed to other pages within the site. If those pages link
back as it usualy happens, the PageRank benefit for the page which
initially received the link is accordingly higher. If we assume
that at a high damping factor the logarithmical basis for PageRank
scalation is 6 and a page receives a PageRank benefit which is
twice as high as the PageRank of the linking page devided by the
number of its outbound links, the linking page could have at least
12 outbound links so that the Toolbar PageRank of the page receiving
the link is still at most one lower than the toolbar PageRank
of the linking page.

A number of 12 outbound links
admittedly seems relatively small. But normally, if a page has
an external inbound link, this is not the only one for that page.
Most likely other pages link to that page and propagate PageRank
to it. And if there are examples where a page receives a single
link from another page and the PageRanks of both pages comply
the PageRank-1 rule although the linking page has many outbound
links, this is first of all an indication for the linking page's
toolbar PageRank being at the upper end of its scale. The linking
page could be a "high" 5 and the page receiving the
link could be a "low" 4. In this way, the linking page
could have up to 72 outbound links. This number rises accordingly
if we assume a higher logarithmical basis for the scalation of
Toolbar PageRank

The Effect of Outbound Links

Since PageRank is based on the linking structure of the whole
web, it is inescapable that if the inbound links of a page influence
its PageRank, its outbound links do also have some impact. To
illustrate the effects of outbound links, we take a look at a
simple example.

We regard a web consisting of
to websites, each having two web pages. One site consists of pages
A and B, the other constists of pages C and D. Initially, both
pages of each site solely link to each other. It is obvious that
each page then has a PageRank of one. Now we add a link which
points from page A to page C. At a damping factor of 0.75, we
therefore get the following equations for the single pages' PageRank
values:

PR(A) = 0.25 + 0.75 PR(B)

PR(B) = 0.25 + 0.375 PR(A)

PR(C) = 0.25 + 0.75 PR(D) + 0.375 PR(A)

PR(D) = 0.25 + 0.75 PR(C)

Solving the equations gives us
the following PageRank values for the first site:

PR(A) = 14/23

PR(B) = 11/23

We therefore get an accumulated
PageRank of 25/23 for the first site. The PageRank values of the
second site are given by

PR(C) = 35/23

PR(D) = 32/23

So, the accumulated PageRank of
the second site is 67/23. The total PageRank for both sites is
92/23 = 4. Hence, adding a link has no effect on the total PageRank
of the web. Additionally, the PageRank benefit for one site equals
the PageRank loss of the other.

The Actual Effect of Outbound Links

As it has already been shown, the PageRank benefit for a closed
system of web pages by an additional inbound link is given by

(d / (1-d)) × (PR(X) / C(X)),

where X is the linking page, PR(X)
is its PageRank and C(X) is the number of its outbound links.
Hence, this value also represents the PageRank loss of a formerly
closed system of web pages, when a page X within this system of
pages now points by a link to an external page.

The validity of the above formula
requires that the page which receives the link from the formerly
closed system of pages does not link back to that system, since
it otherwise gains back some of the lost PageRank. Of course,
this effect may also occur when not the page that receives the
link from the formerly closed system of pages links back directly,
but another page which has an inbound link from that page. Indeed,
this effect may be disregarded because of the damping factor,
if there are enough other web pages in-between the link-recursion.
The validity of the formula also requires that the linking site
has no other external outbound links. If it has other external
outbound links, the loss of PageRank of the regarded site diminishes
and the pages already receiving a link from that page lose PageRank
accordingly.

Even if the actual PageRank values
for the pages of an existing web site were known, it would not
be possible to calculate to which extend an added outbound link
diminishes the PageRank loss of the site, since the above presented
formula regards the status after adding the link.

Intuitive Justification of the Effect of Outbound Links

The intuitive justification for the loss of PageRank by an additional
external outbound link according to the Random Surfer Modell is
that by adding an external outbound link to one page the surfer
will less likely follow an internal link on that page. So, the
probability for the surfer reaching other pages within a site
diminishes. If those other pages of the site have links back to
the page to which the external outbound link has been added, also
this page's PageRank will deplete.

We can conclude that external
outbound links diminish the totalized PageRank of a site and probably
also the PageRank of each single page of a site. But, since links
between web sites are the fundament of PageRank and indespensable
for its functioning, there is the possibility that outbound links
have positive effects within other parts of Google's ranking criteria.
Lastly, relevant outbound links do constitute the quality of a
web page and a webmaster who points to other pages integrates
their content in some way into his own site.

Dangling Links

An important aspect of outbound links is the lack of them on web
pages. When a web page has no outbound links, its PageRank cannot
be distributed to other pages. Lawrence Page and Sergey Brin characterise
links to those pages as dangling links.

The effect of dangling links
shall be illustrated by a small example website. We take a look
at a site consisting of three pages A, B and C. In our example,
the pages A and B link to each other. Additionally, page A links
to page C. Page C itself has no outbound links to other pages.
At a damping factor of 0.75, we get the following equations for
the single pages' PageRank values:

PR(A) = 0.25 + 0.75 PR(B)

PR(B) = 0.25 + 0.375 PR(A)

PR(C) = 0.25 + 0.375 PR(A)

Solving the equations gives us
the following PageRank values:

PR(A) = 14/23

PR(B) = 11/23

PR(C) = 11/23

So, the accumulated PageRank of
all three pages is 36/23 which is just over half the value that
we could have expected if page A had links to one of the other
pages. According to Page and Brin, the number of dangling links
in Google's index is fairly high. A reason therefore is that many
linked pages are not indexed by Google, for example because indexing
is disallowed by a robots.txt file. Additionally, Google meanwhile
indexes several file types and not HTML only. PDF or Word files
do not really have outbound links and, hence, dangling links could
have major impacts on PageRank.

In order to prevent PageRank
from the negative effects of dangling links, pages wihout outbound
links have to be removed from the database until the PageRank
values are computed. According to Page and Brin, the number of
outbound links on pages with dangling links is thereby normalised.
As shown in our illustration, removing one page can cause new
dangling links and, hence, removing pages has to be an iterative
process. After the PageRank calculation is finished, PageRank
can be assigned to the formerly removed pages based on the PageRank
algorithm. Therefore, as many iterations are needed as for removing
the pages. Regarding our illustration, page C could be processed
before page B. At that point, page B has no PageRank yet and,
so, page C will not receive any either. Then, page B receives
PageRank from page A and during the second iteration, also page
C gets its PageRank.

Regarding our example website
for dangling links, removing page C from the database results
in page A and B each having a PageRank of 1. After the calculations,
page C is assigned a PageRank of 0.25 + 0.375 PR(A) = 0.625. So,
the accumulated PageRank does not equal the number of pages, but
at least all pages which have outbound links are not harmed from
the danging links problem.

By removing dangling links from
the database, they do not have any negative effects on the PageRank
of the rest of the web. Since PDF files are dangling links, links
to PDF files do not diminish the PageRank of the linking page
or site. So, PDF files can be a good means of search engine optimisation
for Google

The Effect of the Number of Pages

Since the accumulated PageRank of all pages of the web equals
the total number of web pages, it follows directly that an additional
web page increases the added up PageRank for all pages of the
web by one. But far more interesting than the effect on the added
up PageRank of the web is the impact of additional pages on the
PageRank of actual websites.

To illustrate the effects of
addional web pages, we take a look at a hierachically structured
web site consisting of three pages A, B and C, which are joined
by an additional page D on the hierarchically lower level of the
site. The site has no outbound links. A link from page X which
has no other outbound links and a PageRank of 10 points to page
A. At a damping factor d of 0.75, the equations for the single
pages' PageRank values before adding page D are given by

PR(A) = 0.25 + 0.75 (10 + PR(B)
+ PR(C))

PR(B) = PR(C) = 0.25 + 0.75 (PR(A) / 2)

Solving the equations gives us
the follwing PageRank values:

PR(A) = 260/14

PR(B) = 101/14

PR(C) = 101/14

After adding page D, the equations
for the pages' PageRank values are given by

PR(A) = 0.25 + 0.75 (10 + PR(B)
+ PR(C) + PR(D))

PR(B) = PR(C) = PR(D) = 0.25 + 0.75 (PR(A) / 3)

Solving these equations gives
us the follwing PageRank values:

PR(A) = 266/14

PR(B) = 70/14

PR(C) = 70/14

PR(D) = 70/14

As to be expected since our example
site has no outbound links, after adding page D, the accumulated
PageRank of all pages increases by one from 33 to 34. Further,
the PageRank of page A rises marginally. In contrast, the PageRank
of pages B and C depletes substantially.

The Reduction of PageRank by Additional Pages

By adding pages to a hierarchically structured websites, the consequences
for the already existing pages are nonuniform. The consequences
for websites with a different structure shall be shown by another
example.

We take a look at a website constisting
of three pages A, B and C which are linked to each other in circle.
The pages are then joined by page D which fits into the circular
linking structure. The regarded site has no outbound links. Again,
a link from page X which has no other outbound links and a PageRank
of 10 points to page A. At a damping factor d of 0.75, the equations
for the single pages' PageRank values before adding page D are
given by

PR(A) = 0.25 + 0.75 (10 + PR(C))

PR(B) = 0.25 + 0.75 × PR(A)

PR(C) = 0.25 + 0.75 × PR(B)

Solving the equations gives us
the follwing PageRank values:

PR(A) = 517/37 = 13.97

PR(B) = 397/37 = 10.73

PR(C) = 307/37 = 8.30

After adding page D, the equations
for the pages' PageRank values are given by

PR(A) = 0.25 + 0.75 (10 + PR(D))

PR(B) = 0.25 + 0.75 × PR(A)

PR(C) = 0.25 + 0.75 × PR(B)

PR(D) = 0.25 + 0.75 × PR(C)

Solving these equations gives
us the follwing PageRank values:

PR(A) = 419/35 = 11.97

PR(B) = 323/35 = 9.23

PR(C) = 251/35 = 7.17

PR(D) = 197/35 = 5.63

Again, after adding page D, the
accumulated PageRank of all pages increases by one from 33 to
34. But now, any of the pages which already existed before page
D was added lose PageRank. The more uniform PageRank is distributed
by the links within a site, the more likely will this effect occur.

Since adding pages to a site often
reduces PageRank for already existing pages, it becomes obvious
that the PageRank algorithm tends to privilege smaller web sites.
Indeed, bigger web sites can counterbalance this effect by being
more attractive for other webmasters to link to them, simply because
they have more content.

None the less, it is also possible
to increase the PageRank of existing pages by additional pages.
Therefore, it has to be considered that as few PageRank as possible
is distributed to these additional pages

The Distribution of PageRank Regarding
Search Engine Optimisation

Up to this point, it has been described how the number of pages
and the number of inbound and outbound links, respectively, influence
PageRank. Here, it will mainly be discussed in how far PageRank
can be affected for the purpose of search engine optimisation
by a website's internal linking structure.

In most cases, websites are hierachically
structured to a certain extend, as it is illustrated in our example
of a web site consisting of the pages A, B and C. Normally, the
root page is withal optimised for the most important search phrase.
In our example, the optimised page A has an external inbound link
from page X which has no other outbound links and a PageRank of
10. The pages B and C each receive a link from page A and link
back to it. If we set the damping factor d to 0.5 the equations
for the single pages' PageRank values are given by

PR(A) = 0.5 + 0.5 (10 + PR(B)
+ PR (C))

PR(B) = 0.5 + 0.5 (PR(A) / 2)

PR(C) = 0.5 + 0.5 (PR(A) / 2)

Solving the equations gives us
the follwing PageRank values:

PR(A) = 8

PR(B) = 2.5

PR(C) = 2.5

It is generally not advisable
to solely work on the root page of a site for the purpose of search
engine optimisation. Indeed, it is, in most cases, more reasonable
to optimise each page of a site for different search phrases.

We now assume that the root page
of our example website provides satisfactory results for its search
phrase, but the other pages of the site do not, and therefore
we modify the linking structure of the website. We add links from
page B to page C and vice versa to our formerly hierarchically
structured example site. Again, page A has an external inbound
link from page X which has no other outbound links and a PageRank
of 10. At a damping factor d of 0.5, the equations for the single
pages' PageRank values are given by

PR(A) = 0.5 + 0.5 (10 + PR(B)
/ 2 + PR(C) / 2)

PR(B) = 0.5 + 0.5 (PR(A) / 2 + PR(C) / 2)

PR(C) = 0.5 + 0.5 (PR(A) / 2 + PR(B) / 2)

Solving the equations gives us
the follwing PageRank values:

PR(A) = 7

PR(B) = 3

PR(C) = 3

The result of adding internal
links is an increase of the PageRank values of pages B and C,
so that they likely will rise in search engine result pages for
their targeted keywords. On the other hand, of course, page A
will likely rank lower because of its diminished PageRank.

Generally spoken, PageRank will
distribute for the purpose of search engine optimisation more
equally among the pages of a site, the more the hierarchically
lower pages are interlinked.

Well Directed PageRank Distribution by Concentration of Outbound
Links

It has already been demonstrated that external outbound links
tend to have negative effects on the PageRank of a website's web
pages. Here, it shall be illustrated how this effect can be reduced
for the purpose of search engine optimisation by the systematic
arrangement of external outbound links.

We take a look at another hierarchically
structured example site consisting of the pages A, B, C and D.
Page A has links to the pages B, C and D. Besides a link back
to page A, each of the pages B, C and D has one external outbound
link. None of those external pages which receive links from the
pages B, C and D link back to our example site. If we assume a
damping factor d of 0.5, the equations for the calculation of
the single pages' PageRank values are given by

PR(A) = 0.5 + 0.5 (PR(B) / 2 +
PR(C) / 2 + PR(D) / 2)

PR(B) = PR(C) = PR(D) = 0.5 + 0.5 (PR(A) / 3)

Solving the equations gives us
the follwing PageRank values:

PR(A) = 1

PR(B) = 2/3

PR(C) = 2/3

PR(D) = 2/3

Now, we modify our example site
in a way that page D has all three external outbound links while
pages B and C have no more external outbound links. Besides this,
the general conditions of our example stay the same as above.
None of the external pages which receive a link from pages D link
back to our example site. If we, again, assume a damping factor
d of 0.5, the equations for the calculations of the single pages'
PageRank values are given by

PR(A) = 0.5 + 0.5 (PR(B) + PR(C)
+ PR(D) / 4)

PR(B) = PR(C) = PR(D) = 0.5 + 0.5 (PR(A) / 3)

Solving these equations gives
us the follwing PageRank values:

PR(A) = 17/13

PR(B) = 28/39

PR(C) = 28/39

PR(D) = 28/39

As a result of our modifications,
we see that the PageRank values for each single page of our site
have increased. Regarding search engine optimisation, it is therefore
advisable to concentrate external outbound links on as few pages
as possible, as long as it does not lessen a site's usabilty.

Link Exchanges for the purpose of Search Engine Optimisation

For the purpose of search engine optimisation, many webmasters
exchange links with others to increase link popularity. As it
has already been shown, adding links within closed systems of
web pages has no effects on the accumulated PageRank of those
pages. So, it is questionable if link exchanges have positive
consequences in terms of PageRank at all.

To show the effects of link exchanges,
we take a look at an an example of two hierarchically structured
websites consisting of pages A, B and C and D, E and F, respectively.
Within the first site, page A links to pages B and C and those
link back to page A. The second site is structured accordingly,
so that the PageRank values for its pages do not have to be computed
explicitly. At a damping factor d of 0.5, the equations for the
single pages' PageRank values are given by

PR(A) = 0.5 + 0.5 (PR(B) + PR(C))

PR(B) = PR(C) = 0.5 + 0.5 (PR(A) / 2)

Solving the equations gives us
the follwing PageRank values for the first site

PR(A) = 4/3

PR(B) = 5/6

PR(C) = 5/6

and accordingly for the second
site

PR(D) = 4/3

PR(E) = 5/6

PR(F) = 5/6

Now, two pages of our example
sites start a link exchange. Page A links to page D and vice versa.
If we leave the general conditions of our example the same as
above and, again, set the damping factor d to 0.5, the equations
for the calculations of the single pages' PageRank values are
given by

PR(A) = 0.5 + 0.5 (PR(B) + PR(C)
+ PR(D) / 3)

PR(B) = PR(C) = 0.5 + 0.5 (PR(A) / 3)

PR(D) = 0.5 + 0.5 (PR(E) + PR(F) + PR(A) / 3)

PR(E) = PR(F) = 0.5 + 0.5 (PR(D) / 3)

Solving these equations gives
us the follwing PageRank values:

PR(A) = 3/2

PR(B) = 3/4

PR(C) = 3/4

PR(D) = 3/2

PR(E) = 3/4

PR(F) = 3/4

We see that the link exchange
makes pages A and D benefit in terms of PageRank while all other
pages lose PageRank. Regarding search engine optimisation, this
means that the exactly opposite effect compared to interlinking
hierachically lower pages internally takes place. A link exchange
is thus advisable, if one page (e.g. the root page of a site)
shall be optimised for one important key phrase.

A basic premise for the positive
effects of a link exchange is that both involved pages propagate
a similar amount of PageRank to each other. If one of the involved
pages has a significantly higher PageRank or fewer outbound links,
it is likely that all of its site's pages lose PageRank. Here,
an important influencing factor is the size of a site. The more
pages a web site has, the more PageRank from an inbound link is
distributed to other pages of the site, regardless of the number
of outbound links on the page that is involved in the link exchange.
This way, the page involved in a link exchange itself benefits
lesser from the link exchange and cannot propagate as much PageRank
to the other page involved in the link exchange. All the influencing
factors should be weighted up against each other bevor one trades
links.

Finally, it shall be noted that
it is possible that all pages of a site benefit from a link exchange
in terms of PageRank, whereby also the other site taking part
in the link exchange does not lose PageRank. This may occur, when
the page involved in the link exchange already has a certain number
of external outbound links which don't link back to that site.
In this case, less PageRank is lost by the already existing outbound
links.

The Yahoo Bonus and its Impact
on Search Engine Optimization

Many experts in search engine optimization assume that certain
websites obtain a special PageRank evaluation from the Google
search engine which needs a manual intervention and does not derive
from the PageRank algorithm directly. Mostly, the directories
Yahoo and Open Directory Project (dmoz.org) are considered to
get this special treatment. In the context of search engine optimization,
this assumption would have the consequence that an entry into
the above mentioned directories had a big impact on a site's PageRank.

An approach that is often mentioned
when talking about influencing the PageRank of certain websites
is to assign higher starting values to them before the iterative
computation of PageRank begins. Such proceeding shall be reviewed
by looking at a simple example web consisting of two pages, whereby
each of these pages solely links to the other. We assign an initial
PageRank of 10 to one page and a PageRank of 1 to the other. The
damping factor d is set to 0.1, because the lower d is, the faster
the PageRank values converge during the iterations. So, we get
the following equations for the computation of the pages' PageRank
values:

PR(A) = 0.9 + 0.1 PR(B)

PR(B) = 0.9 + 0.1 PR(A)

During the iterations, we get
the following PageRank values:

Iteration PR (A) PR (B)

0 1 10

1 1.9 1.09

2 1.009 1.0009

3 1.00009 1.000009

It is obvious that despite assigning different starting values
to the two pages each of the PageRank values converges to 1, just
as it would have happened if no initial values were assigned.
Hence, starting values have no effect on PageRank if a sufficient
number of iterations takes place. Indeed, if the computation is
performed with only few iterations, the starting values would
influence PageRank. But in this case, we have to consider that
in our example the PageRank relation between the two pages reverses
after the first iteration. However, it shall be noted that for
our computation the actual PageRank values within one iteration
have been used and not the ones from the previous iteration. If
those values would have been used, the PageRank relation had alternated
after each iteration.

Modification of the PageRank
Algorithm

If assigning special starting values at the begin of the PageRank
calculations has no effect on the results of the computation,
this does not mean that it is not possible to influence the PageRank
of websites or web pages by an intervention in the PageRank algorithm.
Lawrence Page, for instance, describes a method for a special
evaluation of web pages in his PageRank patent specifications
(United States Patent 6,285,999). The starting point for his consideration
is that the random surfer of the Random Surfer Model may get bored
and stop following links with a constant probabilty, but when
he restarts, he won't take a random jump to any page of the web
but will rather jump to certain web pages with a higher probability
than to others. This behaviour is closer to the behaviour of a
real user, who would more likely use, for instance, directories
like Yahoo or ODP as a starting point for surfing.

If a special evaluation of certain
web pages shall take place, the original PageRank algorithm has
to be modified. With another expected value implemented, the algorithm
is given as follows:

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

Here, (1-d) is the probability
for the random surfer no longer following links. E(A) is the probability
for the random surfer going to page A after he has stopped following
links, weighted by the number of web pages. So, E is another expected
value whose average over all pages is 1. In this way, the average
of the PageRank values of all pages of the web will continue to
converge to 1. Thus, the PageRank values do not vaccilate because
of the special evaluation of web pages and the impact of PageRank
on the general ranking of web pages remains stable.

In our example, we set the probability
for the random surfer going to page A after he has stopped following
links to 0.1. The probability for him going to page B is set to
0.9. Since our web consists of two pages E(A) equals 0.2 and E(B)
equals 1.8. At a damping factor d of 0.5 we get the following
equations for the calculation of the single pages' PageRank values:

PR(A) = 0.2 × 0.5 + 0.5
× PR(B)

PR(B) = 1.8 × 0.5 + 0.5 × PR(A)

If we solve these equations we
get the following PageRank values:

PR(A) = 11/15

PR(B) = 19/15

The sum of the PageRank values
remains 2. The higher probability for the random surfer jumping
to page B is reflected by its higher PageRank. Indeed, the uniform
interlinking between both pages prevents our example pages' PageRank
values from a more significant impact of our intervention.

So, it is possible to implement
the special evaluation of certain web pages into the PageRank
algorithm without having to change it fundamentally. It is questionable,
indeed, what criteria is used for the evaluation. Lawrence Page
suggests explicitly the utilization of real usage data in his
PageRank patent specifications. Google, meanwhile, collects usage
date by means of the Google Toolbar. And Google would not even
need as much data, as if the whole ranking was solely based on
usage data. A limited sample would be sufficient to determine
the 1,000 or 10,000 most important pages on the web. The PageRank
algorithm can then fill the holes in usage data and is thereby
able to deliver a more accurate picture of the web.

Of course, all statements regarding
the influence of real usage data on PageRank are pure speculation.
Even if there is a special evaluation of certain web pages at
all will in the end, stay a secret of the people at Google.

Nonetheless, Assigning Starting Values?

Although, assigning special starting values to pages at the begin
of PageRank calculations has no effect on PageRank values it can,
nonetheless, be reasonable.

We take a look at our example
web consisting of the pages A, B and C, whereby page A links to
the pages B and C, page B links to page C and page C links to
page A. In this case, the damping factor d is set to 0.75. So,
we get the following equations for the iterative computation of
the single pages' PageRank values:

PR(A) = 0.25 + 0.75 PR(C)

PR(B) = 0.25 + 0.75 (PR(A) / 2)

PR(C) = 0.25 + 0.75 (PR(A) / 2 + PR(B))

Basically, it is not necessary
to assign starting values to the single pages before the computation
begins. They simply start with a value of 0 and we get the following
PageRank values during the iterations:

Iteration PR(A) PR(B) PR(C)

0 0 0 0

1 0.25 0.34375 0.60156

2 0.70117 0.51294 0.89764

3 0.92323 0.59621 1.04337

4 1.03253 0.63720 1.11510

5 1.08632 0.65737 1.15040

6 1.11280 0.66730 1.16777

7 1.12583 0.67219 1.17633

8 1.13224 0.67459 1.18054

9 1.13540 0.67578 1.18261

10 1.13696 0.67636 1.18363

11 1.13772 0.67665 1.18413

12 1.13810 0.67679 1.18438

13 1.13828 0.67686 1.18450

14 1.13837 0.67689 1.18456

15 1.13842 0.67691 1.18459

16 1.13844 0.67692 1.18460

17 1.13845 0.67692 1.18461

18 1.13846 0.67692 1.18461

19 1.13846 0.67692 1.18461

20 1.13846 0.67692 1.18461

21 1.13846 0.67692 1.18461

22 1.13846 0.67692 1.18462

If we assign 1
to each page before the computation starts, we get the following
PageRank values during the iterations:

Iteration PR(A) PR(B) PR(C)

0 1 1 1

1 1 0.625 1.09375

2 1.07031 0.65137 1.13989

3 1.10492 0.66434 1.16260

4 1.12195 0.67073 1.17378

5 1.13034 0.67388 1.17928

6 1.13446 0.67542 1.18199

7 1.13649 0.67618 1.18332

8 1.13749 0.67656 1.18398

9 1.13798 0.67674 1.18430

10 1.13823 0.67684 1.18446

11 1.13835 0.67688 1.18454

12 1.13840 0.67690 1.18458

13 1.13843 0.67691 1.18460

14 1.13845 0.67692 1.18461

15 1.13845 0.67692 1.18461

16 1.13846 0.67692 1.18461

17 1.13846 0.67692 1.18461

18 1.13846 0.67692 1.18461

19 1.13846 0.67692 1.18462

If we now assign a starting value to each page, which is closer
to its effective PageRank (1.1 for page A, 0.7 for page B and
1.2 for page C), we get the following results:

Iteration PR(A) PR(B) PR(C)

0 1.1 0.7 1.2

1 1.15 0.68125 1.19219

2 1.14414 0.67905 1.18834

3 1.14126 0.67797 1.18645

4 1.13984 0.67744 1.18552

5 1.13914 0.67718 1.18506

6 1.13879 0.67705 1.18483

7 1.13863 0.67698 1.18472

8 1.13854 0.67695 1.18467

9 1.13850 0.67694 1.18464

10 1.13848 0.67693 1.18463

11 1.13847 0.67693 1.18462

12 1.13847 0.67692 1.18462

13 1.13846 0.67692 1.18462

So, the closer the assigned starting values are to the effective
results we would get by solving the equations, the faster do the
PageRank values converge in the iterative computation. Less iterations
are needed, which can be useful for providing more up to date
search results, especially regarding the growth rate of the web.
Starting point for an accurate presumption of the actual PageRank
distribution may be the PageRank values of a former PageRank calculation.
All the pages which are new in the index could get an initial
PageRank of 1, which will then be a lot closer to the effective
PageRank value after the first few iterations.

Additional Factors
Influencing PageRank

It has been widely discussed if additional criteria beyond the
link structure of the web have been implemented in the PageRank
algorithm since the scientific work on PageRank has been published
by Lawrence Page and Sergey Brin. Lawrence Page himself outlines
the following potential influencing factors in his patent specifications
for PageRank:

Visibility of
a link

Position of a link within a document

Distance between web pages

Importance of a linking page

Up-to-dateness of a linking page

First of all, the implementation of additional criteria in PageRank
would result in a better approximation of human usage regarding
the Random Surfer Model. Considering the visibility of a link
and its position within a document implies that a user does not
click on links completely at haphazard, but rather follows links
which are highly and immediately visible regardless of their anchor
text. The other criteria would give Google more flexibility in
determing in how far an inbound link of a page should be considered
important, than the methods which have been described so far.

Whether or not
the above mentioned factors are actually implemented in PageRank
can not be proved empirically and shall not be discussed here.
It shall rather be illustrated in which way additional influencing
factors can be implemented in the PageRank algorithm and which
options the Google search engine thereby gets in terms of influencing
PageRank values.

Modification of the PageRank Algorithm

To implement additional factors in PageRank, the original PageRank
algorithm has again to be modified. Since we have to assume that
PageRank calculations are still based on numerous iterations and
for the purpose of short computation times, we have to consider
to keep the number of database queries during the iterations as
small as possible. Therefore, the following modification of the
PageRank algorithm shall be assumed:

PR(A) = (1-d)
+ d (PR(T1)×L(T1,A) + ... + PR(Tn)×L(Tn,A))

Here, L(Ti,A)
represents the evaluation of a link which points from page Ti
to page A. L(Ti,A) withal replaces the PageRank weighting of page
Ti by the number of outbound links on page Ti which was given
by 1/C(Ti). L(Ti,A) may consist of several factors, each of them
having to be determined only once and then being merged to one
value before the iterative PageRank calculation begins. So, the
number of database queries during the iterations stays the same,
although, admittedly, a much larger database has to be queried
at each step in comparison to the computation by use of the original
algorithm, since now there is an evaluation of each link instead
of an evaluation of pages (by the number of their outbound links).

Different Evaluation of Links within a Document

Two of the criteria for the evaluation of links mentioned by Lawrence
Page in his PageRank patent specifications are the visibilty of
a link and its position within a document. Regarding the Random
Surfer Model, those criteria reflect the probability for the random
surfer clicking on a link on a specific web page. In the original
PageRank algorithm, this probability is given by the term (1/C(Ti)),
whereby the probability is equal for each link on one page.

Assigning different
probabilities to each link on a page can, for instance, be realized
as follows:

We take a look
at a web consisting of three pages A, B anc C, where each of these
pages has outbound links to both of the other pages. Links are
weighted by two evaluation criteria X and Y. X represents the
visibility of a link. X equals 1 if a link is not particularly
emphasized, and 2 if the link is, for instance, bold or italic.
Y represents the position of a link within a document. Y equals
1 if the link is on the lower half of the page, and 3 if the link
is on the upper half of the page. If we assume a multiplicative
correlation between X and Y, the links in our example are evaluated
as follows:

X(A,B) ×
Y(A,B) = 1 × 3 = 3

X(A,C) × Y(A,C) = 1 × 1 = 1

X(B,A) × Y(B,A) = 2 × 3 = 6

X(B,C) × Y(B,C) = 2 × 1 = 2

X(C,A) × Y(C,A) = 2 × 3 = 6

X(C,B) × Y(C,B) = 2 × 1 = 2

For the purpose
of determinig the single factors L, the evaluated links must not
simply be weighted by the number of outbound links on one page,
but in fact by the total of evaluated links on the page. Thereby,
we get the following weighting quotients Z(Ti) for the single
pages Ti:

Z(A) = X(A,B)
× Y(A,B) + X(A,C) × Y(A,C) = 4

Z(B) = X(B,A) × Y(B,A) + X(B,C) × Y(B,C) = 8

Z(C) = X(C,A) × Y(C,A) + X(C,B) × Y(C,B) = 8

The evaluation
factors L(T1,T2) for a link which is pointing from page T1 to
page T2 are hence given by

L(T1,T2) = X(T1,T2)
× Y(T1,T2) / Z(T1)

Their values regarding
our example are as follows:

L(A,B) = 0.75

L(A,C) = 0.25

L(B,A) = 0.75

L(B,C) = 0.25

L(C,A) = 0.75

L(C,B) = 0.25

At a damping factor
d of 0.5, we get the following equations for the calculation of
PageRank values:

PR(A) = 0.5 +
0.5 (0.75 PR(B) + 0.75 PR(C))

PR(B) = 0.5 + 0.5 (0.75 PR(A) + 0.25 PR(C))

PR(C) = 0.5 + 0.5 (0.25 PR(A) + 0.25 PR(B))

Solving these
equations gives us the follwing PageRank values for our example:

PR(A) = 819/693

PR(B) = 721/693

PR(C) = 539/693

First of all,
we see that page A has the highest PageRank of all three pages.
This is caused by page A receiving the relatively higher evaluated
link from page B as well as from page C.

Furthermore, we
see that even by the evaluation of single links, the sum of the
PageRank values of all pages equals 3 (2079/693) and thereby the
total number of pages. So, the PageRank values computed by our
modified PageRank algorithm can be used for the general ranking
of web pages by Google without any normalisation being needed.

Different Evaluation of Links by Page Specific Criteria

Besides the unequal evaluation of links within a document, Lawrence
Page mentions the possibility of evaluating links according to
criteria which are based upon the linking page. At first glance,
this does not seem necessary since it is the main principle of
PageRank to rank pages the higher, the more high ranking pages
link to them. But, at the time of their scientific work on PageRank,
Page and Brin have already recognized that their algorithm is
vulnerable to artificial inflation of PageRank.

An artificial
influence on PageRank might be exerted by webmasters who generate
a multitude of web pages whose links distribute PageRank in a
way that single pages within that system receive a special importance.
Those pages can have a high PageRank without being linked to from
other pages with high PageRank. So, not only the concept of PageRank
is undermined, but also the search engine's index is spammed with
an innumerable amount of web pages which were solely created to
influence PageRank.

In his patent
specifications for PageRank, Lawrence Page presents the evaluation
of links by the distance between pages as a means to avoid the
artificial inflation of PageRank, because the bigger the distance
between two pages, the less likely has one webmaster control over
both. A criterium for the distance between two pages may be if
they are on the same domain or not. In this way, internal links
would be weighted less than external links. In the end, any general
measure of the distance between links can be used to determine
such a weighting. This comprehends if pages are on the same server
or not and also the geographical distance between servers.

As another indicator
for the importance of a document, Lawrence Page mentions the up-to-dateness
of the documents which link to it. This argument considers that
the information on a page is less likely outdated, the more pages
which have been modified recently link to it. In contrast, the
original PageRank concept, just like any method of measuring link
popularity, favours older documents which gained their inbound
links in the course of their existence and have at a higher probability
been modified less recently than new documents. Basically, recently
modified documents may be given a higher evaluation by weighting
the factor (1-d). In this way, both those recently modified documents
and the pages they link to receive a higher PageRank. But, if
a page has been modified recently, is not necessarily an indicator
for the importance of the information presented on it. So, as
suggested by Lawrence Page, it is advisable not to favour recently
modified pages but only their outbound links.

Finally, Page
mentions the importance of the web location of a page as an indicator
of the importance of its outbound links. As an example for an
important web location he names the root page of a domain, but,
in the end, Google could exert influence on PageRank absolutely
arbitrarily.

To implement the
evaluation of the linking page into PageRank, the evaluation factor
of the modified algorithm must consist of several single factors.
For a link that points from page Ti to page A, it can be given
as follows:

L(Ti,A) = K(Ti,A)
× K1(Ti) × ... × Km(Ti)

where K(Ti,A)
is the above presented weighting of a single link within a page
by its visibility or position. Additionally, an evaluation of
page Ti by m criteria which are represented by the factors Kj(Ti)
takes place.

To implement the
evaluation of the linking pages, not only the algorithm but also
the proceedings of PageRank calculation have to be modified. This
shall be illustrated by an example.

We take a look
at a web consisting of three pages A, B and C, whereby page A
links to the pages B and C, page B links to page C and page C
links to page A. The outbound links of one page are evaluated
equally, so there is no weighting by visibilty or position. But
now, the pages are evaluated by one criterium. In this way, an
inbound link from page C shall be considered four times as important
as an inbound link from one of the other pages. After weighting
by the number of pages, we get the following evaluation factors:

K(A) = 0.5

K(B) = 0.5

K(C) = 2

At a damping factor
d of 0.5, the equations for the computation of the PageRank values
are given by

PR(A) = 0.5 +
0.5 × 2 PR(C)

PR(B) = 0.5 + 0.5 × 0.5 × 0.5 PR(A)

PR(C) = 0.5 + 0.5 (0.5 PR(B) + 0.5 × 0.5 PR(A))

Solving the equations
gives us the follwing PageRank values:

PR(A) = 4/3

PR(B) = 2/3

PR(C) = 5/6

At the current
modifications of the PageRank algorithm, the accumulated PageRank
of all pages no longer equals the number of pages. The reason
therefore is that the weighting of the page evaluation by the
number of pages was not appropriate. To determine the proper weighting,
the web's linking structure would have to be anticipated, which
is not possible in case of the actual WWW. Therefore, the PageRank
calculated by an evaluation of linking pages has to be normalized
if there shall not be any unfounded effects on the general ranking
of pages by Google. Within the iterative calculation, a normalization
would have to take place after each iteration to minimize unintentional
distortions.

In the case of
a small web, the evaluation of pages often causes severe distortions.
In the case of the actual WWW, these distortions should normally
equalise by the number of pages. Indeed, it is to be expected
that the evaluation of the distance between pages will cause distortions
on PageRank, since pages with many inbound links surely tend to
be linked to from different geographical regions. But such effects
can be anticipated by experience from previous calculation periods,
so that a normalisation would only have to be marginal.

In either case,
implementing additional factors in PageRank is possible. Indeed,
the computation of PageRank values would take more time

Theme-based PageRank

For many years now, the topic- or theme-based homogeneity of websites
has been dicussed as a possible ranking criterion of search engines.
There are various theoretical approaches for the implementation
of themes in search engine algorithms which all have in common
that web pages are no longer ranked only based on their own content,
but also based on the content of other web pages. For example,
the content of all pages of one website may take influence on
the ranking of one single page of this website. On the other hand,
it is also conceivable that one page's ranking is based on the
content of those pages which link to it or which it links to itself.

The potential
implementation of a theme-based ranking in the Google search engine
is discussed controversially. In search engine optimization forums
and on websites on this topic we can over and over again find
advice that inbound links from sites with a similar theme to our
own have a larger influence on PageRank than links from unrelated
sites. This hypothesis shall be discussed here. Therefore, we
first of all take a look at two relatively new approaches for
the integration of themes in the PageRank technique: on the one
hand the "intelligent surfer" by Matthew Richardson
and Pedro Domingos and on the other hand the Topic-Sensitive PageRank
by Taher Haveliwala. Subsequently, we take a look at the possibility
of using content analyses in order to compare the text of web
pages, which can be a basis for weighting links within the PageRank
technique.

The "Intelligent Surfer" by Richardson and Domingos

Matthew Richardson and Pedro Domingos resort to the Random Surfer
Model in order to explain their approach for the implementation
of themes in the PageRank technique. Instead of a surfer who follws
links completely at random, they suggest a more intelligent surfer
who, on the one hand, only follows links which are related to
an original search query and, on the other hand, also after "getting
bored" only jumps to a page which relates to the original
query.

So, to Richardson
and Domingos' "intelligent surfer" only pages are relevant
that contain the search term of an initial query. But since the
Random Surfer Model does nothing but illustrate the PageRank technique,
the question is how an "intelligent" behaviour of the
Random Surfer influences PageRank. The answer is that for every
term occuring on the web a separate PageRank calculation has to
be conducted and each calculation is solely based on links between
pages which contain that term.

Computing PageRank
this way causes some problems. They especially appear for search
terms that do not occur so often on the web. To make it into the
PageRank calculations for a specific search term, that term has
not only to appear on someone's page, but also on the pages that
link to it. So, the search results would often be based on small
subsets of the web and may omit relevant sites. In addition, using
such small subsets of the web, the algorithms are more vulnerable
to spam by automatically generating numerous pages.

Additionally,
there are serious problems regarding scalability. Richardson and
Domingos estimate the memory and computing time requirements for
several 100,000 terms 100-200 times higher compared to the original
PageRank calculations. Regarding the large number of small subsets
of the web, these numbers appear to be realistic.

The higher memory
requirements should not be so much of a problem because Richardson
and Domingos correctly state that the term specific PageRank values
constitute only a fraction of the data volume of Google's inverse
index. However, the computing time requirements are indeed a large
problem. If we assume just five hours for a conventional PageRank
calculation, then this would last about 3 weeks based on Richardson
and Domingos' model, which makes it unsuitable for actual employment.

Taher Haveliwala's Topic-Sensitive PageRank

The approach of Taher Havilewala seems to be more practical for
actual employment. Just like Richardson and Domingos, also Havilewala
suggests the computation of different PageRanks for different
topics. However, the Topic-Sensitive PageRank does not consist
of hundreds of thousands of PageRanks for different terms, but
of a few PageRanks for different topics. Topic-Sensitive PageRank
is based on the link structure of the whole web, whereby the topic
sensitivity implies that there is a different weighting for each
topic.

The basic principle
of Haveliwala's approach has already been described in our section
on the "Yahoo-Bonus", where we have discussed the possibility
to assign a particular imporance to certain web pages. In the
words of the Random Surfer Model, this is realized by increasing
the probability for the Random Surfer jumping to a page after
"getting bored". Via links, this manual intervention
in the PageRank technique has an influence on the PageRank of
each page on the web. More precisely, we have reached taking influence
on PageRank by implementing another value E in the PageRank algorithm:

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

Haveliwala's Topic-Sensitive-PageRank
goes one step further. Instead of assigning a universally higher
value to a website or a web page, Haveliwala differentiates on
the basis of different topics. For each of these topics, he identifies
other authority pages. On the basis of this evaluation, different
PageRanks are calculated - each separately, but for the entire
web.

For his experiments
on Topic-Sensitive PageRank, Haveliwala has chosen the 16 top-level
categories of the Open Directory Project both for the identification
of topics and for the intervention in PageRank. More precisely,
Haveliwala assigns a higher value E to the pages of those ODP
categories for which he calculates PageRank. If, for example,
he calculates the PageRank for the topic health, all the ODP pages
in the health category receive a relatively higher value E and
they pass this value in the form of PageRank on to the pages which
are linked from there. Of course, this PageRank is passed on to
other pages and, if we assume that health-related websites tend
to link more often to other websites within that topic, pages
on the topic health generally receive a higher PageRank.

Haveliwala confirms
the incompleteness of choosing the Open Directory Project in order
to identify topics, which for example results in a high degree
of dependence on ODP editors and in a rather rough subdivision
into topics. But, as Haveliwala states, his method shows good
results and it can surely be improved without big effort.

However, one crucial
point in Haveliwala's work on Topic-Sensitive-PageRank is the
identification of the user's preferences. Having a Topic-Specific
PageRank is useless as long as we do not know in which topics
an actual user is interested. In the end, search results must
be based on the PageRank that matches the user's preferences best.
The Topic-Sensitive PageRank can only be used if these are known.

Indeed, Haveliwala
does supply some practicable approaches for the identification
of user preferences. He describes, for example, the search in
context by highlighting terms on a web page. In this way, the
content of that web page could be an indicator for waht the user
is looking for. At this point, we want to note the potential of
the Google Toolbar. The Toolbar submits data regarding search
terms and pages that a user has visited to Google. This data can
be used to create user profiles which can then be a basis for
the identification of the user's preferences. However, even without
using such techniques, it is imaginable that a user simply chooses
the topic he is interested in before he does a query.

The Weighting of Links Based on Content Analyses

That it is possible to weight single links within the PageRank
technique has been shown on the previous page. The thought behind
weighting links based on content analyses is to avoid the corrumption
of PageRank. By weighting links this way, it is theoretically
possible to diminish the influence of links between thematically
unrelated page, which have been set for the sole purpose of boosting
PageRank of one page. Indeed, it is questionable if it is possible
to realize such weighting based on content analyses.

The fundamentals
of content analyses are based on Gerard Salton's work in the 1960s
and 1970s. In his vector space model of information retrieval,
documents are modeled as vectors which are built upon terms and
their weighting within the document. These term vectors allow
comparisons between the content of documents by, for instance,
calculating the cosine measure (the inner product) of the vectors.
In its basic form, the vector space model has some weaknesses.
For instance, often the assumption that if and in how far the
same words appear in two documents is an indicator for their similarity
is criticized. However, numerous enhancements have been developed
that solve most of the problems of the vector space model.

One person who
excelled at publications which are based on Salton's vector space
model is Krishna Bharat. This is interesting because Bharat meanwhile
is a member of Google's staff and, particularly, because he is
deemded to be the developer of "Google News" (news.google.com).
Google News is a service that crawls news websites, evaluates
articles and then provides them categorized and grouped in different
subjects on the Google New website. According to Google, all these
procedures are completely automated. Therefore, other criteria
like, for example, the time when an article is published, are
taken into account, but if there is no manual intervention, the
clustering of articles is most certainly only possible, if the
contents of the articles are actually compared to each other.
The questions is: How can this be realized?

In their publication
on a term vector database, Raymie Stata, Krishna Bharat and Farzin
Maghoul describe how the contents of web pages can be compared
based on term vectors and, particularly, they describe how some
of the problems with the vector space model can be solved. Firstly,
not all terms in documents are suitable for content analsysis.
Very frequent terms provide only little discrimination across
vectors and, so, the most frequent third of all terms is eliminated
from the database. Infrequent terms, on the other hand, do not
provide a good basis for measuring similarity. Such terms are,
for example, misspellings. They appear only on few pages which
are likely unrelated in terms of their theme, but because they
are so infrequent, the term vectors of the pages appear to be
closely related. Hence, also the least frequent third of all terms
is eliminated from the database.

Even if only one
third of all terms is included in the term vectors, this selection
is still not very efficient. Stata, Bharat and Maghoul perform
another filtering, so that each term vector is based on a maximum
of 50 terms. But these are not the 50 most frequent terms on a
page. They weight a term by deviding the number of times it appears
on a page by the number of times it appears on all pages, and
those 50 terms with the highest weight are included in the term
vector of a page. This selection actually allows a real differentiation
between the content of pages.

The methods described
above are standards for the vector space model. If, for example,
the inner product of two term vectors is rather high, the contents
of the according pages tend to be similar. This may allow content
comparisons in many areas, but it is doubtful if it is a good
basis for weighting links within the PageRank technique. Most
of all, synonyms and terms that describe similar things can not
be identified. Indeed, there are algorithms for word stemming
which work good for the english language, but in other languages
word stemming is much more complicated. Different languages are
a general problem. Unless, for instance, brand names or loan words
are used, texts in different languages normally do not contain
the same terms. And if they do, these terms normally have a completely
different meaning, so that comparing content in different languages
is not possible. However, Stata, Bharat and Maghoul provide a
method of resolution for these problems.

Stata, Bharat
und Maghoul present a concrete application for their Term Vector
Database by classifying pages thematically. Bharat has also published
on this issue together with Monika Henzinger, presently Google's
Research Director, and they called it "topic distillation".
Topic distillation is based on calculating so-called topic vectors.
Topic vectors are term vectors, but they do not only include terms
of one page but rather the terms of many pages which are on the
same topic. So, in order to create topic vectors, they have to
know a certain amount of web pages which are on several pre-defined
topics. To achieve this, they resort to web directories.

For their application,
Stata, Bharat und Maghoul have crawled about 30,000 links within
each of the then 12 main categories of Yahoo to create topic vectors
which include about 10,000 terms each. Then, in order to identify
the topic of any other web page, they matched the according term
vector with all the topic vectors which were created from the
Yahoo crawl. The topic of a web page derived from the topic vector
which matched the term vector of the web page best. That such
a classification of web pages works can again be observed by the
means of Google News. Google News does not only merge articles
to one news topic, but also arranges them to the categories World,
U.S., Business, Sci/Tech, Sports, Entertainment and Health. As
long as this categorization is not based on the structure of the
website where the articles come from (which is unlikely), the
actual topic of an article has in fact to be computed.

At the time he
published on term vectors, Krishna Bharat did not work on PageRank
but rather on Kleinberg's algorithm, so that he was more interested
in filtering off-topic links than in weighting links. But from
classifying pages to weighting links based on content comparisons,
there is only a small step. Instead of matching the term vectors
of two pages, it is much more efficient to match the topics of
two pages. We can, for instance, create a "topic affinity
vector" for each page based on the degree of affinity of
the page's term vector and all the topic vectors. The better the
topic affinity vectors of two pages match, the more likely are
they on the same topic and the higher should a link between them
be weighted.

Using topic vectors
has one big advantage over comparing term vectors directly: A
topic vector can include terms in different languages by being
based on, for instance, the links on different national Yahoo
versions. Deviant site structures of the national versions can
most certainly be adapted manually. Even better may be using the
ODP because the structure of the sub-categories of the "World"
category is based on the main OPD structure. In this way, measuring
topic similarities between pages in different languages can be
realized, so that a really useful weighting of links based on
text analyses appears to be possible.

Is there an Actual Implementation of Themes in PageRank?

That both the approach of Havelivala and the approach of Richardson
and Domingos are not utilized by Google is obvious. One would
notice it using Google. However, a weighting of links based on
text analyses would not be apparent immediately. It has been shown
that it is theoretically possible. But it is doubtful that it
is actually implemented.

We do not want
to claim that we have shown the only way of weighting links on
the basis of text analyses. Indeed, there are certainly dozens
of others. However, the approach that we provided here is based
on publications of important members of Google's staff and, thus,
we want to rest a critical evaluation on it.

Like always, when
talking about PageRank, there is the question if our approach
is sufficienly scalable. On the one hand, it causes additional
memory requirements. After all, Stata, Bharat and Maghoul describe
the system architecture of a term vector database which is different
from Google's inverse index, since it maps from page ids to terms
and, so, can hardly be integrated in the existing architecture.
At the actual size of Google's index, the additional memory requirements
should be several hundred GB to a few TB. However, this should
not be so much of a problem since Google's index is most certainly
several times bigger. In fact, the time requirements for building
the database and for computing the weigtings appear to be the
critical part.

Building a term
verctor database should be approximately as time-consuming as
building an inverse index. Of course, many procecces can probably
be used for building both but if, for instance, the weighting
of terms in the term vectors has to differ from the weighting
of terms in the inverse index, the time requirements remain substantial.
If we assume that, like in our approach, content analyses are
based on computing the inner products of topic affinity vectors
which have to be calculated by matching term vectors and topic
vectors, this process should be approximately as time-consuming
as computing PageRank. Moreover, we have to consider that the
PageRank calculations themselves beome more complicated by weighting
links.

So, the additional
time requirements are definitely not negligible. This is why we
have to ask ourselves if weighting links based on text analyses
is useful at all. Links between thematically unrelated page, which
have been set for the sole purpose of boosting PageRank of one
page, may be annoying, but most certainly they are only a small
fraction of all links. Additionally, the web itself is completely
inhomogeneous. Google, Yahoo or the ODP do not owe their high
PageRank solely to links from other search engines or directories.
A huge part of the links on the web are simply not set for the
purpose of showing visitors ways to more thematically related
information. Indeed, the motivation for placing links is manifold.
Moreover, the problably most popular websites are completely inhomogeneous
in terms of theme. Think about portals like Yahoo or news websites
which contain articles that cover almost any subject of life.
A strong weighting of links as it has been described here could
influence those website's PageRanks significantly.

If the PageRank
technique shall not become totally futile, a weighting of links
can only take place to a small extent. This, of course, raises
the question if the efforts it requires are justifiable. After
all, there are certainly other ways to eliminate spam which often
comes to the top of search results through thematically unrelated
and probably bought links.

PR0 - Google's
PageRank 0 Penalty

By the end of 2001, the Google search engine introduced a new
kind of penalty for websites that use questionable search engine
optimization tactics: A PageRank of 0. In search engine optimization
forums it is called PR0 and this term shall also be used here.
Characteristically for PR0 is that all or at least a lot of pages
of a website show a PageRank of 0 in the Google Toolbar, even
if they do have high quality inbound links. Those pages are not
completely removed from the index but they are always at the end
of search results and, thus, they are hardly to be found.

A PageRank of
0 does not always mean a penalty. Sometimes, websites which seam
to be penalized simply lack inbound links with an sufficiently
high PageRank. But if pages of a website which have formerly been
placed well in search results, suddenly show the dreaded white
PageRank bar, and if there have not been any substantial changes
regarding the inbound links of that website, this means - according
to the prevailing opinion - certainly a penalty by Google.

We can do nothing
but speculate about the causes for PR0 because Google representatives
rarely publish new information on Google's algorithms. But, non
the less, we want to give a theoretical approach for the way PR0
may work because of its serious effects on search engine optimization.

The Background of PR0

Spam has always been one of the biggest problems that search engines
had to deal with. When spam is detected by search engines, the
usual proceeding is the banishment of those pages, websites, domains
or even IP addresses from the index. But, removing websites manually
from the index always means a large assignment of personnel. This
causes costs and definitely runs contrary to Google's scalability
goals. So, it appears to be necessary to filter spam automatically.

Filtering spam
automatically carries the risk of penalizing innocent webmasters
and, hence, the filters have to react rather sensibly on potential
spam. But then, a lot of spam can pass the filters and some additional
measures may be necessary. In order to filter spam effectively,
it might be useful to take a look at links.

That Google uses
link analysis in order to detect spam has been confirmed more
or less clearly in WebmasterWorld's Google News Forum by a Google
employee who posts as "GoogleGuy". Over and over again,
he advises webmasters to avoid "linking to bad neighbourhoods".
In the following, we want to specify the "linking to bad
neighbourhoods" and, to become more precisely, we want to
discuss how an identification of spam can be realized by the analysis
of link structures. In particular, it shall be shown how entire
networks of spam pages, which may even be located on a lot of
different domains, can be detected.

BadRank as the Opposite of PageRank

The theoretical approach for PR0 as it is presented here was initially
brought up by Raph Levien (www.advogato.org/person/raph). We want
to introduce a technique that - just like PageRank - analyzes
link structures, but, that unlike PageRank does not determine
the general importance of a web page but rather measures its negative
characteristics. For the sake of simplicity this technique shall
be called "BadRank".

BadRank is in
priciple based on "linking to bad neighbourhoods". If
one page links to another page with a high BadRank, the first
page gets a high BadRank itself through this link. The similarities
to PageRank are obvious. The difference is that BadRank is not
based on the evaluation of inbound links of a web page but on
its outbound links. In this sense, BadRank represents a reversion
of PageRank. In a direct adaptation of the PageRank algorithm,
BadRank would be given by the following formula:

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

where

BR(A) is the BadRank
of page A,

BR(Ti) is the BadRank of pages Ti which are outbound links of
page A,

C(Ti) is here the number of inbound links of page Ti and

d is the again necessary damping factor.

In the previously discussed modifications of the PageRank algorithm,
E(A) represented the special evaluation of certain web pages.
Regarding the BadRank algorithm, this value reflects if a page
was detected by a spam filter or not. Without the value E(A),
the BadRank algorithm would be useless because it was nothing
but another analysis of link structures which would not take any
further criteria into account.

By means of the
BadRank algorithm, first of all, spam pages can be evaluated.
A filter assigns a numeric value E(A) to them, which can, for
example, be based on the degree of spamming or maybe even better
on their PageRank. Thereby, again, the sum of all E(A) has to
equal the total number of web pages. In the course of an iterative
computation, BadRank is not only transfered to pages which link
to spam pages. In fact, BadRank is able to identify regions of
the web where spam tends to occur relatively often, just as PageRank
identifies regions of the web which are of general importance.

Of course, BadRank
and PageRank have significant differences, especially, because
of using outbound and inbound links, respectively. Our example
shows a simple, hierarchically structured website that reflects
common link structures pretty well. Each page links to every page
which is on a higher hierachical level and on its branch of the
website's tree structure. Each page links to pages which are arranged
hierarchically directly below them and, additionally, pages on
the same branch and the same hierarchical level link to each other.

The following
table shows the distribution of inbound and outbound links for
the hierarchical levels of such a site.

Level inbound Links outbound Links

0 6 2

1 4 4

2 2 3

As to be expected, regarding inbound links, a hierarchical gradation
from the index page downwards takes place. In contrast, we find
the highest number of outbound links on the website's mid-level.
We can see similar results, when we add another level of pages
to our website while the above described linking rules stay the
same.

Level inbound Links outbound Links

0 14 2

1 8 4

2 4 5

3 2 4

Again, there is a concentration of outbound links on the website's
mid-level. But most of all, the outbound links are much more evenly
distributed than the inbound links.

If we assign a
value of 100 to the index page's E(A) in our original example,
while all other values E equal 1 and if the damping factor d is
0.85, we get the following BadRank values:

Page BadRank

A 22.39

B/C 17.39

D/E/F/G 12.21

First of all, we see that the BadRank distributes from the index
page among all other pages of the website. The combination of
PageRank and BadRank will be discussed in detail below, but, no
matter how the combination will be realized, it is obvious that
both can neutralize each other very well. After all, we can assume
that also the page's PageRank decreases, the lower the hierarchy
level is, so that a PR0 can easily be achieved for all pages.

If we now assume
that the hierarchically inferior page G links to a page X with
a constant BadRank BR(X)=10, whereby the link from page G is the
only inbound link for page X, and if all values E for our example
website equal 1, we get, at a damping factor d of 0.85, the following
values:

Page BadRank

A 4.82

B 7.50

C 14.50

D 4.22

E 4.22

F 11.22

G 17.18

In this case, we see that the distribution of the BadRank is less
homogeneous than in the first scenario. Non the less, a distribution
of BadRank among all pages of the website takes place. Indeed,
the relatively low BadRank of the index page A is remarkable.
It could be a problem to neutralize its PageRank which should
be higher compared to the rest of the pages. This effect is not
really desirable but it reflects the experiences of numerous webmasters.
Quite often, we can see the phenomenom that all pages except for
the index page of a website show a PR0 in the Google Toolbar,
whereby the index page often has a Toolbar PageRank between 2
and 4. Therefore, we can probably assume that this special variant
of PR0 is not caused by the detection of the according website
by a spam filter, but the site rather received a penalty for "linking
to bad neighbourhoods". Indeed, it is also possible that
this variant of PR0 occurs when only hierarchical inferior pages
of a website get trapped in a spam filter.

The Combination of PageRank and BadRank to PR0

If we assume that BadRank exists in the form presented here, there
is now the question in which way BadRank and PageRank can be combined,
in order to penalize as much spammers as possible while at the
same time penalizing as few innocent webmasters as possible.

Intuitively, implementing
BadRank directly in the actual PageRank computations seems to
make sense. For instance, it is possible to calculate BadRank
first and, then, divide a page's PageRank through its BadRank
each time in the course of the iterative calculation of PageRank.
This would have the advantage, that a page with a high BadRank
could pass on just a little PageRank or none at all to the pages
it links to. After all, one can argue that if one page links to
a suspect page, all the other links on that page may also be suspect.

Indeed, such a
direct connection between PageRank and BadRank is very risky.
Most of all, the actual influence of BadRank on PageRank cannot
be estimated in advance. It is to be considered that we would
create a lot of pages which cannot pass on PageRank to the pages
they link to. In fact, these pages are dangling links, and as
it has been discussed in the section on outbound links, it is
absolutely necessary to avoid dangling links while computing PageRank.

So, it would be
advisable to have separate iterative calculations for PageRank
and BadRank. Combining them afterwards can, for instance, be based
on simple arithmetical operations. In principle, a subtraction
would have the desirable consequence that relatively small BadRank
values can hardly have a large influence on relatively high PageRank
values. But, there would certainly be a problem to achieve PR0
for a large number of pages by using the subtraction. We would
rather see a PageRank devaluation for many pages.

Achieving the
effects that we know as PR0 seems easier to be realized by dividing
PageRank through BadRank. But this would imply that BadRank receives
an extremely high importance. However, since the average BadRank
equals 1, a big part of BadRank values is smaller than 1 and,
so, a normalization is necessary. Probably, normalizing and scaling
BadRank to values between 0 and 1 so that "good" pages
have values close to 1, and "bad" pages have values
close to 0 and, subsequently, multiplying these values with PageRank
would supply the best results.

A very effective
and easy to realize alternative would probably be a simple stepped
evaluation of PageRank and BadRank. It would be reasonable that
if BadRank exceeds a certain value it will always lead to a PR0.
The same could happen when the relation of PageRank to BadRank
is below a certain value. Additionally, it would make sense that
if BadRank and/or the relation of BadRank to PageRank is below
a certain value, BadRank takes no influence at all.

Only if none of
these cases occurs, an actual combination of PageRank and BadRank
- for instance by dividing PageRank through BadRank - would be
necessary. In this way, all unwanted effects could be avoided.

A Critical View on BadRank and PR0

How Google would realize the combination of PageRank and BadRank
is of rather minor importance. Indeed, a separate computation
and a subsequent combination of both has the consequence that
it may not be possible to see the actual effect of a high BadRank
by looking at the Toolbar. If a page has a high PageRank in the
original sense, the influence of its BadRank can be negligible.
But if another page links to it, this could have quite serious
consequences.

An even bigger
problem is the direct reversion of the PageRank algorithm as we
have presented it here: Just as an additional inbound for one
page can do nothing but increasing this page's PageRank, an additional
outbound link can only increase its BadRank. This is because of
the addition of BadRank values in the BadRank formula. So, it
does not matter how many "good" outbound links a page
has - one link to a spam page can be enough to lead to a PR0.

Indeed, this problem
may appear in exceptional cases only. By our direct reversion
of the PageRank algorithm, the BadRank of a page is divided by
its inbound links and single links to pages with high BadRank
transfer only a part of that BadRank in each case. Google's Matt
Cutts' remark on this issue is: "If someone accidentally
does a link to a bad site, that may not hurt them, but if they
do twenty, that's a problem." (searchenginewatch.com/sereport/02/11-searchking.html)

However, as long
as all links are weighted uniformly within the BadRank computation,
there is another problem. If two pages differ widely in PageRank
and both have a link to the same page with a high BadRank, this
may lead to the page with the higher PageRank suffering far less
from the transferred BadRank than the page with the low PageRank.
We have to hope that Google knows how to deal with such problems.
Nevertheless it shall be noted that, regarding the procedure presented
here, outbound links can do nothing but harm.

Of course, all
statements regarding how PR0 works are pure speculation. But in
principle, the analysis of link structures similarly to the PageRank
technique should be the way how only Google understands to deal
with spam.