PageRank is a link analysis algorithm that assigns a numerical
weighting to each element of a hyperlinked set of documents,
such as the World Wide Web, with the purpose of "measuring" its
relative importance within the set. The algorithm may be applied
to any collection of entities with reciprocal quotations and
references. The numerical weight that it assigns to any given
element E is also called the PageRank of E and denoted by PR(E).
PageRank was developed at Stanford University by Larry Page
(hence the name Page-Rank) and later Sergey Brin as part of
a research project about a new kind of search engine. The project
started in 1995 and led to a functional prototype, named Google,
in 1998. Shortly after, Page and Brin founded Google Inc.,
the company behind the Google search engine. While just one
of many factors which determine the ranking of Google search
results, PageRank continues to provide the basis for all of
Google's web search tools.
The name PageRank is a trademark of Google. The PageRank process
has been patented (U.S. Patent 6,285,999 ). The patent is not
assigned to Google but to Stanford University.
General description
Google describes PageRank:
“ PageRank relies on the uniquely democratic nature of the web by using its vast
link structure as an indicator of an individual page's value. In essence, Google
interprets a link from page A to page B as a vote, by page A, for page B. But,
Google looks at more than the sheer volume of votes, or links a page receives;
it also analyzes the page that casts the vote. Votes cast by pages that are themselves "important" weigh
more heavily and help to make other pages "important". ”
In other words, a PageRank results from a "ballot" among all
the other pages on the World Wide Web about how important a
page is. A hyperlink to a page counts as a vote of support.
The PageRank of a page is defined recursively and depends on
the number and PageRank metric of all pages that link to it
("incoming links"). A page that is linked to by many pages
with high PageRank receives a high rank itself. If there are
no links to a web page there is no support for that page.
Google assigns a numeric weighting from 0-1 for each webpage
on the Internet; this PageRank denotes your site’s importance
in the eyes of Google. The scale for PageRank is logarithmic
like the Richter Scale and roughly based upon quantity of inbound
links as well as importance of the page providing the link.
Numerous academic papers concerning PageRank have been published
since Page and Brin's original paper.[3] In practice, the PageRank
concept has proven to be vulnerable to manipulation, and extensive
research has been devoted to identifying falsely inflated PageRank
and ways to ignore links from documents with falsely inflated
PageRank.
Alternatives to the PageRank algorithm include the HITS algorithm
proposed by Jon Kleinberg, the IBM CLEVER project and the TrustRank
algorithm.
PageRank algorithm
PageRank is a probability distribution used to represent the
likelihood that a person randomly clicking on links will arrive
at any particular page. PageRank can be calculated for any-size
collection of documents. It is assumed in several research
papers that the distribution is evenly divided between all
documents in the collection at the beginning of the computational
process. The PageRank computations require several passes,
called "iterations", through the collection to adjust approximate
PageRank values to more closely reflect the theoretical true
value.
A probability is expressed as a numeric value between 0 and
1. A 0.5 probability is commonly expressed as a "50% chance" of
something happening. Hence, a PageRank of 0.5 means there is
a 50% chance that a person clicking on a random link will be
directed to the document with the 0.5 PageRank.
Simplified PageRank algorithm
Assume a small universe of four web pages: A, B, C and D. The
initial approximation of PageRank would be evenly divided between
these four documents. Hence, each document would begin with
an estimated PageRank of 0.25.
If pages B, C, and D each only link to A, they would each confer
0.25 PageRank to A. All PageRank PR( ) in this simplistic system
would thus gather to A because all links would be pointing
to A.
But then suppose page B also has a link to page C, and page
D has links to all three pages. The value of the link-votes
is divided among all the outbound links on a page. Thus, page
B gives a vote worth 0.125 to page A and a vote worth 0.125
to page C. Only one third of D's PageRank is counted for A's
PageRank (approximately 0.083).
In other words, the PageRank conferred by an outbound link
L( ) is equal to the document's own PageRank score divided
by the normalized number of outbound links (it is assumed that
links to specific URLs only count once per document).
In the general case, the PageRank value for any page u can
be expressed as:
i.e. the PageRank value for a page u is dependent on the PageRank
values for each page v out of the set Bu (this set contains
all pages linking to page u), divided by the number L(v) of
links from page v.
PageRank algorithm including damping factor
The PageRank theory holds that even an imaginary surfer who
is randomly clicking on links will eventually stop clicking.
The probability, at any step, that the person will continue
is a damping factor d. Various studies have tested different
damping factors, but it is generally assumed that the damping
factor will be set around 0.85.[4]
The damping factor is subtracted from 1 (and in some variations
of the algorithm, the result is divided by the number of documents
in the collection) and this term is then added to the product
of (the damping factor and the sum of the incoming PageRank
scores).
That is,
or (N = the number of documents in collection)
So any page's PageRank is derived in large part from the PageRanks
of other pages. The damping factor adjusts the derived value
downward. The second formula above supports the original statement
in Page and Brin's paper that "the sum of all PageRanks is
one".[3] Unfortunately, however, Page and Brin gave the first
formula, which has led to some confusion.
Google recalculates PageRank scores each time it crawls the
Web and rebuilds its index. As Google increases the number
of documents in its collection, the initial approximation of
PageRank decreases for all documents.
The formula uses a model of a random surfer who gets bored
after several clicks and switches to a random page. The PageRank
value of a page reflects the chance that the random surfer
will land on that page by clicking on a link. It can be understood
as a Markov chain in which the states are pages, and the transitions
are all equally probable and are the links between pages.
If a page has no links to other pages, it becomes a sink and
therefore terminates the random surfing process. However, the
solution is quite simple. If the random surfer arrives at a
sink page, it picks another URL at random and continues surfing
again.
When calculating PageRank, pages with no outbound links are
assumed to link out to all other pages in the collection. Their
PageRank scores are therefore divided evenly among all other
pages. In other words, to be fair with pages that are not sinks,
these random transitions are added to all nodes in the Web,
with a residual probability of usually d = 0.85, estimated
from the frequency that an average surfer uses his or her browser's
bookmark feature.
So, the equation is as follows:
where p1,p2,...,pN are the pages under consideration, M(pi)
is the set of pages that link to pi, L(pj) is the number of
outbound links on page pj, and N is the total number of pages.
The PageRank values are the entries of the dominant eigenvector
of the modified adjacency matrix. This makes PageRank a particularly
elegant metric: the eigenvector is
where R is the solution of the equation
where the adjacency function \ell(p_i,p_j) is 0 if page pj
does not link to pi, and normalised such that, for each j
i.e. the elements of each column sum up to 1.
This is a variant of the eigenvector centrality measure used
commonly in network analysis.
The values of the PageRank eigenvector are fast to approximate
(only a few iterations are needed) and in practice it gives
good results.
As a result of Markov theory, it can be shown that the PageRank
of a page is the probability of being at that page after lots
of clicks. This happens to equal t - 1 where t is the expectation
of the number of clicks (or random jumps) required to get from
the page back to itself.
The main disadvantage is that it favors older pages, because
a new page, even a very good one, will not have many links
unless it is part of an existing site (a site being a densely
connected set of pages, such as Wikipedia). The Google Directory
(itself a derivative of the Open Directory Project) allows
users to see results sorted by PageRank within categories.
The Google Directory is the only service offered by Google
where PageRank directly determines display order. In Google's
other search services (such as its primary Web search) PageRank
is used to weight the relevance scores of pages shown in search
results.
Several strategies have been proposed to accelerate the computation
of PageRank.[5]
Various strategies to manipulate PageRank have been employed
in concerted efforts to improve search results rankings and
monetize advertising links. These strategies have severely
impacted the reliability of the PageRank concept, which seeks
to determine which documents are actually highly valued by
the Web community.
Google is known to actively penalize link farms and other schemes
designed to artificially inflate PageRank. How Google identifies
link farms and other PageRank manipulation tools are among
Google's trade secrets.
PageRank variations Google Toolbar
The Google Toolbar's PageRank feature displays a visited page's
PageRank as a whole number between 0 and 10. The most popular
websites have a PageRank of 10. The least have a PageRank of
0. Google has not disclosed the precise method for determining
a Toolbar PageRank value. Google representative Matt Cutts
has publicly indicated that the Toolbar PageRank values are
republished about once every three months, indicating that
the Toolbar PageRank values are historical rather than real-time
values.[6]
Google directory PageRank
The Google Directory PageRank is an 8-unit measurement. These
values can be viewed in the Google Directory. Unlike the Google
Toolbar which shows the PageRank value by a mouseover of the
greenbar, the Google Directory does not show the PageRank as
a numeric value but only as a green bar.
False or spoofed PageRank
While the PR shown in the Toolbar is considered to be derived
from an accurate PageRank value (at some time prior to the
time of publication by Google) for most sites, it must be noted
that this value is also easily manipulated. A current flaw
is that any low PageRank page that is redirected, via a 302
server header or a "Refresh" meta tag, to a high PR page causes
the lower PR page to acquire the PR of the destination page.
In theory a new, PR0 page with no incoming links can be redirected
to the Google home page - which is a PR 10 - and by the next
PageRank update the PR of the new page will be upgraded to
a PR10. This spoofing technique, also known as 302 Google Jacking,
is a known failing or bug in the system. Any page's PR can
be spoofed to a higher or lower number of the webmaster's choice
and only Google has access to the real PR of the page. Spoofing
is generally detected by running a Google search for a URL
with questionable PR, as the results will display the URL of
an entirely different site (the one redirected to) in its results.
Manipulating PageRank
For search-engine optimization purposes, some companies offer
to sell high PageRank links to webmasters.[7] As links from
higher-PR pages are believed to be more valuable, they tend
to be more expensive. It can be an effective and viable marketing
strategy to buy link advertisements on content pages of quality
and relevant sites to drive traffic and increase a webmaster's
link popularity. However, Google has publicly warned webmasters
that if they are or were discovered to be selling links for
the purpose of conferring PageRank and reputation, their links
will be devalued (ignored in the calculation of other pages'
PageRanks). The practice of buying and selling links is intensely
debated across the Webmastering community. Google advises webmasters
to use the nofollow HTML attribute value on sponsored links.
According to Matt Cutts, Google is concerned about webmasters
who try to game the system, and thereby reduce the quality
of Google search results.[7]
Other uses of PageRank
A version of PageRank has recently been proposed as a replacement
for the traditional ISI impact factor,[8] and implemented at
eigenfactor.org. Instead of merely counting total citation
to a journal, the "importance" of each citation is determined
in a PageRank fashion.
A similar new use of PageRank is to rank academic doctoral
programs based on their records of placing their graduates
in faculty positions. In PageRank terms, academic departments
link to each other by hiring their faculty from each other
(and from themselves).[9]
PageRank has also been used to automatically rank WordNet synsets
according to how strongly they possess a given semantic property,
such as positivity or negativity.[10]
A dynamic weighting method similar to PageRank has been used
to generate customized reading lists based on the link structure
of Wikipedia.[11]
A Web crawler may use PageRank as one of a number of importance
metrics it uses to determine which URL to visit next during
a crawl of the web. One of the early working papers[12] which
were used in the creation of Google is Efficient crawling through
URL ordering,[13] which discusses the use of a number of different
importance metrics to determine how deeply, and how much of
a site Google will crawl. PageRank is presented as one of a
number of these importance metrics, though there are others
listed such as the number of inbound and outbound links for
a URL, and the distance from the root directory on a site to
the URL.
Google's "rel='nofollow'" proposal
In early 2005, Google implemented a new value, "nofollow",
for the rel attribute of HTML link and anchor elements, so
that website builders and bloggers can make links that Google
will not consider for the purposes of PageRank — they are links
that no longer constitute a "vote" in the PageRank system.
The nofollow relationship was added in an attempt to help combat
spamdexing.
As an example, people could create many message-board posts
with links to their website to artificially inflate their PageRank.
Now, however, the message-board administrator can modify the
code to automatically insert "rel='nofollow'" to all hyperlinks
in posts, thus preventing PageRank from being affected by those
particular posts.
This method of avoidance, however, also has various drawbacks,
such as reducing the link value of actual comments.
