PageRank

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 $PR(E).$

The name PageRank is a trademark of Google. The PageRank process has been patented (US patent|6285999). The patent is assigned to Stanford University and not to Google. Google has exclusive license rights on the patent from Stanford University. The university received 1.8M shares in exchange for the patent. The shares were sold in 2005 for $336M.

Description

Google describes PageRank:

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 0-10 for each webpage on the Internet; this PageRank denotes a site’s importance in the eyes of Google. The PageRank is derived from a theoretical probability value on a logarithmic scale like the Richter Scale. The PageRank of a particular page is roughly based upon the quantity of inbound links as well as the PageRank of the pages providing the links. It is known that other factors, e.g. relevance of search words on the page and actual visits to the page reported by the Google toolbar also influence the PageRank. In order to prevent manipulation, spoofing and spamdexing, Google provides no specific details about how other factors influence PageRank.

Numerous academic papers concerning PageRank have been published since Page and Brin's original paper.cite web|url=http://dbpubs.stanford.edu:8090/pub/1998-8 | title=The Anatomy of a Large-Scale Hypertextual Web Search Engine | work=Brin, S.; Page, L | year=1998] 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.

Other link-based ranking algorithms for Web pages include the HITS algorithm invented by Jon Kleinberg (used by Teoma and now Ask.com), the IBM CLEVER project, and the TrustRank algorithm.

History

PageRank was developed at Stanford University by Larry Page (hence the name "Page"-Rank [cite book | author = David Vise and Mark Malseed | year = 2005 | title = The Google Story | url = http://www.thegooglestory.com/ | isbn = ISBN 0-553-80457-X | pages = 37] ) 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.Google Technology. [http://www.google.com/technology/] ]

PageRank is based on citation analysis that was developed in the 1950s by Eugene Garfield at the University of Pennsylvania, and Google's founders cite Garfield's work in their original paper. By following links from one page to another, virtual communities of webpages are found. Web link analysis was first developed by Jon Kleinberg and his team while working on the CLEVER project at IBM's Almaden Research Center.

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.

implified 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.

In the original form of PageRank initial values were simply 1. This meant that the sum of all pages was the total number of pages on the web. Later versions of PageRank (see the below formulas) would assume a probability distribution between 0 and 1. Here we're going to simply use a probability distribution hence the initial value 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.

:$PR(A)=\; PR(B)\; +\; PR(C)\; +\; PR(D).,$This is 0.75.

Again, 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).

:$PR(A)=\; frac\{PR(B)\}\{2\}+\; frac\{PR(C)\}\{1\}+\; frac\{PR(D)\}\{3\}.,$

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).

:$PR(A)=\; frac\{PR(B)\}\{L(B)\}+\; frac\{PR(C)\}\{L(C)\}+\; frac\{PR(D)\}\{L(D)\}.\; ,$

In the general case, the PageRank value for any page u can be expressed as:

:$PR(u)\; =\; sum\_\{v\; in\; B\_u\}\; frac\{PR(v)\}\{L(v)\}$,

i.e. the PageRank value for a page u is dependent on the PageRank values for each page v out of the set B_u (this set contains all pages linking to page u), divided by the number "L"("v") of links from page v.

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. [cite conference | author = Sergey Brin and Lawrence Page | year = 1998 | title = The anatomy of a large-scale hypertextual Web search engine | url = http://www-db.stanford.edu/~backrub/google.html | booktitle = Proceedings of the seventh international conference on World Wide Web 7 | pages = 107-117 (Section 2.1.1 Description of PageRank Calculation)| location = Brisbane, Australia ]

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,

:$PR(A)=\; 1\; -\; d\; +\; d\; left(\; frac\{PR(B)\}\{L(B)\}+\; frac\{PR(C)\}\{L(C)\}+\; frac\{PR(D)\}\{L(D)\}+,cdots\; ight)$

or ("N" = the number of documents in collection)

:$PR(A)=\; \{1\; -\; d\; over\; N\}\; +\; d\; left(\; frac\{PR(B)\}\{L(B)\}+\; frac\{PR(C)\}\{L(C)\}+\; frac\{PR(D)\}\{L(D)\}+,cdots\; ight)\; .$

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". 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:

:$PR(p\_i)\; =\; frac\{1-d\}\{N\}\; +\; d\; sum\_\{p\_j\; in\; M(p\_i)\}\; frac\{PR\; (p\_j)\}\{L(p\_j)\}$

where $p\_1,\; p\_2,\; ...,\; p\_N$ are the pages under consideration, $M(p\_i)$ is the set of pages that link to $p\_i$, $L(p\_j)$ is the number of outbound links on page $p\_j$, 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:$mathbf\{R\}\; =$

egin{bmatrix}{(1-d)/ N} \{(1-d) / N} \vdots \{(1-d) / N}end{bmatrix}

+ d

egin{bmatrix}ell(p_1,p_1) & ell(p_1,p_2) & cdots & ell(p_1,p_N) \ell(p_2,p_1) & ddots & & vdots \vdots & & ell(p_i,p_j) & \ell(p_N,p_1) & cdots & & ell(p_N,p_N)end{bmatrix}

mathbf{R}

where the adjacency function $ell(p\_i,p\_j)$ is 0 if page $p\_j$ does not link to $p\_i$, and normalised such that, for each "j"

:$sum\_\{i\; =\; 1\}^N\; ell(p\_i,p\_j)\; =\; 1,$

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. [cite web | url=http://citeseer.ist.psu.edu/719287.html | title=Fast PageRank Computation via a Sparse Linear System (Extended Abstract) | work= Gianna M. Del Corso, Antonio Gullí, Francesco Romani]

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. In December 2007 Google started "actively" penalizing sites selling paid text links. How Google identifies link farms and other PageRank manipulation tools are among Google's trade secrets.

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. [Cutt, Matts. [http://www.mattcutts.com/blog/whats-an-update/ What’s an update?] Blog post (September 8, 2005)]

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 green bar, the Google Directory does not show the PageRank as a numeric value but only as a green bar.

False or spoofed PageRank

While the PageRank 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 PageRank page causes the lower PageRank page to acquire the PageRank 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 PageRank can be spoofed to a higher or lower number of the webmaster's choice and only Google has access to the real PageRank of the page. Spoofing is generally detected by running a Google search for a URL with questionable PageRank, 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. 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 Webmaster's 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 and relevancy of Google search results.cite web|publisher=mattcutts.com/blog|date=April 14, 2007|accessdate=2007-05-28|title=How to report paid links|url=http://www.mattcutts.com/blog/how-to-report-paid-links/]

The intentional surfer model

The original PageRank algorithm reflects the so-called random surfer model, meaning that the PageRank of a particular page is derived from the theoretical probability of visiting that page when clicking on links at random. However, real users do not randomly surf the web, but follow links according to their interest and intention. A page ranking model that reflects the importance of a particular page as a function of how many actual visits it receives by real users is called the "intentional surfer model" [Citation | author = Jøsang, A. | contribution = Trust and Reputation Systems | title = Foundations of Security Analysis and Design IV, FOSAD 2006/2007 Tutorial Lectures. | year = 2007 | publisher = Springer LNCS 4677 | editor = Aldini, A. | pages = 209-245 | doi = 10.1007/978-3-540-74810-6 | url = http://www.unik.no/people/josang/papers/Jos2007-FOSAD.pdf] . The Google toolbar sends information to Google for every page visited, and thereby provides a basis for computing PageRank based on the intentional surfer model. The introduction of the nofollow attribute by Google to combat spamdexing has the side effect that webmasters also use it on every outgoing link to increase own PageRank. This causes a loss of actual links for the Web crawlers to follow, thereby making the original PageRank algorithm based on the random surfer model more unreliable. Using information provided by the Google toolbar partly compensates for the loss of information caused by the nofollow attribute, so that the PageRank of a page is based on a combination of the random and the intentional surfer models.

Other uses

A version of PageRank has recently been proposed as a replacement for the traditional ISI impact factor, [cite journal
author = Johan Bollen, Marko A. Rodriguez, and Herbert Van de Sompel.
year = 2006
month = December
title = Journal Status
journal = Scientometrics
volume = 69
issue = 3
url = http://www.arxiv.org/abs/cs.GL/0601030] and implemented at [http://www.eigenfactor.org 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). [cite journal | author = Benjamin M. Schmidt and Matthew M. Chingos | title=Ranking Doctoral Programs by Placement: A New Method | year=2007 | journal=PS: Political Science and Politics | volume=40 | issue=July | pages=523–529 | url=http://www.people.fas.harvard.edu/~chingos/rankings_paper.pdf ]

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. [cite web | author = Andrea Esuli and Fabrizio Sebastiani | title=PageRanking WordNet synsets: An Application to Opinion-Related Properties | work=In Proceedings of the 35th Meeting of the Association for Computational Linguistics, Prague, CZ, 2007, pp. 424-431 | url=http://nmis.isti.cnr.it/sebastiani/Publications/ACL07.pdf | accessmonthday=June 30 | accessyear=2007]

A dynamic weighting method similar to PageRank has been used to generate [http://www.wikiosity.com customized reading lists] based on the link structure of Wikipedia. [cite journal|author = Wissner-Gross, A. D. | title = [http://www.alexwg.org/ICALT2006.pdf Preparation of topical readings lists from the link structure of Wikipedia] | journal = Proceedings of the IEEE International Conference on Advanced Learning Technology | location = Rolduc, Netherlands | year = 2006 | pages = 825 | doi = 10.1109/ICALT.2006.1652568 ]

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 [cite web | title=Working Papers Concerning the Creation of Google | work=Google | url=http://dbpubs.stanford.edu:8091/diglib/pub/projectdir/google.html | accessmonthday=November 29 | accessyear=2006] which were used in the creation of Google is "Efficient crawling through URL ordering", [cite journal|author = Cho, J., Garcia-Molina, H., and Page, L. | title = [http://dbpubs.stanford.edu:8090/pub/1998-51 Efficient crawling through URL ordering] | journal = Proceedings of the seventh conference on World Wide Web | location = Brisbane, Australia | year = 1998] 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.

The PageRank may also be used as a [http://de.scientificcommons.org/23846375 methodology] to measure the apparent impact of a community like the Blogosphere on the overall Web itself. This approach uses therefore the PageRank to measure the distribution of attention in reflection of the Scale-free network paradigm.

=Google's "rel='nofollow'" proposal=

In early 2005, Google implemented a new value, "nofollow" [cite web | title=Preventing Comment Spam | work=Google | url=http://googleblog.blogspot.com/2005/01/preventing-comment-spam.html | accessmonthday=Janurary 01 | accessyear=2005] , for the rel attribute of HTML link and anchor elements, so that website developers 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. With the nofollow value message-board administrator can modify their 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. (See: Spam in blogs#nofollow)

Internal Pagerank

Internal Pagerank is nothing but Pagerank of internal pages of the website. Webmasters like to find out Internal Pagerank of blogposts to leave comments on, specially for Dofollow blogs to increase backlinks to their websites and blogs.

Internal pages with good pagerank can have significant impact on Google rankings of that specific page.

ee also

*SimRank - a measure of object-to-object similarity based on random-surfer model
*EigenTrust — a decentralized PageRank algorithm
*Google bomb
*
*Google search
*Hilltop algorithm
*Power method — the iterative eigenvector algorithm used to calculate PageRank
*Search Engine Optimization
*TrustRank

References

Further reading

*
*
*
* cite conference | last = Cheng | first = Alice | coauthors = Eric J. Friedman
title = Manipulability of PageRank under Sybil Strategies
booktitle = Proceedings of the First Workshop on the Economics of Networked Systems (NetEcon06)
place = Ann Arbor, Michigan | date = 2006-06-11
url = http://www.cs.duke.edu/nicl/netecon06/papers/ne06-sybil.pdf | accessdate = 2008-01-22
* cite conference | last = Altman | first = Alon | coauthors = Moshe Tennenholtz
title = Ranking Systems: The PageRank Axioms
booktitle = Proceedings of the 6th ACM conference on Electronic commerce (EC-05)
place = Vancouver, BC | date = 2005
url = http://stanford.edu/~epsalon/pagerank.pdf | accessdate = 2008-02-05

External links

* [http://www.google.com/technology/ Our Search: Google Technology] by Google
* [http://www.ams.org/featurecolumn/archive/pagerank.html How Google Finds Your Needle in the Web's Haystack] by the American Mathematical Society
* [http://patft.uspto.gov/netacgi/nph-Parser?patentnumber=6,285,999 Original PageRank U.S. Patent- Method for node ranking in a linked database] - September 4, 2001
* [http://patft1.uspto.gov/netacgi/nph-Parser?Sect1=PTO1&Sect2=HITOFF&d=PALL&p=1&u=%2Fnetahtml%2FPTO%2Fsrchnum.htm&r=1&f=G&l=50&s1=6,799,176.PN.&OS=PN/6,799,176&RS=PN/6,799,176 PageRank U.S. Patent - Method for scoring documents in a linked database] - September 28, 2004
* [http://patft.uspto.gov/netacgi/nph-Parser?Sect1=PTO2&Sect2=HITOFF&u=%2Fnetahtml%2FPTO%2Fsearch-adv.htm&r=1&p=1&f=G&l=50&d=PTXT&S1=7,058,628.PN.&OS=pn/7,058,628&RS=PN/7,058,628 PageRank U.S. Patent - Method for node ranking in a linked database] - June 6, 2006
* [http://patft.uspto.gov/netacgi/nph-Parser?Sect1=PTO2&Sect2=HITOFF&u=%2Fnetahtml%2FPTO%2Fsearch-adv.htm&r=1&p=1&f=G&l=50&d=PTXT&S1=7,269,587.PN.&OS=pn/7,269,587&RS=PN/7,269,587 PageRank U.S. Patent - Scoring documents in a linked database] - September 11, 2007
* [http://www.econ.upenn.edu/~clausen/research/rep-cost-attack.pdf Cost Attack of PageRank]