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Encyclopedia > PageRank  Mathematical PageRanks (out of 100) for a simple network (PageRanks reported by google are rescaled logarithmically). Page C has a higher PageRank than Page E, even though it has fewer links to it: the link it has is much higher valued. A web surfer who chooses a random link on every page (but with 15% likelihood jumps to a random page on the whole web) is going to be on Page E for 8.1% of the time. (The 15% likelihood of jumping to an arbitrary page corresponds to a damping factor of 85%.) Without damping, all web surfers would eventually end up on Pages A, B, or C, and all other pages would have PageRank zero. Page A is assumed to link to all pages in the web, because it has no outgoing links.

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). Image File history File links Size of this preview: 725 Ã— 599 pixelsFull resolutionâ€Ž (2,910 Ã— 2,406 pixels, file size: 499 KB, MIME type: image/jpeg) I, the copyright holder of this work, hereby release it into the public domain. ... Image File history File links Size of this preview: 725 Ã— 599 pixelsFull resolutionâ€Ž (2,910 Ã— 2,406 pixels, file size: 499 KB, MIME type: image/jpeg) I, the copyright holder of this work, hereby release it into the public domain. ... Network analysis is the analysis of networks through network theory (or more generally graph theory). ... // A hyperlink, is a reference or navigation element in a document to another section of the same document or to another document that may be on a (different) website. ... WWWs historical logo designed by Robert Cailliau The World Wide Web (commonly shortened to the Web) is a system of interlinked, hypertext documents accessed via the Internet. ... In mathematics, computing, linguistics, and related disciplines, an algorithm is a finite list of well-defined instructions for accomplishing some task that, given an initial state, will terminate in a defined end-state. ... Look up reciprocal in Wiktionary, the free dictionary. ...

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. â€œ(TM)â€ redirects here. ... For other uses, see Patent (disambiguation). ...

 “ 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". ”

Google assigns a numeric weighting from 0-10 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. A logarithmic scale is a scale of measurement that uses the logarithm of a physical quantity instead of the quantity itself. ... The Richter magnitude test scale (or more correctly local magnitude ML scale) assigns a single number to quantify the size of an earthquake. ...

Numerous academic papers concerning PageRank have been published since Page and Brin's original paper. 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. The HITS (hypertext induced topic selection) algorithm is an algorithm for rating, and therefore also ranking, Web pages. ... Jon Kleinberg. ... The CLEVER project was a research project in Web search led by Jon Kleinberg at IBMs Almaden Research Center. ... TrustRank is a link analysis technique for semi-automatically separating useful webpages from spam. ...

## 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. In probability theory, every random variable may be attributed to a function defined on a state space equipped with a probability distribution that assigns a probability to every subset (more precisely every measurable subset) of its state space in such a way that the probability axioms are satisfied. ...

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. Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ...

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. $PR(A)= PR(B) + PR(C) + PR(D).,$

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

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 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.

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

or (N = the number of documents in collection) $PR(A)= {1 - d over N} + d left( frac{PR(B)}{L(B)}+ frac{PR(C)}{L(C)}+ frac{PR(D)}{L(D)}+,cdots right) .$

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. In mathematics, a Markov chain, named after Andrey Markov, is a discrete-time stochastic process with the Markov property. ...

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. // Uniform Resource Locator (URL) formerly known as Universal Resource Locator, is a technical, Web-related term used in two distinct meanings: In popular usage and many technical documents, it is a synonym for Uniform Resource Identifier (URI); Strictly, the idea of a uniform syntax for global identifiers of network-retrievable...

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 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 In linear algebra, the eigenvectors (from the German eigen meaning own) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ... In mathematics and computer science, the adjacency matrix for a finite graph on n vertices is an n × n matrix in which entry aij is the number of edges from vi to vj in . ... $mathbf{R} = begin{bmatrix} PR(p_1) PR(p_2) vdots PR(p_N) end{bmatrix}$

where R is the solution of the equation $mathbf{R} = begin{bmatrix} {(1-d)/ N} {(1-d) / N} vdots {(1-d) / N} end{bmatrix} + d begin{bmatrix} ell(p_1,p_1) & ell(p_1,p_2) & cdots & ell(p_1,p_N) ell(p_2,p_1) & ddots & & vdots vdots & & ell(p_i,p_j) & ell(p_N,p_1) & cdots & & ell(p_N,p_N) end{bmatrix} mathbf{R}$

where the adjacency function $ell(p_i,p_j)$ is 0 if page pj does not link to pi, 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. Eigenvector centrality is a measure of the importance of a node in a network. ... Network analysis is the analysis of networks through network theory (or more generally graph theory). ...

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. It has been suggested that this article or section be merged with Markov property. ... In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are...

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. Wikipedia (IPA: , or ( ) is a multilingual, web-based, free content encyclopedia project, operated by the Wikimedia Foundation, a non-profit organization. ... The Open Directory Project (ODP), also known as dmoz (from , its original domain name), is a multilingual open content directory of World Wide Web links owned by Netscape that is constructed and maintained by a community of volunteer editors. ...

Several strategies have been proposed to accelerate the computation of PageRank.

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.

## PageRank variations

An example of the PageRank indicator as found on the 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. Image File history File links No higher resolution available. ... Google Toolbar is an Internet browser toolbar available for Internet Explorer and Mozilla Firefox (with slightly different features). ... Matt Cutts (left, with Chris Hooley) Matt Cutts works for the quality group in Google, specializing in search engine optimization issues. ...

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. This page is a summary of services and tools provided by Google Inc. ...

### Other uses of PageRank

A version of PageRank has recently been proposed as a replacement for the traditional ISI impact factor, 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. The Impact factor, often abbreviated IF, is a measure of the citations to science and social science journals. ...

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).

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. WordNet is a semantic lexicon for the English language. ... In metadata a Synonym ring or synset, is a group of data elements that are considered semantically equivalent for the purposes of information retrieval. ...

A dynamic weighting method similar to PageRank has been used to generate customized reading lists based on the link structure of Wikipedia.

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 which were used in the creation of Google is Efficient crawling through URL ordering, 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. For the search engine of the same name, see WebCrawler. ...

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. (See: Spam in blogs#rel="nofollow") Spam in blogs (also called simply blog spam or comment spam) is a form of spamdexing. ...

The Hilltop algorithm is a patented algorithm created by Krishna Bharath and George A. Mihaila of the University of Toronto. ... Google has often adopted a light-hearted approach in a variety of circumstances. ... TrustRank is a link analysis technique for semi-automatically separating useful webpages from spam. ... The power method is a iterative approximative method for calculating the eigenvectors of a matrix. ... PR0 indicates a PageRank of zero (out of a maximum of ten) on the Google search engine. ... Example of a Google bomb. ... This article or section does not cite its references or sources. ... A typical search results page Search engine optimization (SEO) is the process of improving the volume and quality of traffic to a web site from search engines via natural (organic or algorithmic) search results. ... Results from FactBites:

 Display Page Rank on your your website - Google PageRank Checker (397 words) Google PageRank started to update on some datacenter since 28 september and since 2 october is visible on goole toolbars. PageRank is a family of algorithms for assigning numerical weightings to hyperlinked documents (or web pages) indexed by a search engine. PageRank is a numeric value that represents how important a page is on the web.
 Pagerank Explained. Google's PageRank and how to make the most of it. (6205 words) It's the original one that was published when PageRank was being developed, and it is probable that Google uses a variation of it but they aren't telling us what it is. It doesn't matter though, as this equation is good enough. For a page's calculation, its existing PageRank (if it has any) is abandoned completely and a fresh calculation is done where the page relies solely on the PageRank "voted" for it by its current inbound links, which may have changed since the last time the page's PageRank was calculated. Page B now has a new PageRank value, but it can't be accurate because the calculation used the new PageRank value of the inbound link from page A, which is inaccurate.
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