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- Published: 2009-09-17
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- Author: m0serious
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PageRank is a link analysis algorithm, named after Larry Page, used by the Google Internet search engine 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 referred to as the PageRank of E and denoted by
The name "PageRank" is a trademark of Google, and the PageRank process has been patented ( ). However, 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 1.8 million shares of Google in exchange for use of the patent; the shares were sold in 2005 for $336 million.
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. 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.
PageRank has been influenced by citation analysis, early developed by Eugene Garfield in the 1950s at the University of Pennsylvania, and by Hyper Search, developed by Massimo Marchiori at the University of Padua. In the same year PageRank was introduced (1998), Jon Kleinberg published his important work on HITS. Google's founders cite Garfield, Marchiori, and Kleinberg in their original paper. The technology in RankDex would be patented by 1999 and used later when Li founded Baidu in China. Li's work would be referenced by some of Larry Page's US patents for his Google search methods.
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.
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 formulas below) would assume a probability distribution between 0 and 1. Here a simple probability distribution will be used- 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.
: This is 0.75.
Suppose that page B has a link to page C as well as to page A, while 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
:
In other words, the PageRank conferred by an outbound link is equal to the document's own PageRank score divided by the normalized number of outbound links L( ) (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.
To be more specific, in the latter formula, 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. Basically, the two formulas do not differ fundamentally from each other. A PageRank which has been calculated by using the former formula has to be multiplied by the total number of web pages to get the according PageRank that would have been calculated by using the latter formula. Even Page and Brin mixed up the two formulas in their most popular paper "The Anatomy of a Large-Scale Hypertextual Web Search Engine", where they claim the latter formula to form a probability distribution over web pages with the sum of all pages' PageRanks being one.
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. 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 are the pages under consideration, is the set of pages that link to , is the number of outbound links on page , 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 :
\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 is 0 if page does not link to , and normalized such that, for each j
:,
i.e. the elements of each column sum up to 1, so the matrix is a stochastic matrix (for more details see the computation section below). Thus this is a variant of the eigenvector centrality measure used commonly in network analysis.
Because of the large eigengap of the modified adjacency matrix above, the values of the PageRank eigenvector are fast to approximate (only a few iterations are needed).
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 where 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.
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 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.
At each time step, the computation, as detailed above, yields :, or in matrix notation :, (*) where and is the column vector of length containing only ones.
The matrix is defined as : i.e., :, where denotes the adjacency matrix of the graph and is the diagonal matrix with the outdegrees in the diagonal.
The computation ends when for some small :, i.e., when convergence is assumed.
The solution exists and is unique for . This can be seen by noting that is by construction a stochastic matrix and hence has an eigenvalue equal to one because of the Perron-Frobenius theorem.
Note that in Eq. (***) the matrix on the right-hand side in the parenthesis can be interpreted as :, where is an initial probability distribution. In the current case :.
Finally, if has columns with only zero values, they should be replaced with the initial probability vector . In other words :, where the matrix is defined as :, with : In this case, the above two computations using only give the same PageRank if their results are normalized: : .
The Google Toolbar's PageRank is updated approximately 4 times a year, so often shows out of date values. It was last updated on 3 April 2010.
With the introduction of Google Places into the mainstream organic SERP, PageRank plays little to no role in ranking a business in the Local Business Results . While the theory of citations is still computed in their algorithm, PageRank is not a factor since Google ranks business listings and not web pages.
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 been used to rank spaces or streets to predict how many people (pedestrians or vehicles) come to the individual spaces or streets. . In lexical semantics it has been used to perform Word Sense Disambiguation and to automatically rank WordNet synsets according to how strongly they possess a given semantic property, such as positivity or negativity.
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 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.
The PageRank may also be used as a 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.
In any ecosystem, a modified version of PageRank may be used to determine species that are essential to the continuing health of the environment.
As an example, people could previously create many message-board posts with links to their website to artificially inflate their PageRank. With the nofollow value, message-board administrators 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 legitimate comments. (See: Spam in blogs#nofollow)
In an effort to manually control the flow of PageRank among pages within a website, many webmasters practice what is known as PageRank Sculpting - which is the act of strategically placing the nofollow attribute on certain internal links of a website in order to funnel PageRank towards those pages the webmaster deemed most important. This tactic has been used since the inception of the nofollow attribute, but the technique has been thought by many to have lost its effectiveness.
Category:Google Category:Search engine optimization Category:Reputation management Category:Internet search algorithms Category:American inventions Category:Crowdsourcing Category:Markov models Category:Link analysis
This text is licensed under the Creative Commons CC-BY-SA License. This text was originally published on Wikipedia and was developed by the Wikipedia community.