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,
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". 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
is 0 if page pi does not link to pj, and normalised such that, for each i
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.
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 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 weigh 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 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.
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". 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
is 0 if page pi does not link to pj, and normalised such that, for each ii.e. the elements of each column sum up to 1. 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 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 weigh 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 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.
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