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The damping factor

PageRank theory assumes, an imaginary surfer who is randomly clicking hyperlinks will at some stage desist from clicking. The probability, at any step, the person will continue is the damping factor d. Various studies have tested different damping factors, but it's generally assumed that damping factor will be set around 0.85.

PageRank is based on the random surfer model. Essentially, the damping factor is a decay factor. What it represents is the chance that a user will stopping clicking links and get bored with the current page and then request another random page (as with directly typing in a new URL rather than following a link on the current page). This was originally set to about 85% or 0.85. If the damping factor is 85% then there is assumed to be about a 15% chance that a typical users won't follow any links on the page and instead navigate to a new random URL.

Because of the size of the actual web, the Google search engine uses an approximate, iterative computation of Page Rank values. This means that each page is assigned an initial starting value and the Page Ranks of all pages are then calculated in several computation circles based on the equations determined by the Page ranking algorithm. So if a page has 1000 PR "points" and 10 outbound links and the damping factor is 85%, those links do NOT each get passed 100 PR points... 15% of the overall PR basically vaporizes and the remaining points, 850 PR points in this example, would get passed out over the 10 outbound links (assuming all are followed). So each link would get passed 85 PR points.


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The damping factor is subtracted from 1 (and in some variations of the algorithm, the result is divided by the number of documents (N) 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)/N + d(PR(B)/L(B) + PR(C)/L(C) + PR(D)/L(D) + ...)

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 original paper, however, gave the following formula, which has led to some confusion:

PR(A) = 1 - d + d(PR(B)/L(B) + PR(C)/L(C) + PR(D)/L(D) + ...)

The difference between them is that the PageRank values in the first formula sum to one, while in the second formula each PageRank is multiplied by N and the sum becomes N. A statement in Page and Brin's paper that "the sum of all PageRanks is one" and claims by other Google employees support the first variant of the formula above. Page and Brin confused the two formulas in their most popular paper "The Anatomy of a Large-Scale Hypertextual Web Search Engine", where they mistakenly claimed that the latter formula formed a probability distribution over web pages.

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, which are all equally probable, 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.

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