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So any page’s PageRank is derived in large part from the PageRanks of other pages. The damping fac-

tor adjusts the derived value downward. The second formula 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

R =

(1-

d

) /

N

(1-

d

) /

N

(1-

d

) /

N

.

.

. +

d

l

(

1

,

1

)

l

(

2

,

1

)

l

(

N

,

1

)

.

.

.

l

(

1

,

2

)

l

(

1

,

N

)

l

(

N

,

N

)

l

(

i

,

j

)

. . .

. . .

.

.

.

R

R =

PR

(

1

)

PR

(

2

)

PR

(

N

)

.

.

.

PR

( )

=

1 -

d

N

+

d

j M

( )

PR

(

j

)

L

(

j

)

196

SEO Strategies

Part II

13 1 8:57 196

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