The Effect of the Number of Pages:
Since the accumulated PageRank of all pages of
the web equals the total number of web pages, it follows directly
that an additional web page increases the added up PageRank for
all pages of the web by one. But far more interesting than the effect
on the added up PageRank of the web is the impact of additional
pages on the PageRank of actual websites.
 To
illustrate the effects of addional web pages, we take a look at
a hierachically structured web site consisting of three pages A,
B and C, which are joined by an additional page D on the hierarchically
lower level of the site. The site has no outbound links. A link
from page X which has no other outbound links and a PageRank of
10 points to page A.
At a damping factor d of 0.75, the equations for
the single pages' PageRank values before adding page D are given
by:
- PR(A) = 0.25 + 0.75 (10 + PR(B) + PR(C))
- PR(B) = PR(C) = 0.25 + 0.75 (PR(A) / 2)
Solving the equations gives us the follwing PageRank values:
- PR(A) = 260/14
- PR(B) = 101/14
- PR(C) = 101/14
After adding page D, the equations for the pages'
PageRank values are given by
- PR(A) = 0.25 + 0.75 (10 + PR(B) + PR(C) + PR(D))
- PR(B) = PR(C) = PR(D) = 0.25 + 0.75 (PR(A) / 3)
Solving these equations gives us the follwing PageRank
values:
- PR(A) = 266/14
- PR(B) = 70/14
- PR(C) = 70/14
- PR(D) = 70/14
As to be expected since our example site has no
outbound links, after adding page D, the accumulated PageRank of
all pages increases by one from 33 to 34. Further, the PageRank
of page A rises marginally. In contrast, the PageRank of pages B
and C depletes substantially.
The Reduction of PageRank by Additional
Pages:
By adding pages to a hierarchically structured
websites, the consequences for the already existing pages are nonuniform.
The consequences for websites with a different structure shall be
shown by another example.
 We
take a look at a website constisting of three pages A, B and C which
are linked to each other in circle. The pages are then joined by
page D which fits into the circular linking structure. The regarded
site has no outbound links. A
gain, a link from page X which has no other outbound
links and a PageRank of 10 points to page A.
At a damping factor d of 0.75, the equations for
the single pages' PageRank values before adding page D are given
by:
- PR(A) = 0.25 + 0.75 (10 + PR(C))
- PR(B) = 0.25 + 0.75 × PR(A)
- PR(C) = 0.25 + 0.75 × PR(B)
Solving the equations gives us the follwing PageRank values:
- PR(A) = 517/37 = 13.97
- PR(B) = 397/37 = 10.73
- PR(C) = 307/37 = 8.30
After adding page D, the equations for the pages'
PageRank values are given by
- PR(A) = 0.25 + 0.75 (10 + PR(D))
- PR(B) = 0.25 + 0.75 × PR(A)
- PR(C) = 0.25 + 0.75 × PR(B)
- PR(D) = 0.25 + 0.75 × PR(C)
Solving these equations gives us the follwing PageRank
values:
- PR(A) = 419/35 = 11.97
- PR(B) = 323/35 = 9.23
- PR(C) = 251/35 = 7.17
- PR(D) = 197/35 = 5.63
Again, after adding page D, the accumulated PageRank
of all pages increases by one from 33 to 34. But now, any of the
pages which already existed before page D was added lose PageRank.
The more uniform PageRank is distributed by the links within a site,
the more likely will this effect occur.
Since adding pages to a site often reduces PageRank
for already existing pages, it becomes obvious that the PageRank
algorithm tends to privilege smaller web sites. Indeed, bigger web
sites can counterbalance this effect by being more attractive for
other webmasters to link to them, simply because they have more
content.
None the less, it is also possible to increase
the PageRank of existing pages by additional pages. Therefore, it
has to be considered that as few PageRank as possible is distributed
to these additional pages.
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