CS315 – Link Analysis
Three generations of Search Engines
Anchor text
Link analysis for ranking


Pagerank
HITS
1st Generation: Content Similarity
Content Similarity Ranking:
The more rare words two documents share,
the more similar they are
Documents are treated as “bags of words”
(no effort to “understand” the contents)
Similarity is measured by vector angles
t3
Query Results are ranked
by sorting the angles
between query and documents
d
2
θ
d1
t1
t2
But we also have links (los links!)
Page B
Page A
Anchor text
hyperlink
Assumption 1:
A hyperlink from a page denotes vote of confidence
to second page (quality signal)
Assumption 2:
The anchor text of the hyperlink
describes the target page (textual context)
2nd Generation: Add Popularity
A hyperlink
from a page in site A
to some page in site B
is considered a popularity vote from
site A to site B
Score of a page =
number of in-links
www.aa.com
1
www.bb.com
2
www.cc.com
1
Query Processing


First retrieve all pages meeting
the text query (say venture
capital).
Order these by the link popularity
(of the page or the site)
www.dd.com
2
www.zz.com
0
3rd Generation: Add Reputation
Each page starts with some basic “reputation” (e.g., =1)
and repeatedly distributes equal fractions to its links
(while receiving from them)
until some “equilibrium”
The reputation “PageRank” of a page P =
the sum
of a fair fraction of the reputations PR(W )  PR(W1 )  PR(W 2 ) ...PR(W n )
O(W1)
O(W 2 )
O(W n )
of all pages Pj that point to P
Beautiful Math behind it



PR = principal eigenvector
of the web’s link matrix
PR equivalent to the chance
of randomly surfing to the page
Idea similar to academic co-citations
Roots of PR: Citation Analysis
Citation frequency

The kind of background work Deans are doing at tenure time
Co-citation coupling frequency


Co-citations with a given author measures “impact”
Are you co-cited with influential publications?
Bibliographic coupling frequency

Articles that co-cite the same articles are related
Citation indexing

Who is author cited by?
PageRank PR
– Complete Definition
W is a web page
Wi are the web pages that have a link to W
O(Wi) is the number of out-links from Wi
t is the teleportation probability (e.g. 0.15)
N is the size of the web (that we have seen)
t
PR(W1 ) PR(W 2 )
PR(W n )
PR(W )   (1 t)(

...
)
N
O(W1 )
O(W 2 )
O(W n )
W1
W1
W2
W2
W3
W3
W.
PageRank: Iterative Computation
t
PR(W1 ) PR(W 2 )
PR(W n )
PR(W )   (1 t)(

...
)
N
O(W1 )
O(W 2 )
O(W n )
t is normally set to 0.15,
but for this example, for simplicity let’s set it to 0.5
Set initial PR values to 1
Solve the following equations iteratively:
PR(A)  0.5 /3  0.5PR(C)
PR(B)  0.5 /3  0.5(PR(A) /2)
PR(C)  0.5 /3  0.5(PR(A) /2  PR(B))
Example
Computation
of PR
in Excel
Pagerank – Matrix Multiplication Equivalent Def.
Imagine a browser doing a random walk on web pages:



Start at a random page P
At each step,
walk with equal probability out of the current page
along one of the links on that page,
1/3
Continue doing this random
1/3
P
walk for a long time
1/3
“In the steady state”
each page has a long-term visit rate:
Use this rate as the page’s score.
Not quite enough
The web is full of dead-ends.


Random walk can get stuck in dead-ends.
Makes no sense to talk about long-term visit rates.
??
Teleporting
At a dead end,
jump to a random web page.
At any non-dead end,



With probability, say, 15%,
jump to a random web page.
With remaining probability (85%),
go out on a random link.
t=0.15 is the “teleporting” parameter.
Result of teleporting
Now cannot get stuck locally.
There exists a computable long-term rate
at which any page is visited

This not obvious, but it has been proven!
How do we compute this visit rate?
Markov chains: abstractions of random walks
A Markov chain consists of n states,
and an nn transition probability matrix P.
At each step, we are in exactly one of the states.
For 1  i,j  n,
the matrix entry Pij
tells us the probability of j being the next state,
given we are currently in state i.
Clearly, for all i, n
 Pij  1.
j 1
i
Pij
j
Computing PR with Markov chains
Example (next two slides):
Represent the teleporting random walk
with teleporting parameter t=15%
as a Markov chain, for this graph:
A
B
C
D
Computing P with Matrix Multiplication
Start with Adjacency matrix A of the Web Graph

If there is hyperlink from i to j, Aij = 1, else Aij = 0
If

a row has all 0’s,
 replace each element by 1/N
Else



divide each 1 by the number of 1’s in the row
Multiply the matrix by 1-t
Add t/N to every entry of the resulting matrix
A
B
C
D
P=
Computing all Pageranks
P=
Theorem:
Regardless of where we start, we
eventually reach the steady state a.
Start with any distribution
(say x=(1 0 … 0)).




After one step, we’re at xP;
after two steps at xP2 ,
then xP3 and so on.
“Eventually” means for “large” k, xPk = a.
Algorithm: multiply x by increasing
powers of P
until the product looks stable.
A
B
C
D
Pagerank summary
Preprocessing:



Given graph of links, build matrix P.
From it compute a.
The entry ai is a number between 0 and 1: the pagerank of page i.
Query processing:
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

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Retrieve pages meeting query.
Rank them by their pagerank.
Order is query-independent
If PR(A) > PR(B) for some query, it beats it in every query
How is Pagerank used?
http://www.google.com/corporate/tech.html
PageRank Technology:
PageRank reflects our view of the importance of web pages by
considering more than 500 million variables and 2 billion terms. Pages
that we believe are important pages receive a higher PageRank and
are more likely to appear at the top of the search results.
This claim has recently changed:
“Today we use more than 200 signals, including PageRank, to order
websites, and we update these algorithms on a weekly basis”
Pagerank is dead, long live Pagerank!