1. Link Analysis
Rajendra Akerkar
Vestlandsforsking, Norway

2. Objectives
 To review common approaches to l k analysis
h link l
 To calculate the popularity of a site based on link
analysis
 To model human judgments indirectly
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3. Outline
1.
1 Early Approaches to Link Analysis
2. Hubs and Authorities: HITS
3. Page Rank
4. Stability
5. Probabilistic Link Analysis
6. Limitation of Link Analysis
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4. Early Approaches
Basic Assumptions
• Hyperlinks contain information about the human
judgment of a site
• The more incoming links to a site the more it is
site,
judged important
Bray 1996
y
• The visibility of a site is measured by the number
of other sites pointing to it
• The luminosity of a site is measured by the number
of other sites to which it points
 Limitation: failure to capture the relative
importance of different parents (children) sites
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5. Early Approaches
Mark 1988
• To calculate the score S of a document at vertex v
1
S(v) = s(v) + Σ S(w)
| ch[v] | w Є |ch(v)|
v: a vertex i the h
in h hypertext graph G = (V E)
h (V,
S(v): the global score
s(v): the score if the document is isolated
ch(v): children of the document at vertex v
• Limitation:
- Require G to be a directed acyclic g p (
q y graph (DAG)
)
- If v has a single link to w, S(v) > S(w)
- If v has a long path to w and s(v) < s(w),
then S(v) > S (w)
 unreasonable
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6. Early Approaches
Marchiori (1997)
• Hyper information should complement textual
information to obtain the overall information
S(v) = s(v) + h(v)
( ) ( ) ( )
- S(v): overall information
- s(v): textual information
- h(v): hyper information
r(v, w)
• h(v) = Σ F S(w)
w Є |ch[v]|
- F a fading constant F Є (0 1)
F: constant, (0,
- r(v, w): the rank of w after sorting the children
of v by S(w)
 a remedy of the previous approach (Mark 1988)
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7. HITS - Kleinberg’s Algorithm
• HITS – Hypertext Induced Topic Selection
yp p
• For each vertex v Є V in a subgraph of interest:
a(v) - the authority of v
h(v) - the hubness of v
• A site is very a thoritati e if it recei es man
er authoritative receives many
citations. Citation from important sites weight
more than citations from less-important sites
• Hubness shows the importance of a site. A good
hub is a site that links to many authoritative sites
y
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8. Authority and Hubness
2 5
3 1 1 6
4
7
a(1) = h(2) + h(3) + h(4) h(1) = a(5) + a(6) + a(7)
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9. Authority d H b
A th it and Hubness Convergence
C g
• R
Recursive d
i dependency:
d
a(v)  Σ h(w)
w Є pa[v]
h(v)  Σ w Є ch[v] a(w)
• Using Linear Algebra, we can prove:
a(v) and h(v) converge
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10. HITS Example
Find a base subgraph:
• Start with a root set R {1, 2, 3, 4}
• {1, 2, 3, 4} - nodes relevant to
the topic
• Expand the root set R to include
all the children and a fixed
number of parents of nodes in R
 A new set S (base subgraph) 
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11. HITS Example
BaseSubgraph( R d)
R,
1. S  r
2. for each v in R
3. do S  S U ch[v]
4. P  pa[v]
5. if |P| > d
f
6. then P  arbitrary subset of P having size d
7. SSUP
8.
8 return S
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12. HITS Example
Hubs and authorities: two n-dimensional a and h
HubsAuthorities(G)
|V|
1 1  [1,…,1] Є R
2 a0  h 0  1
3 t 1
4 repeat
5 for each v in V
6 do at (v)  Σ h (w)
w Є pa[v] t -1
7 ht (v)  Σ w Є pa[v] a (w)
8 a t  at / || at || t -1
9 h t  ht / || ht ||
10 t  t+1
11 until || a t – at -1 || + || h t – ht -1 || < ε
1 1
12 return (a t , h t )
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13. HITS Example Results
y
Authority
Hubness
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Authority and hubness weights
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14. HITS Improvements
Brarat and Henzinger (1998)
• HITS problems
1) The document can contain many identical links to
the same document in another host
2) Links are generated automatically (e.g. messages
posted on newsgroups)
• Solutions
1) Assign weight to identical multiple edges, which
are inversely proportional to their multiplicity
2) Prune irrelevant nodes or regulating the influence
of a node with a relevance weight
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15. PageRank
 Introduced by Page et al (
y g (1998))
 The weight is assigned by the rank of parents
 Difference with HITS
 HITS takes Hubness & Authority weights
 The page rank is proportional to its parents’ rank, but inversely
proportional to its parents’ outdegree
15

16. Matrix Notation
Adjacent M i
Adj Matrix
A=
* http://www.kusatro.kyoto-u.com
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17. Matrix Notation
 Matrix Notation
r=αBr=Mr
α : eigenvalue
r : eigenvector of B
Ax=λx
B=
| A - λI | x = 0
Finding Pagerank
 to find eigenvector of B with an associated eigenvalue α
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18. Matrix Notation
PageRank: eigenvector of P relative to max eigenvalue
B = P D P-1
D: diagonal matrix of eigenvalues {λ1, … λn}
P: regular matrix that consists of eigenvectors
PageRank r1 =
normalized
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19. Matrix Notation
• Confirm the result
# of inlinks from high ranked page
hard to explain about 5&2 6&7
5&2,
• Interesting Topic
How do you create your
homepage highly ranked?
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20. Markov Chain Notation
 Random surfer model
 Description of a random walk through the Web graph
 Interpreted as a transition matrix with asymptotic probability that a
surfer is currently browsing that page
rt = M rtt-1
1
M: transition matrix for a first-order Markov chain (stochastic)
Does it converge to some sensible solution ( too)
g (as )
regardless of the initial ranks ?
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21. Problem
 “Rank Sink” Problem
 In general, many Web pages have no inlinks/outlinks
 It results in dangling edges in the graph
E.g.
no parent  rank 0
MT converges to a matrix
whose last column is all zero
no children  no solution
MT converges to zero matrix
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22. Modification
 Surfer will restart browsing by p
g y picking a new Web p g at
g page
random
M=(B+E)
E : escape matrix
M : stochastic matrix
 S ll problem?
Still bl
 It is not guaranteed that M is primitive
 If M is stochastic and primitive, PageRank converges to
corresponding stationary distribution of M
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23. PageRank Algorithm
* Page et al 1998
al,
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24. Distribution of the Mixture Model
 The probability distribution that results from combining the
p y g
Markovian random walk distribution & the static rank source
distribution
r = εe + (1- ε)x
1
ε: probability of selecting non-linked page
PageRank
Now, transition matrix [εH + (1- ε)M] is primitive and stochastic
rt converges to the dominant eigenvector
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25. Stability
 Whether the link analysis algorithms based on eigenvectors
are stable in the sense that results don’t change significantly?
 Th connectivity of a portion of the graph is changed
The f f h h h d
arbitrary
 How will it affect the results of algorithms?
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26. Stability of HITS
Ng et al (2001)
• A bound on the number of hyperlinks k that can added or deleted
from one page without affecting the authority or hubness weights
• It is possible to perturb a symmetric matrix by a quantity that grows
as δ that produces a constant perturbation of the dominant
eigenvector
δ: eigengap λ1 – λ2
g g p
d: maximum outdegree of G
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27. Stability of PageRank
Ng et al (2001)
V: the set of vertices touched by the perturbation
 The parameter ε of the mixture model has a stabilization role
 If the set of pages affected by the p
pg y perturbation have a small rank, the overall
,
change will also be small
tighter bound by
Bianchini et al (2001)
δ(j) >= 2 depends on the edges incident on j
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28. SALSA
 SALSA (Lempel, Moran 2001)
 Probabilistic extension of the HITS algorithm
 Random walk is carried out by following hyperlinks both in the
forward and in the backward direction
 Two separate random walks
 Hub walk
 Authority walk
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29. Forming a Bipartite Graph in SALSA
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30. Random Walks
 Hub walk
 Follow a Web link from a page uh to a page wa (a forward link)
and then
 Immediately traverse a backlink going from wa to vh, where (u w)
(u,w)
Є E and (v,w) Є E
 Authority Walk
y
 Follow a Web link from a page w(a) to a page u(h) (a backward
link) and then
 Immediately traverse a f
d l forward link going b k f
dl k back from vh to wa
where (u,w) Є E and (v,w) Є E
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31. Computing Weights
 Hub weight computed from the sum of the product of the
inverse degree of the in-links and the out-links
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32. Why We Care
 Lempel and Moran (2001) showed theoretically that SALSA
weights are more robust that HITS weights in the presence of the
Tightly Knit Community (TKC) Effect.
 This effect occurs when a small collection of pages (related to a given topic)
is connected so that every hub links to every authority and includes as a special
case the mutual reinforcement effect
 The pages in a community connected in this way can be ranked
highly by HITS, higher than pages in a much larger collection
HITS
where only some hubs link to some authorities
 TKC could be exploited by spammers hoping to increase their
page weight ( li k f
i h (e.g. link farms)
)
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33. A Similar Approach
 Rafiei and Mendelzon (2000) and Ng et al. (2001)
propose similar approaches using reset as in
PageRank
 Unlike PageRank, in this model the surfer will
follow a forward link on odd steps but a
backward li k
b k d link on even steps t
 The stability properties of these ranking
distributions are similar to those of PageRank (Ng
et al. 2001)
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34. Overcoming TKC
 Similarity downweight sequencing and sequential
y g g
clustering (Roberts and Rosenthal 2003)
 Consider the underlying structure of clusters
y g
 Suggest downweight sequencing to avoid the
Tight Knit Community problem
g yp
 Results indicate approach is effective for few
tested queries, but still untested on a large scale
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35. PHITS and More
 PHITS: Cohn and Chang (2000)
 Only the principal eigenvector is extracted using SALSA, so the
authority along the remaining eigenvectors is completely
neglected
 Account for more eigenvectors of the co-citation matrix
 See also Lempel, Moran (2003)
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36. Limits of Link Analysis
 META tags/ invisible text
 Search engines relying on meta tags in documents are often
misled (intentionally) by web developers
P f
Pay-for-place
l
 Search engine bias : organizations pay search engines and page
rank
 Advertisements: organizations pay high ranking pages for
advertising space
 W h a primary effect of increased visibility to end users and a secondary
With ff f d bl d d d
effect of increased respectability due to relevance to high ranking page
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37. Limits of Link Analysis
 Stability
 Adding even a small number of nodes/edges to the graph has a
significant impact
 T i drift – similar t TKC
Topic d ift i il to
 A top authority may be a hub of pages on a different topic
resulting in increased rank of the authority p g
g y page
 Content evolution
 Adding/removing links/content can affect the intuitive
authority rank of a page requiring recalculation of page ranks
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38. Further Reading
 R. Akerkar d P. Lingras, Building an I ll
R Ak k and P L B ld Intelligent W b Th
Web: Theory
and Practice, Jones & Bartlett, 2008
 http://www.jbpub.com/catalog/9780763741372/
p j p g
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