Introduction
 Early search engines mainly compare content
similarity of the query and the indexed pages. I.e.,
 They use information retrieval methods, cosine, TF-IDF, ...
 From 1996, it became clear that content similarity
alone was no longer sufficient.
 The number of pages grew rapidly in the mid-late 1990’s.
 Try “classification technique”, Google estimates: 10
million relevant pages.
 How to choose only 30-40 pages and rank them suitably
to present to the user?
 Content similarity is easily spammed.
 A page owner can repeat some words and add many
related words to boost the rankings of his pages and/or
to make the pages relevant to a large number of
queries.

Introduction (cont …)
 Starting around 1996, researchers began to work on
the problem. They resort to hyperlinks.
 In Feb, 1997, Yanhong Li (Robin Li), Scotch Plains, NJ, filed
a hyperlink based search patent. The method uses words in
anchor text of hyperlinks.
 Web pages on the other hand are connected through
hyperlinks, which carry important information.
 Some hyperlinks: organize information at the same site.
 Other hyperlinks: point to pages from other Web sites. Such
out-going hyperlinks often indicate an implicit conveyance of
authority to the pages being pointed to.
 Those pages that are pointed to by many other pages
are likely to contain authoritative information.

 During 1997-1998, two most influential hyperlink
based search algorithms PageRank and HITS were
reported.
 Both algorithms are related to social networks.
They exploit the hyperlinks of the Web to rank pages
according to their levels of “prestige” or “authority”.
 HITS: Jon Kleinberg (Cornel University), at Ninth Annual
ACM-SIAM Symposium on Discrete Algorithms, January
1998
 PageRank: Sergey Brin and Larry Page, PhD students
from Stanford University, at Seventh International World
Wide Web Conference (WWW7) in April, 1998.
 PageRank powers the Google search engine.

 Apart from search ranking, hyperlinks are also useful
for finding Web communities.
 A Web community is a cluster of densely linked pages
representing a group of people with a special interest.
 Beyond explicit hyperlinks on the Web, links in other
contexts are useful too, e.g.,
 for discovering communities of named entities (e.g., people
and organizations) in free text documents, and
 for analyzing social phenomena in emails..

Road map
 Introduction
 PageRank
 HITS
 Summary

Social network analysis
 Social network is the study of social entities (people
in an organization, called actors), and their
interactions and relationships.
 The interactions and relationships can be
represented with a network or graph,
 each vertex (or node) represents an actor and
 each link represents a relationship.
 From the network, we can study the properties of its
structure, and the role, position and prestige of each
social actor.
 We can also find various kinds of sub-graphs, e.g.,
communities formed by groups of actors.

Social network and the Web
 Social network analysis is useful for the Web
because the Web is essentially a virtual society, and
thus a virtual social network,
 Each page: a social actor and
 each hyperlink: a relationship.
 Many results from social network can be adapted
and extended for use in the Web context.
 We study two types of social network analysis,
centrality and prestige, which are closely related to
hyperlink analysis and search on the Web.

Centrality
 Important or prominent actors are those that
are linked or involved with other actors
extensively.
 A person with extensive contacts (links) or
communications with many other people in
the organization is considered more important
than a person with relatively fewer contacts.
 The links can also be called ties. A central
actor is one involved in many ties.

Degree Centrality

Closeness Centrality

Betweenness Centrality
 If two non-adjacent actors j and k want to
interact and actor i is on the path between j
and k, then i may have some control over the
interactions between j and k.
 Betweenness measures this control of i over
other pairs of actors. Thus,
 if i is on the paths of many such interactions, then
i is an important actor.

Betweenness Centrality (cont …)
 Undirected graph: Let pjk be the number of
shortest paths between actor j and actor k.
 The betweenness of an actor i is defined as the
number of shortest paths that pass i (pjk(i))
normalized by the total number of shortest paths.
∑<kj jk
jk
p
ip )(
(4)

Betweenness Centrality (cont …)

Prestige
 Prestige is a more refined measure of prominence of
an actor than centrality.
 Distinguish: ties sent (out-links) and ties received (in-links).
 A prestigious actor is one who is object of extensive
ties as a recipient.
 To compute the prestige: we use only in-links.
 Difference between centrality and prestige:
 centrality focuses on out-links
 prestige focuses on in-links.
 We study three prestige measures. Rank prestige
forms the basis of most Web page link analysis
algorithms, including PageRank and HITS.

Degree prestige

Proximity prestige
 The degree index of prestige of an actor i only
considers the actors that are adjacent to i.
 The proximity prestige generalizes it by considering
both the actors directly and indirectly linked to actor
i.
 We consider every actor j that can reach i.
 Let Ii be the set of actors that can reach actor i.
 The proximity is defined as closeness or distance
of other actors to i.
 Let d(j, i) denote the distance from actor j to actor i.

Proximity prestige (cont …)

Rank prestige
 In the previous two prestige measures, an important
factor is considered,
 the prominence of individual actors who do the “voting”
 In the real world, a person i chosen by an important
person is more prestigious than chosen by a less
important person.
 For example, if a company CEO votes for a person is much
more important than a worker votes for the person.
 If one’s circle of influence is full of prestigious
actors, then one’s own prestige is also high.
 Thus one’s prestige is affected by the ranks or statuses of
the involved actors.

Rank prestige (cont …)
 Based on this intuition, the rank prestige PR(i) is
define as a linear combination of links that point to i:

Road map
 Introduction
 PageRank
 HITS
 Summary

Co-citation and Bibliographic Coupling
 Another area of research concerned with links is
citation analysis of scholarly publications.
 A scholarly publication cites related prior work to
acknowledge the origins of some ideas and to compare the
new proposal with existing work.
 When a paper cites another paper, a relationship is
established between the publications.
 Citation analysis uses these relationships (links) to perform
various types of analysis.
 We discuss two types of citation analysis, co-
citation and bibliographic coupling. The HITS
algorithm is related to these two types of analysis.

Co-citation
 If papers i and j are both cited by paper k, then they
may be related in some sense to one another.
 The more papers they are cited by, the stronger their
relationship is.

Co-citation
 Let L be the citation matrix. Each cell of the matrix is
defined as follows:
 Lij = 1 if paper i cites paper j, and 0 otherwise.
 Co-citation (denoted by Cij) is a similarity measure
defined as the number of papers that co-cite i and j,
 Cii is naturally the number of papers that cite i.
 A square matrix C can be formed with Cij, and it is
called the co-citation matrix.
,
1
∑=
=
n
k
kjkiij LLC

Bibliographic coupling
 Bibliographic coupling operates on a similar
principle.
 Bibliographic coupling links papers that cite the
same articles
 if papers i and j both cite paper k, they may be related.
 The more papers they both cite, the stronger their
similarity is.

Bibliographic coupling (cont …)

Road map
 Introduction
 PageRank
 HITS
 Summary

PageRank
 The year 1998 was an eventful year for Web
link analysis models. Both the PageRank and
HITS algorithms were reported in that year.
 The connections between PageRank and
HITS are quite striking.
 Since that eventful year, PageRank has
emerged as the dominant link analysis
model,
 due to its query-independence,
 its ability to combat spamming, and
 Google’s huge business success.

PageRank: the intuitive idea
 PageRank relies on the democratic nature of the
Web by using its vast link structure as an indicator of
an individual page's value or quality.
 PageRank interprets a hyperlink from page x to
page y as a vote, by page x, for page y.
 However, PageRank looks at more than the sheer
number of votes; it also analyzes the page that
casts the vote.
 Votes casted by “important” pages weigh more heavily and
help to make other pages more "important."
 This is exactly the idea of rank prestige in social
network.

More specifically
 A hyperlink from a page to another page is an
implicit conveyance of authority to the target page.
 The more in-links that a page i receives, the more
prestige the page i has.
 Pages that point to page i also have their own
prestige scores.
 A page of a higher prestige pointing to i is more important
than a page of a lower prestige pointing to i.
 In other words, a page is important if it is pointed to by
other important pages.

PageRank algorithm
 According to rank prestige, the importance of
page i (i’s PageRank score) is the sum of the
PageRank scores of all pages that point to i.
 Since a page may point to many other pages, its
prestige score should be shared.
 The Web as a directed graph G = (V, E). Let the
total number of pages be n. The PageRank score
of the page i (denoted by P(i)) is defined by:
,
)(
)(
),(
∑∈
=
Eij jO
jP
iP Oj is the number
of out-link of j

Matrix notation
 We have a system of n linear equations with n
unknowns. We can use a matrix to represent them.
 Let P be a n-dimensional column vector of PageRank
values, i.e., P = (P(1), P(2), …, P(n))T
.
 Let A be the adjacency matrix of our graph with
 We can write the n equations with (PageRank)




∈
=
otherwise
Ejiif
OA iij
0
),(
1
PAP T
=
(14)
(15)

Solve the PageRank equation
 This is the characteristic equation of the
eigensystem, where the solution to P is an
eigenvector with the corresponding eigenvalue of 1.
 It turns out that if some conditions are satisfied, 1 is
the largest eigenvalue and the PageRank vector P is
the principal eigenvector.
 A well known mathematical technique called power
iteration can be used to find P.
 Problem: the above Equation does not quite suffice
because the Web graph does not meet the
conditions.
PAP T
= (15)

Using Markov chain
 To introduce these conditions and the
enhanced equation, let us derive the same
Equation (15) based on the Markov chain.
 In the Markov chain, each Web page or node in
the Web graph is regarded as a state.
 A hyperlink is a transition, which leads from one
state to another state with a probability.
 This framework models Web surfing as a
stochastic process.
 It models a Web surfer randomly surfing the
Web as state transition.

Random surfing
 Recall we use Oi to denote the number of
out-links of a node i.
 Each transition probability is 1/Oi if we
assume the Web surfer will click the
hyperlinks in the page i uniformly at random.
 The “back” button on the browser is not used and
 the surfer does not type in an URL.

Transition probability matrix
 Let A be the state transition probability matrix,,
 Aij represents the transition probability that the surfer
in state i (page i) will move to state j (page j). Aij is
defined exactly as in Equation (14).




















=
nnnn
n
n
AAA
AAA
AAA
...
...
...
...
...
...
.
21
22221
11211
A

Let us start
 Given an initial probability distribution vector that a
surfer is at each state (or page)
 p0 = (p0(1), p0(2), …, p0(n))T
(a column vector) and
 an n×n transition probability matrix A,
we have
 If the matrix A satisfies Equation (17), we say that
A is the stochastic matrix of a Markov chain.
∑=
=
n
i
ip
1
0 1)(
∑=
=
n
j
ijA
1
1
(16)
(17)

Back to the Markov chain
 In a Markov chain, a question of common
interest is:
 Given p0 at the beginning, what is the probability
that m steps/transitions later the Markov chain will
be at each state j?
 We determine the probability that the system
(or the random surfer) is in state j after 1 step
(1 transition) by using the following reasoning:
∑=
=
n
i
ij ipAjp
1
01 )()1()( (18)

State transition

Stationary probability distribution
 By a Theorem of the Markov chain,
 a finite Markov chain defined by the stochastic
matrix A has a unique stationary probability
distribution if A is irreducible and aperiodic.
 The stationary probability distribution means
that after a series of transitions pk will converge
to a steady-state probability vector π
regardless of the choice of the initial
probability vector p0, i.e.,
πp =
∞→
k
k
lim (21)

PageRank again
 When we reach the steady-state, we have pk
= pk+1 =π, and thus
π =AT
π.
 π is the principal eigenvector of AT
with
eigenvalue of 1.
 In PageRank, π is used as the PageRank
vector P. We again obtain Equation (15),
which is re-produced here as Equation (22):
PAP T
= (22)

Is P = π justified?
 Using the stationary probability distribution π
as the PageRank vector is reasonable and
quite intuitive because
 it reflects the long-run probabilities that a random
surfer will visit the pages.
 A page has a high prestige if the probability of
visiting it is high.

Back to the Web graph
 Now let us come back to the real Web context
and see whether the above conditions are
satisfied, i.e.,
 whether A is a stochastic matrix and
 whether it is irreducible and aperiodic.
 None of them is satisfied.
 Hence, we need to extend the ideal-case
Equation (22) to produce the “actual
PageRank” model.

A is a not stochastic matrix
 A is the transition matrix of the Web graph
 It does not satisfy equation (17)
 because many Web pages have no out-links, which
are reflected in transition matrix A by some rows of
complete 0’s.
 Such pages are called the dangling pages (nodes).




∈
=
otherwise
Ejiif
OA iij
0
),(
1
∑=
=
n
j
ijA
1
1

An example Web hyperlink graph




















=
02121000
000000
313103100
000010
00021021
00021210
A

Fix the problem: two possible ways
1. Remove those pages with no out-links during the
PageRank computation as these pages do not
affect the ranking of any other page directly.
2. Add a complete set of outgoing links from each
such page i to all the pages on the Web.




















=
02121000
616161616161
313103100
000010
00021021
00021210
A
Let us use the
second way

A is a not irreducible
 Irreducible means that the Web graph G is
strongly connected.
Definition: A directed graph G = (V, E) is
strongly connected if and only if, for each pair
of nodes u, v ∈ V, there is a path from u to v.
 A general Web graph represented by A is not
irreducible because
 for some pair of nodes u and v, there is no path
from u to v.
 In our example, there is no directed path from
nodes 3 to 4.

A is a not aperiodic
 A state i in a Markov chain being periodic
means that there exists a directed cycle that
the chain has to traverse.
Definition: A state i is periodic with period k >
1 if k is the smallest number such that all
paths leading from state i back to state i have
a length that is a multiple of k.
 If a state is not periodic (i.e., k = 1), it is
aperiodic.
 A Markov chain is aperiodic if all states are
aperiodic.

An example: periodic
 Fig. 5 shows a periodic Markov chain with k = 3. Eg,
if we begin from state 1, to come back to state 1 the
only path is 1-2-3-1 for some number of times, say h.
Thus any return to state 1 will take 3h transitions.

Deal with irreducible and aperiodic
 It is easy to deal with the above two problems
with a single strategy.
 Add a link from each page to every page and
give each link a small transition probability
controlled by a parameter d.
 Obviously, the augmented transition matrix
becomes irreducible and aperiodic

Improved PageRank
 After this augmentation, at a page, the
random surfer has two options
 With probability d, he randomly chooses an out-
link to follow.
 With probability 1-d, he jumps to a random page
 Equation (25) gives the improved model,
where E is eeT
(e is a column vector of all 1’s)
and thus E is a n×n square matrix of all 1’s.
PA
E
P ))1(( T
d
n
d +−= (25)

Follow our example




















=+−
061610619061061061
157610619061061061
15761061061061061
061610619061157157
061610611211061157
06161061061157061
)1( T
d
n
d A
E

The final PageRank algorithm
 (1-d)E/n + dAT
is a stochastic matrix
(transposed). It is also irreducible and
aperiodic
 If we scale Equation (25) so that eT
P = n,
 PageRank for each page i is
PAeP T
dd +−= )1(
∑=
+−=
n
j
ji jPAddiP
1
)()1()(
(27)
(28)

The final PageRank (cont …)
 (28) is equivalent to the formula given in the
PageRank paper
 The parameter d is called the damping
factor which can be set to between 0 and 1.
d = 0.85 was used in the PageRank paper.
∑∈
+−=
Eij jO
jP
ddiP
),(
)(
)1()(

Compute PageRank
 Use the power iteration method

Advantages of PageRank
 Fighting spam. A page is important if the pages
pointing to it are important.
 Since it is not easy for Web page owner to add in-links into
his/her page from other important pages, it is thus not easy
to influence PageRank.
 PageRank is a global measure and is query
independent.
 PageRank values of all the pages are computed and saved
off-line rather than at the query time.
 Criticism: Query-independence. It could not
distinguish between pages that are authoritative in
general and pages that are authoritative on the
query topic.

Road map
 Introduction
 PageRank
 HITS
 Summary

HITS
 HITS stands for Hypertext Induced Topic
Search.
 Unlike PageRank which is a static ranking
algorithm, HITS is search query dependent.
 When the user issues a search query,
 HITS first expands the list of relevant pages
returned by a search engine and
 then produces two rankings of the expanded set
of pages, authority ranking and hub ranking.

Authorities and Hubs
Authority: Roughly, a authority is a page with
many in-links.
 The idea is that the page may have good or
authoritative content on some topic and
 thus many people trust it and link to it.
Hub: A hub is a page with many out-links.
 The page serves as an organizer of the
information on a particular topic and
 points to many good authority pages on the topic.

Examples

The key idea of HITS
 A good hub points to many good authorities, and
 A good authority is pointed to by many good hubs.
 Authorities and hubs have a mutual reinforcement
relationship. Fig. 8 shows some densely linked
authorities and hubs (a bipartite sub-graph).

The HITS algorithm: Grab pages
 Given a broad search query, q, HITS collects
a set of pages as follows:
 It sends the query q to a search engine.
 It then collects t (t = 200 is used in the HITS
paper) highest ranked pages. This set is called
the root set W.
 It then grows W by including any page pointed to
by a page in W and any page that points to a
page in W. This gives a larger set S, base set.

The link graph G
 HITS works on the pages in S, and assigns every
page in S an authority score and a hub score.
 Let the number of pages in S be n.
 We again use G = (V, E) to denote the hyperlink
graph of S.
 We use L to denote the adjacency matrix of the
graph.


 ∈
=
otherwise
Ejiif
Lij
0
),(1

The HITS algorithm
 Let the authority score of the page i be a(i),
and the hub score of page i be h(i).
 The mutual reinforcing relationship of the two
scores is represented as follows:
∑∈
=
Eij
jhia
),(
)()(
∑∈
=
Eji
jaih
),(
)()(
(31)
(32)

HITS in matrix form
 We use a to denote the column vector with all
the authority scores,
a = (a(1), a(2), …, a(n))T
, and
 use h to denote the column vector with all the
authority scores,
h = (h(1), h(2), …, h(n))T
,
 Then,
a = LT
h
h = La
(33)
(34)

Computation of HITS
 The computation of authority scores and hub scores
is the same as the computation of the PageRank
scores, using power iteration.
 If we use ak and hk to denote authority and hub
vectors at the kth iteration, the iterations for
generating the final solutions are

The algorithm

Relationships with co-citation and
bibliographic coupling
 Recall that co-citation of pages i and j,
denoted by Cij, is
 the authority matrix (LT
L) of HITS is the co-citation
matrix C
 bibliographic coupling of two pages i and j,
denoted by Bij is
 the hub matrix (LLT
) of HITS is the bibliographic
coupling matrix B
ij
T
n
k
kjkiij LLC )(
1
LL== ∑=
,)(
1
ij
T
n
k
jkikij LLB LL== ∑=

Strengths and weaknesses of HITS
 Strength: its ability to rank pages according to the
query topic, which may be able to provide more
relevant authority and hub pages.
 Weaknesses:
 It is easily spammed. It is in fact quite easy to influence
HITS since adding out-links in one’s own page is so easy.
 Topic drift. Many pages in the expanded set may not be
on topic.
 Inefficiency at query time: The query time evaluation is
slow. Collecting the root set, expanding it and performing
eigenvector computation are all expensive operations

Road map
 Introduction
 PageRank
 HITS
 Summary

Summary
 In this chapter, we introduced
 Social network analysis, centrality and prestige
 Co-citation and bibliographic coupling
 PageRank, which powers Google
 HITS
 Yahoo! and MSN have their own link-based
algorithms as well, but not published.
 Important to note: Hyperlink based ranking is not
the only algorithm used in search engines. In fact, it
is combined with many content based factors to
produce the final ranking presented to the user.

Summary
 Links can also be used to find communities,
which are groups of content-creators or
people sharing some common interests.
 Web communities
 Email communities
 Named entity communities
 Focused crawling: combining contents and
links to crawl Web pages of a specific topic.
 Follow links and
 Use learning/classification to determine whether a
page is on topic.

Cs583 link-analysis

More Related Content

What's hot (18)

Similar to Cs583 link-analysis (20)

More from Borseshweta (7)

Recently uploaded (20)

Cs583 link-analysis