Understanding PageRank and HITS Algorithms for Information Retrieval, Exams of Computer Science

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Node Ranking

Lidan Wang and Brad Skaggs

CMSC 828G

1

T^

i Topics^ „^ What is PageRank„^ What is PageRank^ „^ PageRank algorithm [Brin and Page 2002]^ „^ Topic-sensitive PageRank [Richardson et al. 2002, Haveliwala2002]^ „^ HITS [Kleinberg 2002]^ „^ Other variations^ „^ Applications„^ Applications

P^

R^

k i t iti

PageRank intuition^ „^ How can we tell a page’s importance?„^ How can we tell a page s importance?^ „^ Relies on the link structure of the web to indicate anpage’s value.^ „^ A hyperlink from page A to page B as a vote, for pageB by page A.A page is “important” if:„^ A^ page is “important” if:^ „^ There are many pages pointing to it.^ „^ These pages are considered valuable.

E^

l

Example

Taken from http://en.wikipedia.org/wiki/PageRank

Li k

t^

(1)

Links as votes (1)^ „^ A link from page B to page A denotes endorsement:„^ A^ link from page B to page A denotes endorsement:^ „^ Page B considers page A an authority on a subject^ „^ An authority value or PageRank(PR) value is assigned toevery page (document)every page (document)

PR(A) = 0.75A

B

D

0.25 C

Li k

t^

(2)

Links as votes (2) „^ PR conferred by an outgoing link is the document

’s own PR

„^ PR conferred by an outgoing link is the document s own PRdivided by the number of outgoing links:

PR(A) = PR(B)/2 + PR(C) + PR(D)/

A^

B

PR(B)/2PR(D)/

D

PR(C) C

PR(D)/

PR(A) = PR(t1)/O(t1)+

+PR(tn)/O(tn); t

D^ = page linking to A

C

„^ PR(A)

= PR(t1)/O(t1)+…+PR(tn)/O(tn); t

= page linking to A,i^

O(t^ ): number of links from ti

i

M t i

ti

Matrix notion „Hyperlink matrix

H =[H ] in ⎧

A^

B

„Hyperlink matrix

H =[H

] inij

which the entry in the i

th^ row

thand j column is: H^ = ij^

1 if Oi

( i ,^ j )^

∈^ E ⎧⎪ ⎨ ⎪^0 otherwise ⎩

C

⎩

H^ =

10 2 1 2 ⎡ ⎢ 0 0 1⎢

⎤ ⎥ ⎥

A^ B^

C

A B

„Let^ PR

= (PR(1), …, PR(n))

T

PR^ =^

T^ H PR

H^

0 0 1⎢ ⎢ (^) 1 0 0⎣

⎥ ⎥⎦ B C Stochastic matrix: all entries >=0;

„We can write equations:

PR^

H^ PR

each row sums up to 1

S l

f^

P^

R^

k

Solve for PageRank

A^

B^

A^ B^

C

„A well known mathematical

k^ +^1 PR H^

T^ kPR^

A^

B^ H

=

10 2 1 2 ⎡ ⎢ 0 0 1⎢ ⎢

⎤ ⎥ ⎥ ⎥

A^ B^

C

A B

„A well known mathematicaltechnique called poweriteration can be used to find PRi e

PR

k +1^ =^

T^ H PR k

C^

⎢^ 1 0 0⎣

⎥⎦

C

(^0) PR

(^1) PR

(^2) PR

(^3) PR

kPR

„i.e. „The solution to PR is aneigenvector with the

PR^

PR^

PR^

PR^

PR

0.^ 0.^ 0.^ 0.^

0.^ 0.^ 0.^ 0.^

eigenvector

with the corresponding eigenvalue 1. „If some conditions are satisfied,

0.^ 0.^ 0.^ 0.^

,

1 is the largest eigenvalue andthe PageRank vector PR is theprincipal eigenvector. „Problem: the Web graph doesnot meet the conditions (later).

CConvergence^ „^ Markov Chain Theorem

:

„^ A finite Markov chain defined by the stochastic matrix H has a uniquestationary probability distribution if H is irreducible and aperiodic. „ After a number of transitions

k^ PR will converge to a steady-state

probability vector

lim^ PR k^ →∞

k^ =^ PR

„^ When the steady-state is reached, then

k^ PR =^ PR

k+1^ =

PR

PR^

T^ H PR k^ →∞ PR^ =H

T^ PR

„^ PR^

is the principal eigenvector of H

T^ with eigenvalue 1.

W b

h

Web graph „ Questions:^ „^ Is^ H

is a stochastic matrix? „^ Is^ H

irreducible? „^ Is^ H

irreducible? „^ Is^ H

and aperiodic? „^ If the above conditions are not satisfied, how to extend thesimplified equation to the actual PageRank formula?

I^ H

t^

h^

ti? (2)

Is^ H

stochastic? (2)

„^ Is the sum of entries in each row equal to 1?„^ Is the sum of entries in each row equal to 1? „^ If web pages have no out-links, i.e. some rows intransition matrix

H^ have all 0’s. They are called the

dangling pages:

⎛^

v

0 H =

0 1/^

1 2^0

v

v^

v

1/5^ 1/

1/5^ 1/

⎜ ⎜ ⎜^ ⎝

v

v

Fixed

I^ H

i^

d^

bl^

d^

i di?

Is^ H

irreducable and aperiodic?

Irreducible:

Aperiodic:

„^ States vi and vj can communicate inthe directed graph, i.e. a path existsbetween them.

Aperiodic: „State vi is periodic. „Aperiodic Markov chain: none

„^ Irreducible Markov chain: all statescommunicate

„Aperiodic Markov chain: noneof the states is periodic v^2 v^1

v^2 v^1

v^0

v^3

v^4

v^0

v^3

v^4

Fi^

l P^

R^

k^

ti^

(1)

Final PageRank equation (1)S t^

th^

d^

f^ h^

t^

ti

„So at a page, the random surfer has two options^ „With probability d, randomly chooses an out-link to follow.^ „With probability 1-d, jumps to a random page

((^ d

E^ )

T^ d )

„Equation below gives the improved model:^ PR

=^ ((^1

E − d )^ n +^ d H

T^ ) PR

„Where

E^ is^

T^ ee (e is a column vector of all 1

’s)^ E

is a

„Where

E^ is^

ee^ (e is a column vector of all 1 s).

E^ is a

nxn square matrix of all 1’s.

Fi^

l P^

R^

k^

ti^

Final PageRank equation (2)The new matrix is stochastic

irreducible and aperiodic

„The

new matrix is stochastic, irreducible and aperiodic

„Scale the equation so that

T ePR

=n:

PR^ =

(1−^

d ) e^ +

T d H

PR

PR ( i )

(^1

d )^ d

H

PR (

j )

n ∑

„PageRank for each page i is:

PR ( i )

=^ (^1 −

d )^ +^

d^ H

PR ( ji^

j )

∑^ j =^1

(1^ d

)^ d^

PR (^ j

=^ (^1 −

d )^ +^

d^

(^ j ) O^ j

∑( j , i )∈ E