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Connected Components, and Pagerank

Lecture slides credit: Lada Adamic, U Michigan Jure Leskovec, Stanford, Andreas Haeberlen, UPenn

COMP4650

Lexing Xie Research School of Computer Science, ANU

Connected components •  Strongly connected components

–  Each node within the component can be reached from every other node in the component by following directed links

n  Strongly connected components n  B C D E n  A n  G H n  F

n  Weakly connected components: every node can be reached from every other node by following links in either direcRon

A

B

C

D E

F G

H

A

B

C

D E

F G

H

n  Weakly connected components n  A B C D E n  G H F

n  In undirected networks one talks simply about ‘connected components’

by Lada Adamic, U Michigan

Strongly Connected Component (SCC)

Proof (*)

Giant component •  if the largest component encompasses a significant fracRon of the

graph, it is called the giant component

by Lada Adamic, U Michigan

Why not? (*)

Bow-‐Tie Structure of the Web

Andrei Broder, Ravi Kumar, Farzin Maghoul, Prabhakar Raghavan, Sridhar Rajagopalan, Raymie Stata, Andrew Tomkins, and Janet Wiener. 2000. Graph structure in the Web. Comput. Netw. 33, 1-‐6 (June 2000), 309-‐320.

Not Everyone Asks/Replies

•  Core: A strongly connected component, in which everyone asks and answers •  IN: Mostly askers. •  OUT: Mostly Helpers

The Web is a bow Re The Java Forum network is an uneven bow Re

Jun Zhang, Mark S. Ackerman, and Lada Adamic. 2007. ExperRse networks in online communiRes: structure and algorithms. In Proceedings of the 16th internaRonal conference on World Wide Web (WWW '07)., 221-‐230.

Minkyoung Kim, Lexing Xie, Peter Christen, Event Diffusion Paherns in Social Media (2012) Intl. Conf. on Weblogs and Social Media (ICWSM), 8 pages, Dublin, Ireland, June 2012

Modern InformaRon Retrieval: The Concepts and Technology behind Search Ricardo Baeza-‐Yates, Berthier Ribeiro-‐Neto Addison-‐Wesley Professional; 2 ediRon (February 10, 2011)

Google's PageRank (Brin/Page 98)

•  A technique for esRmaRng page quality – Based on web link graph

•  Results are combined with IR score – Think of it as: TotalScore = IR score * PageRank –  In pracRce, search engines use many other factors (for example, Google says it uses more than 200)

11 A. Haeberlen , Z. Ives, U Penn

PageRank: IntuiRon

•  Imagine a contest for The Web's Best Page –  IniRally, each page has one vote –  Each page votes for all the pages it has a link to –  To ensure fairness, pages voRng for more than one page must split their vote equally between them

–  VoRng proceeds in rounds; in each round, each page has the number of votes it received in the previous round

–  In pracRce, it's a lihle more complicated -‐ but not much! 12

A

BE

C

DF

G

H

I

J

Shouldn't E's vote be worth more than F's?

How many levels should we consider?

Source: A. Haeberlen , Z. Ives, U Penn

PageRank •  Each page i is given a rank xi •  Goal: Assign the xi such that the rank of each page is governed by the ranks of the pages linking to it:

13

jBj j

i xN

xi

∑∈

=1

Rank of page j Rank of page i

Every page j that links to i

Number of links out

from page j How do we compute the rank values?


14

IteraRve PageRank (simplified)

)()1( 1 kj

Bj j

ki x

Nx

i

∑∈

+ =

nxi

1)0( =Initialize all ranks to be equal, e.g.:

Iterate until convergence


15

Example: Step 0

0.33

0.33

0.33

Initialize all ranks to be equal n

xi1)0( =


16

Example: Step 1

0.17

0.33

0.33

)()1( 1 kj

Bj j

ki x

Nx

i

∑∈

+ =

0.17

Propagate weights across out-edges


17

Example: Step 2

0.17

0.50

0.33

Compute weights based on in-edges )0()1( 1

jBj j

i xN

xi

∑∈

=


18

Example: Convergence

0.2

0.40

0.4

)()1( 1 kj

Bj j

ki x

Nx

i

∑∈

+ =


19

Naïve PageRank Algorithm Restated

•  Let –  N(p) = number outgoing links from page p –  B(p) = number of back-‐links to page p

–  Each page b distributes its importance to all of the pages it points to (so

we scale by 1/N(b)) –  Page p’s importance is increased by the importance of its back set

)()(

1)( bPageRankbN

pPageRankpBb∑∈

=


20

In Linear Algebra formulaRon •  Create an m x m matrix M to capture links:

–  M(i, j) = 1 / nj if page i is pointed to by page j and page j has nj outgoing links

= 0 otherwise

–  IniRalize all PageRanks to 1, mulRply by M repeatedly unRl all values converge:

–  Computes principal eigenvector via power iteraRon

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

)(...

)()(

)'(...

)'()'(

2

1

2

1

mm pPageRank

pPageRankpPageRank

M

pPageRank

pPageRankpPageRank


21

A Brief Example

Google

Amazon Yahoo

0 0.5 0.5 0 0 0.5 1 0.5 0

g' y’ a’

g y a

= *

Total rank sums to number of pages

g y a

1 1 1

=

1 0.5 1.5

, 1

0.75 1.25

, 1

0.67 1.33

, …

Running for multiple iterations:


22

Oops #1 – PageRank Sinks

Google

Amazon Yahoo

0 0 0.5 0.5 0 0.5 0.5 0 0

g' y’ a’

g y a

= *

g y a

1 1 1

=

0.5 1

0.5

, 0.25 0.5 0.25

, 0 0 0

, … ,


'dead end' -‐ PageRank is lost auer each round


23

Oops #2 – PageRank hogs

Google

Amazon Yahoo

0 0 0.5 0.5 1 0.5 0.5 0 0

g' y’ a’

g y a

= *

g y a

1 1 1

=

0.5 2

0.5

, 0.25 2.5 0.25

, 0 3 0

, … ,


PageRank cannot flow out and accumulates


24

Improved PageRank

•  Remove out-‐degree 0 nodes (or consider them to refer back to referrer)

•  Add decay factor d to deal with sinks

•  Typical value: d=0.85

)()(

1)1()( bPageRankbN

ddpPageRankpBb

∑∈

+−=


25

Stopping the Hog

0 0 0.5 0.5 1 0.5 0.5 0 0

g' y’ a’

g y a

= 0.85 *

g y a

=

0.26 2.48 0.26

,

0.15 0.15 0.15

+


… though does this seem right?

Google

Amazon Yahoo

0.57 1.85 0.57

0.39 2.21 0.39

0.32 2.36 0.32

, , , … ,


Random Surfer Model •  PageRank has an intuiRve basis in random walks on graphs

•  Imagine a random surfer, who starts on a random page and, in each step, –  with probability d, klicks on a random link on the page –  with probability 1-‐d, jumps to a random page (bored?)

•  The PageRank of a page can be interpreted as the fracRon of steps the surfer spends on the corresponding page –  TransiRon matrix can be interpreted as a Markov Chain

26 Source: A. Haeberlen , Z. Ives, U Penn

PageRank, random walk on graphs G W = (1-‐d)*G + d*U

Pagerank vector

COMP4650 L Xie, ANU Computer Science

+

–  with probability d, klicks on a random link on the page –  with probability 1-‐d, jumps to a random page (bored?) –  TransiRon matrix W can be interpreted as a Markov Chain

28 COMP4650 L Xie, ANU Computer Science

Search Engine OpRmizaRon (SEO) •  Has become a big business •  White-‐hat techniques

–  Google webmaster tools –  Add meta tags to documents, etc.

•  Black-‐hat techniques –  Link farms –  Keyword stuffing, hidden text, meta-‐tag stuffing, ... –  Spamdexing

•  IniRal soluRon: <a rel="nofollow" href="...">...</a> •  Some people started to abuse this to improve their own rankings

–  Doorway pages / cloaking •  Special pages just for search engines •  BMW Germany and Ricoh Germany banned in February 2006

–  Link buying

29 Source: A. Haeberlen , Z. Ives, U Penn

30

Recap: PageRank •  EsRmates absolute 'quality' or 'importance' of a given

page based on inbound links –  Query-‐independent –  Can be computed via fixpoint iteraRon –  Can be interpreted as the fracRon of Rme a 'random surfer' would spend on the page

–  Several refinements, e.g., to deal with sinks

•  Considered relaRvely stable –  But vulnerable to black-‐hat SEO

•  An important factor, but not the only one –  Overall ranking is based on many factors (Google: >200)


What could be the other 200 factors?

•  Note: This is enRrely speculaRve! 31

Posi9ve Nega9ve

On-‐page

Off-‐page

Source: Web InformaRon Systems, Prof. Beat Signer, VU Brussels

Keyword in Rtle? URL? Keyword in domain name? Quality of HTML code Page freshness Rate of change ...

Links to 'bad neighborhood' Keyword stuffing Over-‐opRmizaRon Hidden content (text has same color as background) AutomaRc redirect/refresh ...

High PageRank Anchor text of inbound links Links from authority sites Links from well-‐known sites Domain expiraRon date ...

Fast increase in number of inbound links (link buying?) Link farming Different pages user/spider Content duplicaRon ...

Summary •  Macro structure of networks

– Strongly and weakly connected components – ApplicaRons in web/discussion forums etc.

•  PageRank algorithm – Computed using power iteraRons – Used for ranking webpages – Can be thought of as random walk on graphs

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