How PageRank Works

Ketan Mayer-PatelUniversity of North Carolina

January 31, 2011

Me vs. Jeff

• High school– Public school in Texas

• College– The University of

California, Berkeley

• Faculty member at...– UNC

• High School– Hoity-toity, private all-

boys school in Jersey

• College– Stanford

• Faculty member at...– Duke

The World Wide Web

• A Simple Request/Response System

Request for web page.

Web page returned.

Making The Request

• How do you make a web request?– Use a browser.• Specify what you want directly.• Follow a link.

– Turns out we very rarely specify documents directly.

– Uniform Resource Locator (URL)• http://server-name.com/path/to/a/page

– Two key characteristics of hyperlinks:• Directional• Unilateral

Web Search In Three Easy Steps

• What’s step one?– Cut a hole in the box.

Web Search In Three Easy Steps

• First, crawl.– Try to find all of the web pages.

• Follow the links.

• Second, index.– Organize what you find.

• Lots of secret sauce here.

• Third, query.– Usually, text query words.– Retrieves a list of related pages.

• Usually because they contain the query text.

Which to list first?

• Possible clues:– Number of times the query term appears– Where it appears• Title, body text, URL, metadata, etc.

– How it appears• Style of text• Role of text

– Position in the document graph• This is what distinguished Google from other search

engines at the time.

PageRank

• Supposedly named after Larry Page

• Part of his research in grad school– Patented while in grad school.– Licensed to Google for ~ 1 million

shares of Google.• Sold for about $300M

Document Graph

Probability Distribution of a Random Walk

• Start walking the graph.• After some reasonably long amount of time,

stop.• What’s the chance that you are on a particular

page. – Larger chance => more important page– Is this actually true?• Maybe, maybe not

Random Walk Example

Random Walk Example

Random Walk Example

Random Walk Example

Random Walk Example

Random Walk Example

Random Walk Example

Trapdoors and Dead Ends

Shangri-La:Can’t ever get here.

Hotel California:Can’t ever leave.

Spider Traps

Fixing Our Random Walk

• What can we do to fix it?– Add a bit more randomness.• At each step, with probability α jump to any random

page.• Otherwise, randomly follow a link.

– Provides a way in to / out of trapdoors / dead ends and spider traps.

Random Walk Scalability

• Problem: Would need to simulate the random walk over and over again to even come close to discovering the underlying probability distribution.– Easy to do for small graphs.– Pain in the ass for large ones.

• Markov Chain– Tool for analyzing stochastic processes.– Power method

Power Method Equation• N : Number of documents• Rk : Page rank of document k

• Lk : Number of outgoing links in k• δ(k,j) : Delta function for links between k and j

δ(k,j) = 1 if and only if there exists a link from document k to document j

€

R j = δ (k, j)RkLkk=1

N

∑

Power Method Equation

• Our definition is circular.– To calculate page rank of a page we need to already

know the page rank of other pages.• Iterative solution.– Start with an initial assignment.

• Basically set the page rank of every page to 1/N.• Why 1/N?

– Calculate an updated value for every page using the current values.

– Keep repeating until the value are stable.

Power Method Equation

• Intuition:– Page rank of a document is the sum of its fair

share of the page ranks of the pages that link to the document.

€

R ji+1 = δ (k, j)

Rki

Lki

k=1

N

∑

Examplei = 0

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.1

€

R ji+1 = δ (k, j)

Rki

Lki

k=1

N

∑

Examplei = 1

0

0.1

0.1

0.125

0.2

0.05

0.1

0.075

0.025

0.125

€

R ji+1 = δ (k, j)

Rki

Lki

k=1

N

∑

Examplei = 10

0

0.154

0.134

0.015

0.071

0.036

0.072

0.051

0.015

0.189

€

R ji+1 = δ (k, j)

Rki

Lki

k=1

N

∑

Something is wrong!

Power Method v2• Dead ends leak.• Spider traps slowly collect everything.• Translating our random walk solution:

– Add a “virtual” link from every document to every other document.– Define a weighting factor α between 0.0 and 1.0

• Distribute α proportion of your page rank over the virtual links• Distribute (1- α) proportion of your page rank over the real links

€

R ji+1 =

αRki

Nk=1

N

∑ + δ (k, j)(1−α )Rk

i

Lki

k=1

N

∑

Power Method v2• Dead ends leak.• Spider traps slowly collect everything.• Translating our random walk solution:

– Add a “virtual” link from every document to every other document.– Define a weighting factor α between 0.0 and 1.0

• Distribute α proportion of your page rank over the virtual links• Distribute (1- α) proportion of your page rank over the real links

€

R ji+1 =

α

N+ (1−α ) δ (k, j)

Rki

Lki

k=1

N

∑

Convergence

• Typical value for α is 0.15.• Convergence typically occurs in about 50

iterations even for large graphs.

Examplei = 10

0.011

0.107

0.112

0.034

0.105

0.061

0.073

0.074

0.024

0.115

€

R ji+1 =

α

N+ (1−α ) δ (k, j)

Rki

Lki

k=1

N

∑

Examplei = 10

0.011

0.107

0.112

0.034

0.105

0.061

0.073

0.074

0.024

0.115

0

0.154

0.134

0.015

0.071

0.036

0.072

0.0510.015

0.189

€

R ji+1 =

α

N+ (1−α ) δ (k, j)

Rki

Lki

k=1

N

∑

Billions and billions

• How do you do this with billions of documents?– Can be implemented using

matrix math.– Special techniques for sparse

matrices.– PageRank roughly equivalent

to first eigenvector.

Gaming The System

• Google Bomb!– Create a lot of links to the page that you want to

be highly ranked.• Create your own spider trap.

– Relatively easy to combat by discounting links that come from the same domain.

• Comment spam.• Porn trap.

Last Notes

• Stanford Sucks!• GO HEELS!

Bad Math

• When originally presented, the final version of the power method equation was shown as:

• The simplification for the first term is wrong and should have been:

€

R ji+1 =

α

N+ (1−α ) δ (k, j)

Rki

Lki

k=1

N

∑

€

R ji+1 = α + (1−α ) δ (k, j)

Rki

Lki

k=1

N

∑