Google's PageRank Algorithm

The Problem Google Models Further thoughts

Google’s PageRank Algorithm Anders O. F. Hendrickson Concordia College Moorhead, MN

Math/CS Colloquium March 23, 2010

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Outline

1

The Problem Too many results Principles behind a solution

2

Google Models Five Models Markov Chains

3

Further thoughts

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Too many results Principles behind a solution

How do you organize an index?

This is two questions, really: How do you organize the topics in the index? How do you organize the pages found for each topic?

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Too many results Principles behind a solution

How do you organize an index?

This is two questions, really: How do you organize the topics in the index? How do you organize the pages found for each topic?

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Too many results Principles behind a solution

How do you organize an index?

This is two questions, really: How do you organize the topics in the index? (Alphabetically) How do you organize the pages found for each topic?

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Too many results Principles behind a solution

How do you organize an index?

This is two questions, really: How do you organize the topics in the index? (Alphabetically) How do you organize the pages found for each topic?

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Too many results Principles behind a solution

How do you organize an index?

This is two questions, really: How do you organize the topics in the index? (Alphabetically) How do you organize the pages found for each topic? (Sequentially)

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Too many results Principles behind a solution

How do you organize a computer search?

For a computer search, you specify the topic, so the first question is irrelevant, but the second still matters. Only one topic is desired. How do you organize the entries found for a topic?

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Too many results Principles behind a solution

How do you organize a computer search?

For a computer search, you specify the topic, so the first question is irrelevant, but the second still matters. Only one topic is desired. How do you organize the entries found for a topic?

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Too many results Principles behind a solution

How do you organize a computer search?

For a computer search, you specify the topic, so the first question is irrelevant, but the second still matters. Only one topic is desired. How do you organize the entries found for a topic?

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Too many results Principles behind a solution

How do you organize a computer search?

For a computer search, you specify the topic, so the first question is irrelevant, but the second still matters. Only one topic is desired. How do you organize the entries found for a topic? (Sequentially)

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Too many results Principles behind a solution

Drawbacks to returning results sequentially

Sequential ordering isn’t really informative. Okay for rare topics, bad for common topics. Slight improvements: Subtopics Markup

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Too many results Principles behind a solution

Drawbacks to returning results sequentially

Sequential ordering isn’t really informative. Okay for rare topics, bad for common topics. Slight improvements: Subtopics Markup

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Too many results Principles behind a solution

Drawbacks to returning results sequentially

Sequential ordering isn’t really informative. Okay for rare topics, bad for common topics. Slight improvements: Subtopics Markup

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Too many results Principles behind a solution

Drawbacks to returning results sequentially

Sequential ordering isn’t really informative. Okay for rare topics, bad for common topics. Slight improvements: Subtopics Markup

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Too many results Principles behind a solution

Drawbacks to returning results sequentially

Sequential ordering isn’t really informative. Okay for rare topics, bad for common topics. Slight improvements: Subtopics Markup

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Too many results Principles behind a solution

What if we’re searching the Web?

There is no sequential ordering! Even “rare” topics find many, many webpages. (Example) Attempted improvements: Subtopics—e.g. Yahoo! Directory Markup slightly possible—e.g. PDF vs. HTML, showing a few relevant lines

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Too many results Principles behind a solution

What if we’re searching the Web?

There is no sequential ordering! Even “rare” topics find many, many webpages. (Example) Attempted improvements: Subtopics—e.g. Yahoo! Directory Markup slightly possible—e.g. PDF vs. HTML, showing a few relevant lines

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Too many results Principles behind a solution

What if we’re searching the Web?

There is no sequential ordering! Even “rare” topics find many, many webpages. (Example) Attempted improvements: Subtopics—e.g. Yahoo! Directory Markup slightly possible—e.g. PDF vs. HTML, showing a few relevant lines

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Too many results Principles behind a solution

What if we’re searching the Web?

There is no sequential ordering! Even “rare” topics find many, many webpages. (Example) Attempted improvements: Subtopics—e.g. Yahoo! Directory Markup slightly possible—e.g. PDF vs. HTML, showing a few relevant lines

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Too many results Principles behind a solution

What if we’re searching the Web?

There is no sequential ordering! Even “rare” topics find many, many webpages. (Example) Attempted improvements: Subtopics—e.g. Yahoo! Directory Markup slightly possible—e.g. PDF vs. HTML, showing a few relevant lines

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Too many results Principles behind a solution

A solution: ask the librarian

An expert will know which books are particularly good or relevant. Likewise, you could ask people for recommendations of good websites. Of course, the quality of the recommendation is only as good as the quality of the recommender.

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Too many results Principles behind a solution

How does this translate to the web? Fundamental Principle Every hyperlink on the Web is a recommendation to visit another page. So one way to figure out which pages are “important” or “good” is to trust the judgment of the other website authors out there who link to it.

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Too many results Principles behind a solution

How does this translate to the web? Fundamental Principle Every hyperlink on the Web is a recommendation to visit another page. So one way to figure out which pages are “important” or “good” is to trust the judgment of the other website authors out there who link to it.

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Modeling the web For simplicity, suppose the web has only six pages, A, B, C, D, E, and F. There are two natural ways to model its hyperlink structure mathematically.

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Modeling the web For simplicity, suppose the web has only six pages, A, B, C, D, E, and F. There are two natural ways to model its hyperlink structure mathematically.

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Modeling the web For simplicity, suppose the web has only six pages, A, B, C, D, E, and F. There are two natural ways to model its hyperlink structure mathematically. Directed Graph A

B

F

C D Anders O. F. Hendrickson

E Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Modeling the web For simplicity, suppose the web has only six pages, A, B, C, D, E, and F. There are two natural ways to model its hyperlink structure mathematically. Directed Graph

Matrix FROM

A

F

 A

TO

B

B

C D

C

E

D Anders O. F. Hendrickson

E

F

       

A

B

C

D

E

F

0 1 0 0 0 0

1 0 1 1 0 0

1 0 0 0 0 0

1 0 0 0 0 1

0 0 0 1 0 0

0 0 0 0 0 0

Google’s PageRank Algorithm



       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 1: Counting hyperlinks FROM

TO

A A



B

      

C D E F

0 1 0 0 0 0

B

C

1 0 1 1 0 0

D

1 0 0 0 0 0

E

1 0 0 0 0 1

F

0 0 0 1 0 0

0 0 0 0 0 0

       

We want to count the number of hyperlinks into a webpage. In other words, we want to count the entries in each row.

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 1: Counting hyperlinks

       

0 1 0 0 0 0

+1 +0 +1 +1 +0 +0

+1 +0 +0 +0 +0 +0

+1 +0 +0 +0 +0 +1

+0 +0 +0 +1 +0 +0

+0 +0 +0 +0 +0 +0

       

We want to count the number of hyperlinks into a webpage. In other words, we want to count the entries in each row.

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 1: Counting hyperlinks

       

0 1 0 0 0 0

+1 +0 +1 +1 +0 +0

+1 +0 +0 +0 +0 +0

+1 +0 +0 +0 +0 +1

+0 +0 +0 +1 +0 +0

+0 +0 +0 +0 +0 +0

       

=3 =1 =1 =2 =0 =1

A B C D E F

We want to count the number of hyperlinks into a webpage. In other words, we want to count the entries in each row.

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Matrix Multiplication As it happens,  this is exactly what matrix multiplication by 1 .  ~e =   ..  does! 1        

0 1 0 0 0 0

1 0 1 1 0 0

1 0 0 0 0 0

1 0 0 0 0 1

0 0 0 1 0 0

Anders O. F. Hendrickson

0 0 0 0 0 0

       

1 1 1 1 1 1





      =      

3 1 1 2 0 1

Google’s PageRank Algorithm

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Problem with Model 1 Look again at the graph. We trust A most, since he has 3 recommendations. So when A recommends B, shouldn’t that count for more than E’s recommendation of D? A

B

F

C D

Anders O. F. Hendrickson

E

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 2: Iteration

       

0 1 0 0 0 0

1 0 1 1 0 0

1 0 0 0 0 0

1 0 0 0 0 1

0 0 0 1 0 0

0 0 0 0 0 0

       

3 1 1 2 0 1

       

Let’s re-run our matrix multiplication, but multiplying each recommendation by the recommender’s previous score. This is M(M ~e) = M 2~e.

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 2: Iteration

       

0 1 0 0 0 0

1 0 1 1 0 0

1 0 0 0 0 0

1 0 0 0 0 1

0 0 0 1 0 0

0 0 0 0 0 0

       

3 1 1 2 0 1





       =      

4 3 1 1 0 2

       

Let’s re-run our matrix multiplication, but multiplying each recommendation by the recommender’s previous score. This is M(M ~e) = M 2~e.

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 2: Iteration

       

0 1 0 0 0 0

1 0 1 1 0 0

1 0 0 0 0 0

1 0 0 0 0 1

0 0 0 1 0 0

0 0 0 0 0 0

       

3 1 1 2 0 1





       =      

4 3 1 1 0 2

       

Let’s re-run our matrix multiplication, but multiplying each recommendation by the recommender’s previous score. This is M(M ~e) = M 2~e.

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 2: Iteration

So let’s iterate! Let ~π0 = ~e and let ~πk +1 = M~πk .     1~ M e=   

0 1 0 0 0 0

1 0 1 1 0 0

1 0 0 0 0 0

1 0 0 0 0 1

0 0 0 1 0 0

Anders O. F. Hendrickson

0 0 0 0 0 0

       

1 1 1 1 1 1

       

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 2: Iteration

So let’s iterate! Let ~π0 = ~e and let ~πk +1 = M~πk .     1~ M e=   

0 1 0 0 0 0

1 0 1 1 0 0

1 0 0 0 0 0

1 0 0 0 0 1

0 0 0 1 0 0

Anders O. F. Hendrickson

0 0 0 0 0 0

       

1 1 1 1 1 1





       =      

3 1 1 2 0 1

Google’s PageRank Algorithm

     = ~π1   

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 2: Iteration

So let’s iterate! Let ~π0 = ~e and let ~πk +1 = M~πk .     2~ M e=   

0 1 0 0 0 0

1 0 1 1 0 0

1 0 0 0 0 0

1 0 0 0 0 1

0 0 0 1 0 0

Anders O. F. Hendrickson

0 0 0 0 0 0

       

3 1 1 2 0 1

       

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 2: Iteration

So let’s iterate! Let ~π0 = ~e and let ~πk +1 = M~πk .     2~ M e=   

0 1 0 0 0 0

1 0 1 1 0 0

1 0 0 0 0 0

1 0 0 0 0 1

0 0 0 1 0 0

Anders O. F. Hendrickson

0 0 0 0 0 0

       

3 1 1 2 0 1





       =      

4 3 1 1 0 2

Google’s PageRank Algorithm

     = ~π2   

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 2: Iteration

So let’s iterate! Let ~π0 = ~e and let ~πk +1 = M~πk .     3~ M e=   

0 1 0 0 0 0

1 0 1 1 0 0

1 0 0 0 0 0

1 0 0 0 0 1

0 0 0 1 0 0

Anders O. F. Hendrickson

0 0 0 0 0 0

       

4 3 1 1 0 2

       

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 2: Iteration

So let’s iterate! Let ~π0 = ~e and let ~πk +1 = M~πk .     3~ M e=   

0 1 0 0 0 0

1 0 1 1 0 0

1 0 0 0 0 0

1 0 0 0 0 1

0 0 0 1 0 0

Anders O. F. Hendrickson

0 0 0 0 0 0

       

4 3 1 1 0 2





       =      

5 4 3 3 0 1

Google’s PageRank Algorithm

     = ~π3   

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 2: Iteration

So let’s iterate! Let ~π0 = ~e and let ~πk +1 = M~πk .     4~ M e=   

0 1 0 0 0 0

1 0 1 1 0 0

1 0 0 0 0 0

1 0 0 0 0 1

0 0 0 1 0 0

Anders O. F. Hendrickson

0 0 0 0 0 0

       

5 4 3 3 0 1

       

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 2: Iteration

So let’s iterate! Let ~π0 = ~e and let ~πk +1 = M~πk .     4~ M e=   

0 1 0 0 0 0

1 0 1 1 0 0

1 0 0 0 0 0

1 0 0 0 0 1

0 0 0 1 0 0

Anders O. F. Hendrickson

0 0 0 0 0 0

       

5 4 3 3 0 1





       =      

10 5 4 4 0 3

Google’s PageRank Algorithm

     = ~π4   

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 2: Iteration

So let’s iterate! Let ~π0 = ~e and let ~πk +1 = M~πk .     5~ M e=   

0 1 0 0 0 0

1 0 1 1 0 0

1 0 0 0 0 0

1 0 0 0 0 1

0 0 0 1 0 0

Anders O. F. Hendrickson

0 0 0 0 0 0

       

10 5 4 4 0 3

       

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 2: Iteration

So let’s iterate! Let ~π0 = ~e and let ~πk +1 = M~πk .     5~ M e=   

0 1 0 0 0 0

1 0 1 1 0 0

1 0 0 0 0 0

1 0 0 0 0 1

0 0 0 1 0 0

Anders O. F. Hendrickson

0 0 0 0 0 0

       

10 5 4 4 0 3





       =      

13 10 5 5 0 4

Google’s PageRank Algorithm

     = ~π5   

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 2: Iteration

So let’s iterate! Let ~π0 = ~e and let ~πk +1 = M~πk .     6~ M e=   

0 1 0 0 0 0

1 0 1 1 0 0

1 0 0 0 0 0

1 0 0 0 0 1

0 0 0 1 0 0

Anders O. F. Hendrickson

0 0 0 0 0 0

       

13 10 5 5 0 4

       

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 2: Iteration

So let’s iterate! Let ~π0 = ~e and let ~πk +1 = M~πk .     6~ M e=   

0 1 0 0 0 0

1 0 1 1 0 0

1 0 0 0 0 0

1 0 0 0 0 1

0 0 0 1 0 0

Anders O. F. Hendrickson

0 0 0 0 0 0

       

13 10 5 5 0 4





       =      

20 13 10 10 0 5

Google’s PageRank Algorithm

     = ~π6   

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 2: Iteration

So let’s iterate! Let ~π0 = ~e and let ~πk +1 = M~πk .     7~ M e=   

0 1 0 0 0 0

1 0 1 1 0 0

1 0 0 0 0 0

1 0 0 0 0 1

0 0 0 1 0 0

Anders O. F. Hendrickson

0 0 0 0 0 0

       

20 13 10 10 0 5

       

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 2: Iteration

So let’s iterate! Let ~π0 = ~e and let ~πk +1 = M~πk .     7~ M e=   

0 1 0 0 0 0

1 0 1 1 0 0

1 0 0 0 0 0

1 0 0 0 0 1

0 0 0 1 0 0

Anders O. F. Hendrickson

0 0 0 0 0 0

       

20 13 10 10 0 5





       =      

33 20 13 13 0 10

Google’s PageRank Algorithm

     = ~π7   

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 2: Iteration

So let’s iterate! Let ~π0 = ~e and let ~πk +1 = M~πk .     8~ M e=   

0 1 0 0 0 0

1 0 1 1 0 0

1 0 0 0 0 0

1 0 0 0 0 1

0 0 0 1 0 0

Anders O. F. Hendrickson

0 0 0 0 0 0

       

33 20 13 13 0 10

       

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 2: Iteration

So let’s iterate! Let ~π0 = ~e and let ~πk +1 = M~πk .     8~ M e=   

0 1 0 0 0 0

1 0 1 1 0 0

1 0 0 0 0 0

1 0 0 0 0 1

0 0 0 1 0 0

Anders O. F. Hendrickson

0 0 0 0 0 0

       

33 20 13 13 0 10





       =      

46 33 20 20 0 13

Google’s PageRank Algorithm

     = ~π8   

The Problem Google Models Further thoughts

Five Models Markov Chains

Problems with Model 2

Four problems: Where should we stop? When we iterate, the numbers get bigger and bigger. How do we know the rankings stabilize? A profligate recommender will have an undue influence on the results. Solution: when B recommends 3 people (A, C, and D), let’s make that worth 31 each instead of 1.

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Problems with Model 2

Four problems: Where should we stop? When we iterate, the numbers get bigger and bigger. How do we know the rankings stabilize? A profligate recommender will have an undue influence on the results. Solution: when B recommends 3 people (A, C, and D), let’s make that worth 31 each instead of 1.

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Problems with Model 2

Four problems: Where should we stop? When we iterate, the numbers get bigger and bigger. How do we know the rankings stabilize? A profligate recommender will have an undue influence on the results. Solution: when B recommends 3 people (A, C, and D), let’s make that worth 31 each instead of 1.

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Problems with Model 2

Four problems: Where should we stop? When we iterate, the numbers get bigger and bigger. How do we know the rankings stabilize? A profligate recommender will have an undue influence on the results. Solution: when B recommends 3 people (A, C, and D), let’s make that worth 31 each instead of 1.

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Problems with Model 2

Four problems: Where should we stop? When we iterate, the numbers get bigger and bigger. How do we know the rankings stabilize? A profligate recommender will have an undue influence on the results. Solution: when B recommends 3 people (A, C, and D), let’s make that worth 31 each instead of 1.

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Problems with Model 2

Four problems: Where should we stop? When we iterate, the numbers get bigger and bigger. How do we know the rankings stabilize? A profligate recommender will have an undue influence on the results. Solution: when B recommends 3 people (A, C, and D), let’s make that worth 31 each instead of 1.

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 3: hyperlink matrix H

Scaling the columns to sum to 1 gives us a hyperlink matrix H:   0 1 1 1 0 0  1 0 0 0 0 0     0 1 0 0 0 0     0 1 0 0 1 0 .    0 0 0 0 0 0  0 0 0 1 0 0

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 3: hyperlink matrix H

Scaling the columns to sum to 1 gives us a hyperlink matrix H:     H=    

0 13 1 21 0 0 1 0 0 0 0 0 0 13 0 0 0 0 0 13 0 0 1 0 0 0 0 0 0 0 0 0 0 21 0 0

Anders O. F. Hendrickson

    .   

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 3: hyperlink matrix H

Scaling the columns to sum to 1 gives us a hyperlink matrix H:     H=    

0 13 1 21 0 0 1 0 0 0 0 0 0 13 0 0 0 0 0 13 0 0 1 0 0 0 0 0 0 0 0 0 0 21 0 0

    .   

So let’s let ~π0 = ~e and let ~πk +1 = H~πk .

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 3: hyperlink matrix H     ~π0 = H ~e =   n   01

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 0 0 0 0 0

1 2

0 0 0 0 1 2

0 0 0 1 0 0

0 0 0 0 0 0

A

B

0        

      

1 6 1 6 1 6 1 6 1 6 1 6





      =      

F

C D

Anders O. F. Hendrickson

E

Google’s PageRank Algorithm

0.17 0.17 0.17 0.17 0.17 0.17

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 3: hyperlink matrix H     ~π1 = H ~e =   n   11

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 0 0 0 0 0

1 2

0 0 0 0 1 2

0 0 0 1 0 0

0 0 0 0 0 0

A

B

1        

      

1 6 1 6 1 6 1 6 1 6 1 6





      =      

F

C D

Anders O. F. Hendrickson

E

Google’s PageRank Algorithm

0.31 0.17 0.06 0.22 0.00 0.08

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 3: hyperlink matrix H     ~π2 = H ~e =   n   21

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 0 0 0 0 0

1 2

0 0 0 0 1 2

0 0 0 1 0 0

0 0 0 0 0 0

A

B

2        

      

1 6 1 6 1 6 1 6 1 6 1 6





      =      

F

C D

Anders O. F. Hendrickson

E

Google’s PageRank Algorithm

0.22 0.31 0.06 0.06 0.00 0.11

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 3: hyperlink matrix H     ~π3 = H ~e =   n   31

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 0 0 0 0 0

1 2

0 0 0 0 1 2

0 0 0 1 0 0

0 0 0 0 0 0

A

B

3        

      

1 6 1 6 1 6 1 6 1 6 1 6





      =      

F

C D

Anders O. F. Hendrickson

E

Google’s PageRank Algorithm

0.19 0.22 0.10 0.10 0.00 0.03

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 3: hyperlink matrix H     ~π4 = H ~e =   n   41

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 0 0 0 0 0

1 2

0 0 0 0 1 2

0 0 0 1 0 0

0 0 0 0 0 0

A

B

4        

      

1 6 1 6 1 6 1 6 1 6 1 6





      =      

F

C D

Anders O. F. Hendrickson

E

Google’s PageRank Algorithm

0.23 0.19 0.07 0.07 0.00 0.05

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 3: hyperlink matrix H     ~π5 = H ~e =   n   51

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 0 0 0 0 0

1 2

0 0 0 0 1 2

0 0 0 1 0 0

0 0 0 0 0 0

A

B

5        

      

1 6 1 6 1 6 1 6 1 6 1 6





      =      

F

C D

Anders O. F. Hendrickson

E

Google’s PageRank Algorithm

0.17 0.23 0.06 0.06 0.00 0.04

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 3: hyperlink matrix H     ~π6 = H ~e =   n   61

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 0 0 0 0 0

1 2

0 0 0 0 1 2

0 0 0 1 0 0

0 0 0 0 0 0

A

B

6        

      

1 6 1 6 1 6 1 6 1 6 1 6





      =      

F

C D

Anders O. F. Hendrickson

E

Google’s PageRank Algorithm

0.17 0.17 0.08 0.08 0.00 0.03

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 3: hyperlink matrix H     ~π7 = H ~e =   n   71

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 0 0 0 0 0

1 2

0 0 0 0 1 2

0 0 0 1 0 0

0 0 0 0 0 0

A

B

7        

      

1 6 1 6 1 6 1 6 1 6 1 6





      =      

F

C D

Anders O. F. Hendrickson

E

Google’s PageRank Algorithm

0.17 0.17 0.06 0.06 0.00 0.04

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 3: hyperlink matrix H     ~π8 = H ~e =   n   81

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 0 0 0 0 0

1 2

0 0 0 0 1 2

0 0 0 1 0 0

0 0 0 0 0 0

A

B

8        

      

1 6 1 6 1 6 1 6 1 6 1 6





      =      

F

C D

Anders O. F. Hendrickson

E

Google’s PageRank Algorithm

0.14 0.17 0.06 0.06 0.00 0.03

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 3: hyperlink matrix H     ~π9 = H ~e =   n   91

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 0 0 0 0 0

1 2

0 0 0 0 1 2

0 0 0 1 0 0

0 0 0 0 0 0

A

B

9        

      

1 6 1 6 1 6 1 6 1 6 1 6





      =      

F

C D

Anders O. F. Hendrickson

E

Google’s PageRank Algorithm

0.14 0.14 0.06 0.06 0.00 0.03

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 3: hyperlink matrix H  ~π10 = H

   ~e =   n  

10 1

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 0 0 0 0 0

1 2

0 0 0 0 1 2

0 0 0 1 0 0

0 0 0 0 0 0

A

B

10        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

F

C D

Anders O. F. Hendrickson



E

Google’s PageRank Algorithm

0.13 0.14 0.05 0.05 0.00 0.03

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 3: hyperlink matrix H  ~π11 = H

   ~e =   n  

11 1

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 0 0 0 0 0

1 2

0 0 0 0 1 2

0 0 0 1 0 0

0 0 0 0 0 0

A

B

11        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

F

C D

Anders O. F. Hendrickson



E

Google’s PageRank Algorithm

0.12 0.13 0.05 0.05 0.00 0.02

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 3: hyperlink matrix H  ~π12 = H

   ~e =   n  

12 1

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 0 0 0 0 0

1 2

0 0 0 0 1 2

0 0 0 1 0 0

0 0 0 0 0 0

A

B

12        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

F

C D

Anders O. F. Hendrickson



E

Google’s PageRank Algorithm

0.11 0.12 0.04 0.04 0.00 0.02

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 3: hyperlink matrix H  ~π13 = H

   ~e =   n  

13 1

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 0 0 0 0 0

1 2

0 0 0 0 1 2

0 0 0 1 0 0

0 0 0 0 0 0

A

B

13        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

F

C D

Anders O. F. Hendrickson



E

Google’s PageRank Algorithm

0.11 0.11 0.04 0.04 0.00 0.02

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 3: hyperlink matrix H  ~π14 = H

   ~e =   n  

14 1

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 0 0 0 0 0

1 2

0 0 0 0 1 2

0 0 0 1 0 0

0 0 0 0 0 0

A

B

14        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

F

C D

Anders O. F. Hendrickson



E

Google’s PageRank Algorithm

0.10 0.11 0.04 0.04 0.00 0.02

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 3: hyperlink matrix H  ~π15 = H

   ~e =   n  

15 1

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 0 0 0 0 0

1 2

0 0 0 0 1 2

0 0 0 1 0 0

0 0 0 0 0 0

A

B

15        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

F

C D

Anders O. F. Hendrickson



E

Google’s PageRank Algorithm

0.09 0.10 0.04 0.04 0.00 0.02

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 3: hyperlink matrix H  ~π16 = H

   ~e =   n  

16 1

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 0 0 0 0 0

1 2

0 0 0 0 1 2

0 0 0 1 0 0

0 0 0 0 0 0

A

B

16        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

F

C D

Anders O. F. Hendrickson



E

Google’s PageRank Algorithm

0.09 0.09 0.03 0.03 0.00 0.02

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 3: hyperlink matrix H  ~π17 = H

   ~e =   n  

17 1

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 0 0 0 0 0

1 2

0 0 0 0 1 2

0 0 0 1 0 0

0 0 0 0 0 0

A

B

17        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

F

C D

Anders O. F. Hendrickson



E

Google’s PageRank Algorithm

0.08 0.09 0.03 0.03 0.00 0.02

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 3: hyperlink matrix H  ~π18 = H

   ~e =   n  

18 1

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 0 0 0 0 0

1 2

0 0 0 0 1 2

0 0 0 1 0 0

0 0 0 0 0 0

A

B

18        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

F

C D

Anders O. F. Hendrickson



E

Google’s PageRank Algorithm

0.06 0.08 0.03 0.03 0.00 0.02

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 3: hyperlink matrix H  ~π19 = H

   ~e =   n  

19 1

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 0 0 0 0 0

1 2

0 0 0 0 1 2

0 0 0 1 0 0

0 0 0 0 0 0

A

B

19        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

F

C D

Anders O. F. Hendrickson



E

Google’s PageRank Algorithm

0.06 0.06 0.03 0.03 0.00 0.01

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 3: hyperlink matrix H  ~π20 = H

   ~e =   n  

20 1

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 0 0 0 0 0

1 2

0 0 0 0 1 2

0 0 0 1 0 0

0 0 0 0 0 0

A

B

20        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

F

C D

E

Problem: All the scores go to zero!

Anders O. F. Hendrickson



Google’s PageRank Algorithm

0.05 0.06 0.02 0.02 0.00 0.01

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 3: hyperlink matrix H  ~π21 = H

   ~e =   n  

21 1

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 0 0 0 0 0

1 2

0 0 0 0 1 2

0 0 0 1 0 0

0 0 0 0 0 0

A

B

21        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

F

C D

E

Problem: All the scores go to zero! Why? Because F is “dangling node.” Anders O. F. Hendrickson



Google’s PageRank Algorithm

0.05 0.05 0.02 0.02 0.00 0.01

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Fix the dangling node A dangling node like F recommends no other page. Instead, let’s “read” F as recommending everyone equally.  H

=

      

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 0 0 0 0 0

1 2

0 0 0 0 1 2

0 0 0 1 0 0

Anders O. F. Hendrickson

0 0 0 0 0 0

       

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Fix the dangling node A dangling node like F recommends no other page. Instead, let’s “read” F as recommending everyone equally.  S

=

      

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 0 0 0 0 0

1 2

0 0 0 0 1 2

0 0 0 1 0 0

Anders O. F. Hendrickson

1 6 1 6 1 6 1 6 1 6 1 6

       

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Fix the dangling node A dangling node like F recommends no other page. Instead, let’s “read” F as recommending everyone equally.  S

=

      

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 0 0 0 0 0

1 2

0 0 0 0 1 2

0 0 0 1 0 0

Anders O. F. Hendrickson

0 0 0 0 0 0





       +      

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

Google’s PageRank Algorithm

0 0 0 0 0 0

1 6 1 6 1 6 1 6 1 6 1 6

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Fix the dangling node A dangling node like F recommends no other page. Instead, let’s “read” F as recommending everyone equally.  S

=

=

      

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0 

  1 H+  6  

1 2

1 0 0 0 0 0 1 1 1 1 1 1

0 0 0 1 0 0

0 0 0 0 1 2

0 0 0 0 0 0





       +      

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

       

0

0

Anders O. F. Hendrickson

0

0

0

1




Google’s PageRank Algorithm

0 0 0 0 0 0

1 6 1 6 1 6 1 6 1 6 1 6

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Fix the dangling node A dangling node like F recommends no other page. Instead, let’s “read” F as recommending everyone equally.  S

=

=

      

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0 

  1 H+  6  

1 2

1 0 0 0 0 0 1 1 1 1 1 1

0 0 0 1 0 0

0 0 0 0 1 2

0 0 0 0 0 0





       +      

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

1 6 1 6 1 6 1 6 1 6 1 6

       

0

0

0

0

0

1




=H+

where ~a is a vector whose 1’s indicate dangling nodes. Anders O. F. Hendrickson

Google’s PageRank Algorithm

1 T ~e~a . n

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 4: S = H + n1 ~e~aT     1 ~π0 = S 0 ~e =   n  

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 2

1 0 0 0 0 0

0 0 0 0 1 2

0 0 0 1 0 0

1 6 1 6 1 6 1 6 1 6 1 6

A

B

0        

      

1 6 1 6 1 6 1 6 1 6 1 6





      =      

F

C D

Anders O. F. Hendrickson

E

Google’s PageRank Algorithm

0.17 0.17 0.17 0.17 0.17 0.17

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 4: S = H + n1 ~e~aT     1 ~π1 = S 1 ~e =   n  

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 2

1 0 0 0 0 0

0 0 0 0 1 2

0 0 0 1 0 0

1 6 1 6 1 6 1 6 1 6 1 6

A

B

1        

      

1 6 1 6 1 6 1 6 1 6 1 6





      =      

F

C D

Anders O. F. Hendrickson

E

Google’s PageRank Algorithm

0.33 0.19 0.08 0.25 0.03 0.11

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 4: S = H + n1 ~e~aT     1 ~π2 = S 2 ~e =   n  

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 2

1 0 0 0 0 0

0 0 0 0 1 2

0 0 0 1 0 0

1 6 1 6 1 6 1 6 1 6 1 6

A

B

2        

      

1 6 1 6 1 6 1 6 1 6 1 6





      =      

F

C D

Anders O. F. Hendrickson

E

Google’s PageRank Algorithm

0.29 0.35 0.08 0.11 0.02 0.14

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 4: S = H + n1 ~e~aT     1 ~π3 = S 3 ~e =   n  

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 2

1 0 0 0 0 0

0 0 0 0 1 2

0 0 0 1 0 0

1 6 1 6 1 6 1 6 1 6 1 6

A

B

3        

      

1 6 1 6 1 6 1 6 1 6 1 6





      =      

F

C D

Anders O. F. Hendrickson

E

Google’s PageRank Algorithm

0.28 0.32 0.14 0.16 0.02 0.08

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 4: S = H + n1 ~e~aT     1 ~π4 = S 4 ~e =   n  

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 2

1 0 0 0 0 0

0 0 0 0 1 2

0 0 0 1 0 0

1 6 1 6 1 6 1 6 1 6 1 6

A

B

4        

      

1 6 1 6 1 6 1 6 1 6 1 6





      =      

F

C D

Anders O. F. Hendrickson

E

Google’s PageRank Algorithm

0.34 0.29 0.12 0.14 0.01 0.09

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 4: S = H + n1 ~e~aT     1 ~π5 = S 5 ~e =   n  

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 2

1 0 0 0 0 0

0 0 0 0 1 2

0 0 0 1 0 0

1 6 1 6 1 6 1 6 1 6 1 6

A

B

5        

      

1 6 1 6 1 6 1 6 1 6 1 6





      =      

F

C D

Anders O. F. Hendrickson

E

Google’s PageRank Algorithm

0.30 0.36 0.11 0.13 0.02 0.09

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 4: S = H + n1 ~e~aT     1 ~π6 = S 6 ~e =   n  

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 2

1 0 0 0 0 0

0 0 0 0 1 2

0 0 0 1 0 0

1 6 1 6 1 6 1 6 1 6 1 6

A

B

6        

      

1 6 1 6 1 6 1 6 1 6 1 6





      =      

F

C D

Anders O. F. Hendrickson

E

Google’s PageRank Algorithm

0.31 0.32 0.13 0.15 0.01 0.08

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 4: S = H + n1 ~e~aT     1 ~π7 = S 7 ~e =   n  

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 2

1 0 0 0 0 0

0 0 0 0 1 2

0 0 0 1 0 0

1 6 1 6 1 6 1 6 1 6 1 6

A

B

7        

      

1 6 1 6 1 6 1 6 1 6 1 6





      =      

F

C D

Anders O. F. Hendrickson

E

Google’s PageRank Algorithm

0.33 0.32 0.12 0.13 0.01 0.09

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 4: S = H + n1 ~e~aT     1 ~π8 = S 8 ~e =   n  

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 2

1 0 0 0 0 0

0 0 0 0 1 2

0 0 0 1 0 0

1 6 1 6 1 6 1 6 1 6 1 6

A

B

8        

      

1 6 1 6 1 6 1 6 1 6 1 6





      =      

F

C D

Anders O. F. Hendrickson

E

Google’s PageRank Algorithm

0.31 0.34 0.12 0.13 0.01 0.08

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 4: S = H + n1 ~e~aT     1 ~π9 = S 9 ~e =   n  

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 2

1 0 0 0 0 0

0 0 0 0 1 2

0 0 0 1 0 0

1 6 1 6 1 6 1 6 1 6 1 6

A

B

9        

      

1 6 1 6 1 6 1 6 1 6 1 6





      =      

F

C D

Anders O. F. Hendrickson

E

Google’s PageRank Algorithm

0.32 0.32 0.13 0.14 0.01 0.08

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 4: S = H + n1 ~e~aT 

~π10

   1 = S 10 ~e =   n  

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 0 0 0 0 0

1 2

0 0 0 0 1 2

0 0 0 1 0 0

1 6 1 6 1 6 1 6 1 6 1 6

A

B

10        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

F

C D

Anders O. F. Hendrickson



E

Google’s PageRank Algorithm

0.32 0.33 0.12 0.13 0.01 0.08

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 4: S = H + n1 ~e~aT 

~π11

   1 = S 11 ~e =   n  

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 0 0 0 0 0

1 2

0 0 0 0 1 2

0 0 0 1 0 0

1 6 1 6 1 6 1 6 1 6 1 6

A

B

11        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

F

C D

Anders O. F. Hendrickson



E

Google’s PageRank Algorithm

0.31 0.33 0.12 0.14 0.01 0.08

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 4: S = H + n1 ~e~aT 

~π12

   1 = S 12 ~e =   n  

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 0 0 0 0 0

1 2

0 0 0 0 1 2

0 0 0 1 0 0

1 6 1 6 1 6 1 6 1 6 1 6

A

B

12        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

F

C D

Anders O. F. Hendrickson



E

Google’s PageRank Algorithm

0.32 0.32 0.12 0.14 0.01 0.08

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 4: S = H + n1 ~e~aT 

~π13

   1 = S 13 ~e =   n  

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 0 0 0 0 0

1 2

0 0 0 0 1 2

0 0 0 1 0 0

1 6 1 6 1 6 1 6 1 6 1 6

A

B

13        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

F

C D

Anders O. F. Hendrickson



E

Google’s PageRank Algorithm

0.32 0.33 0.12 0.14 0.01 0.08

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 4: S = H + n1 ~e~aT 

~π14

   1 = S 14 ~e =   n  

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 0 0 0 0 0

1 2

0 0 0 0 1 2

0 0 0 1 0 0

1 6 1 6 1 6 1 6 1 6 1 6

A

B

14        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

F

C D

Anders O. F. Hendrickson



E

Google’s PageRank Algorithm

0.31 0.33 0.12 0.14 0.01 0.08

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 4: S = H + n1 ~e~aT 

~π15

   1 = S 15 ~e =   n  

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 0 0 0 0 0

1 2

0 0 0 0 1 2

0 0 0 1 0 0

1 6 1 6 1 6 1 6 1 6 1 6

A

B

15        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

F

C D

Anders O. F. Hendrickson



E

Google’s PageRank Algorithm

0.32 0.33 0.12 0.14 0.01 0.08

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 4: S = H + n1 ~e~aT 

~π16

   1 = S 16 ~e =   n  

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 0 0 0 0 0

1 2

0 0 0 0 1 2

0 0 0 1 0 0

1 6 1 6 1 6 1 6 1 6 1 6

A

B

16        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

F

C D

Anders O. F. Hendrickson



E

Google’s PageRank Algorithm

0.31 0.33 0.12 0.14 0.01 0.08

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 4: S = H + n1 ~e~aT 

~π17

   1 = S 17 ~e =   n  

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 0 0 0 0 0

1 2

0 0 0 0 1 2

0 0 0 1 0 0

1 6 1 6 1 6 1 6 1 6 1 6

A

B

17        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

F

C D

Anders O. F. Hendrickson



E

Google’s PageRank Algorithm

0.31 0.33 0.12 0.14 0.01 0.08

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 4: S = H + n1 ~e~aT 

~π18

   1 = S 18 ~e =   n  

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 0 0 0 0 0

1 2

0 0 0 0 1 2

0 0 0 1 0 0

1 6 1 6 1 6 1 6 1 6 1 6

A

B

18        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

F

C D

Anders O. F. Hendrickson



E

Google’s PageRank Algorithm

0.32 0.33 0.12 0.14 0.01 0.08

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 4: S = H + n1 ~e~aT 

~π19

   1 = S 19 ~e =   n  

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 0 0 0 0 0

1 2

0 0 0 0 1 2

0 0 0 1 0 0

1 6 1 6 1 6 1 6 1 6 1 6

A

B

19        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

F

C D

Anders O. F. Hendrickson



E

Google’s PageRank Algorithm

0.31 0.33 0.12 0.14 0.01 0.08

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 4: S = H + n1 ~e~aT 

~π20

   1 = S 20 ~e =   n  

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 0 0 0 0 0

1 2

0 0 0 0 1 2

0 0 0 1 0 0

1 6 1 6 1 6 1 6 1 6 1 6

A

B

20        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

F

C D

E

Hooray! Anders O. F. Hendrickson



Google’s PageRank Algorithm

0.32 0.33 0.12 0.14 0.01 0.08

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 4: S = H + n1 ~e~aT 

~π21

   1 = S 21 ~e =   n  

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 0 0 0 0 0

1 2

0 0 0 0 1 2

0 0 0 1 0 0

1 6 1 6 1 6 1 6 1 6 1 6

A

B

21        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

F

C D

E

Hooray! Anders O. F. Hendrickson



Google’s PageRank Algorithm

0.32 0.33 0.12 0.14 0.01 0.08

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 4: S = H + n1 ~e~aT 

~π22

   1 = S 22 ~e =   n  

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 0 0 0 0 0

1 2

0 0 0 0 1 2

0 0 0 1 0 0

1 6 1 6 1 6 1 6 1 6 1 6

A

B

22        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

F

C D

E

Hooray! Anders O. F. Hendrickson



Google’s PageRank Algorithm

0.31 0.33 0.12 0.14 0.01 0.08

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 4: S = H + n1 ~e~aT 

~π23

   1 = S 23 ~e =   n  

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 0 0 0 0 0

1 2

0 0 0 0 1 2

0 0 0 1 0 0

1 6 1 6 1 6 1 6 1 6 1 6

A

B

23        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

F

C D

E

Hooray! Anders O. F. Hendrickson



Google’s PageRank Algorithm

0.32 0.33 0.12 0.14 0.01 0.08

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 4: S = H + n1 ~e~aT 

~π24

   1 = S 24 ~e =   n  

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 0 0 0 0 0

1 2

0 0 0 0 1 2

0 0 0 1 0 0

1 6 1 6 1 6 1 6 1 6 1 6

A

B

24        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

F

C D

E

Hooray! Anders O. F. Hendrickson



Google’s PageRank Algorithm

0.32 0.33 0.12 0.14 0.01 0.08

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 4: S = H + n1 ~e~aT 

~π25

   1 = S 25 ~e =   n  

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 0 0 0 0 0

1 2

0 0 0 0 1 2

0 0 0 1 0 0

1 6 1 6 1 6 1 6 1 6 1 6

A

B

25        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

F

C D

E

Hooray! Anders O. F. Hendrickson



Google’s PageRank Algorithm

0.32 0.33 0.12 0.14 0.01 0.08

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 4: S = H + n1 ~e~aT 

~π26

   1 = S 26 ~e =   n  

0 1 0 0 0 0

1 3

0 1 3 1 3

0 0

1 0 0 0 0 0

1 2

0 0 0 0 1 2

0 0 0 1 0 0

1 6 1 6 1 6 1 6 1 6 1 6

A

B

26        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

F

C D

E

Hooray! Anders O. F. Hendrickson



Google’s PageRank Algorithm

0.32 0.33 0.12 0.14 0.01 0.08

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Problem with Model 4: cycles     ~π0 = S ~e =   n   01

0 1 0 0 0 0

0 0 1 0 0 0

1 3

0 0 1 3 1 3

0

1 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 1 0

0        

      

1 6 1 6 1 6 1 6 1 6 1 6





      =      

B

A

C

E

F

D

Anders O. F. Hendrickson

Google’s PageRank Algorithm

0.17 0.17 0.17 0.17 0.17 0.17

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Problem with Model 4: cycles     ~π1 = S ~e =   n   11

0 1 0 0 0 0

0 0 1 0 0 0

1 3

0 0 1 3 1 3

0

1 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 1 0

1        

      

1 6 1 6 1 6 1 6 1 6 1 6





      =      

B

A

C

E

F

D

Anders O. F. Hendrickson

Google’s PageRank Algorithm

0.22 0.17 0.17 0.06 0.22 0.17

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Problem with Model 4: cycles     ~π2 = S ~e =   n   21

0 1 0 0 0 0

0 0 1 0 0 0

1 3

0 0 1 3 1 3

0

1 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 1 0

2        

      

1 6 1 6 1 6 1 6 1 6 1 6





      =      

B

A

C

E

F

D

Anders O. F. Hendrickson

Google’s PageRank Algorithm

0.11 0.22 0.17 0.06 0.22 0.22

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Problem with Model 4: cycles     ~π3 = S ~e =   n   31

0 1 0 0 0 0

0 0 1 0 0 0

1 3

0 0 1 3 1 3

0

1 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 1 0

3        

      

1 6 1 6 1 6 1 6 1 6 1 6





      =      

B

A

C

E

F

D

Anders O. F. Hendrickson

Google’s PageRank Algorithm

0.11 0.11 0.22 0.06 0.28 0.22

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Problem with Model 4: cycles     ~π4 = S ~e =   n   41

0 1 0 0 0 0

0 0 1 0 0 0

1 3

0 0 1 3 1 3

0

1 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 1 0

4        

      

1 6 1 6 1 6 1 6 1 6 1 6





      =      

B

A

C

E

F

D

Anders O. F. Hendrickson

Google’s PageRank Algorithm

0.13 0.11 0.11 0.07 0.30 0.28

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Problem with Model 4: cycles     ~π5 = S ~e =   n   51

0 1 0 0 0 0

0 0 1 0 0 0

1 3

0 0 1 3 1 3

0

1 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 1 0

5        

      

1 6 1 6 1 6 1 6 1 6 1 6





      =      

B

A

C

E

F

D

Anders O. F. Hendrickson

Google’s PageRank Algorithm

0.11 0.13 0.11 0.04 0.31 0.30

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Problem with Model 4: cycles     ~π6 = S ~e =   n   61

0 1 0 0 0 0

0 0 1 0 0 0

1 3

0 0 1 3 1 3

0

1 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 1 0

6        

      

1 6 1 6 1 6 1 6 1 6 1 6





      =      

B

A

C

E

F

D

Anders O. F. Hendrickson

Google’s PageRank Algorithm

0.07 0.11 0.13 0.04 0.33 0.31

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Problem with Model 4: cycles     ~π7 = S ~e =   n   71

0 1 0 0 0 0

0 0 1 0 0 0

1 3

0 0 1 3 1 3

0

1 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 1 0

7        

      

1 6 1 6 1 6 1 6 1 6 1 6





      =      

B

A

C

E

F

D

Anders O. F. Hendrickson

Google’s PageRank Algorithm

0.08 0.07 0.11 0.04 0.36 0.33

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Problem with Model 4: cycles     ~π8 = S ~e =   n   81

0 1 0 0 0 0

0 0 1 0 0 0

1 3

0 0 1 3 1 3

0

1 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 1 0

8        

      

1 6 1 6 1 6 1 6 1 6 1 6





      =      

B

A

C

E

F

D

Anders O. F. Hendrickson

Google’s PageRank Algorithm

0.08 0.08 0.07 0.04 0.37 0.36

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Problem with Model 4: cycles     ~π9 = S ~e =   n   91

0 1 0 0 0 0

0 0 1 0 0 0

1 3

0 0 1 3 1 3

0

1 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 1 0

9        

      

1 6 1 6 1 6 1 6 1 6 1 6





      =      

B

A

C

E

F

D

Anders O. F. Hendrickson

Google’s PageRank Algorithm

0.06 0.08 0.08 0.02 0.38 0.37

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Problem with Model 4: cycles 

~π10

   ~e =  =S  n   10 1

0 1 0 0 0 0

0 0 1 0 0 0

1 3

0 0 1 3 1 3

0

1 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 1 0

10        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

B

A

C

E

F

D

Anders O. F. Hendrickson



Google’s PageRank Algorithm

0.05 0.06 0.08 0.03 0.40 0.38

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Problem with Model 4: cycles 

~π11

   ~e =  =S  n   11 1

0 1 0 0 0 0

0 0 1 0 0 0

1 3

0 0 1 3 1 3

0

1 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 1 0

11        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

B

A

C

E

F

D

Anders O. F. Hendrickson



Google’s PageRank Algorithm

0.05 0.05 0.06 0.03 0.41 0.40

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Problem with Model 4: cycles 

~π12

   ~e =  =S  n   12 1

0 1 0 0 0 0

0 0 1 0 0 0

1 3

0 0 1 3 1 3

0

1 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 1 0

12        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

B

A

C

E

F

D

Anders O. F. Hendrickson



Google’s PageRank Algorithm

0.05 0.05 0.05 0.02 0.42 0.41

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Problem with Model 4: cycles 

~π13

   ~e =  =S  n   13 1

0 1 0 0 0 0

0 0 1 0 0 0

1 3

0 0 1 3 1 3

0

1 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 1 0

13        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

B

A

C

E

F

D

Anders O. F. Hendrickson



Google’s PageRank Algorithm

0.04 0.05 0.05 0.02 0.43 0.42

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Problem with Model 4: cycles 

~π14

   ~e =  =S  n   14 1

0 1 0 0 0 0

0 0 1 0 0 0

1 3

0 0 1 3 1 3

0

1 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 1 0

14        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

B

A

C

E

F

D

Anders O. F. Hendrickson



Google’s PageRank Algorithm

0.03 0.04 0.05 0.02 0.44 0.43

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Problem with Model 4: cycles 

~π15

   ~e =  =S  n   15 1

0 1 0 0 0 0

0 0 1 0 0 0

1 3

0 0 1 3 1 3

0

1 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 1 0

15        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

B

A

C

E

F

D

Anders O. F. Hendrickson



Google’s PageRank Algorithm

0.03 0.03 0.04 0.02 0.44 0.44

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Problem with Model 4: cycles 

~π16

   ~e =  =S  n   16 1

0 1 0 0 0 0

0 0 1 0 0 0

1 3

0 0 1 3 1 3

0

1 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 1 0

16        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

B

A

C

E

F

D

Anders O. F. Hendrickson



Google’s PageRank Algorithm

0.03 0.03 0.03 0.01 0.45 0.44

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Problem with Model 4: cycles 

~π17

   ~e =  =S  n   17 1

0 1 0 0 0 0

0 0 1 0 0 0

1 3

0 0 1 3 1 3

0

1 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 1 0

17        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

B

A

C

E

F

D

Anders O. F. Hendrickson



Google’s PageRank Algorithm

0.02 0.03 0.03 0.01 0.45 0.45

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Problem with Model 4: cycles 

~π18

   ~e =  =S  n   18 1

0 1 0 0 0 0

0 0 1 0 0 0

1 3

0 0 1 3 1 3

0

1 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 1 0

18        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

B

A

C

E

F

D

Anders O. F. Hendrickson



Google’s PageRank Algorithm

0.02 0.02 0.03 0.01 0.46 0.45

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Problem with Model 4: cycles 

~π19

   ~e =  =S  n   19 1

0 1 0 0 0 0

0 0 1 0 0 0

1 3

0 0 1 3 1 3

0

1 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 1 0

19        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

B

A

C

E

F

D

Anders O. F. Hendrickson



Google’s PageRank Algorithm

0.02 0.02 0.02 0.01 0.46 0.46

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Problem with Model 4: cycles 

~π20

   ~e =  =S  n   20 1

0 1 0 0 0 0

0 0 1 0 0 0

1 3

0 0 1 3 1 3

0

1 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 1 0

20        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

B

A

C

E

F

D

Anders O. F. Hendrickson



Google’s PageRank Algorithm

0.02 0.02 0.02 0.01 0.47 0.46

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Problem with Model 4: cycles 

~π21

   ~e =  =S  n   21 1

0 1 0 0 0 0

0 0 1 0 0 0

1 3

0 0 1 3 1 3

0

1 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 1 0

21        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

B

A

C

E

F

D

Anders O. F. Hendrickson



Google’s PageRank Algorithm

0.02 0.02 0.02 0.01 0.47 0.47

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Problem with Model 4: cycles 

~π22

   ~e =  =S  n   22 1

0 1 0 0 0 0

0 0 1 0 0 0

1 3

0 0 1 3 1 3

0

1 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 1 0

22        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

B

A

C

E

F

D

Anders O. F. Hendrickson



Google’s PageRank Algorithm

0.01 0.02 0.02 0.01 0.47 0.47

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Problem with Model 4: cycles 

~π23

   ~e =  =S  n   23 1

0 1 0 0 0 0

0 0 1 0 0 0

1 3

0 0 1 3 1 3

0

1 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 1 0

23        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

B

A

C

E

F

D

Anders O. F. Hendrickson



Google’s PageRank Algorithm

0.01 0.01 0.02 0.01 0.48 0.47

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Problem with Model 4: cycles 

~π24

   ~e =  =S  n   24 1

0 1 0 0 0 0

0 0 1 0 0 0

1 3

0 0 1 3 1 3

0

1 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 1 0

24        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

B

A

C

E

F

D

Anders O. F. Hendrickson



Google’s PageRank Algorithm

0.01 0.01 0.01 0.01 0.48 0.48

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Problem with Model 4: cycles 

~π25

   ~e =  =S  n   25 1

0 1 0 0 0 0

0 0 1 0 0 0

1 3

0 0 1 3 1 3

0

1 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 1 0

25        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

B

A

C

E

F

D

Anders O. F. Hendrickson



Google’s PageRank Algorithm

0.01 0.01 0.01 0.00 0.48 0.48

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Problem with Model 4: cycles 

~π26

   ~e =  =S  n   26 1

0 1 0 0 0 0

0 0 1 0 0 0

1 3

0 0 1 3 1 3

0

1 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 1 0

26        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

B

A

C

E

F

D

Anders O. F. Hendrickson



Google’s PageRank Algorithm

0.01 0.01 0.01 0.00 0.48 0.48

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Problem with Model 4: cycles 

~π27

   ~e =  =S  n   27 1

0 1 0 0 0 0

0 0 1 0 0 0

1 3

0 0 1 3 1 3

0

1 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 1 0

27        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

B

A

C

E

F

D

Anders O. F. Hendrickson



Google’s PageRank Algorithm

0.01 0.01 0.01 0.00 0.49 0.48

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Problem with Model 4: cycles 

~π28

   ~e =  =S  n   28 1

0 1 0 0 0 0

0 0 1 0 0 0

1 3

0 0 1 3 1 3

0

1 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 1 0

28        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

B

A

C

E

F

D

Anders O. F. Hendrickson



Google’s PageRank Algorithm

0.01 0.01 0.01 0.00 0.49 0.49

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Problem with Model 4: cycles 

~π29

   ~e =  =S  n   29 1

0 1 0 0 0 0

0 0 1 0 0 0

1 3

0 0 1 3 1 3

0

1 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 1 0

29        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

B

A

C

E

F

D

Anders O. F. Hendrickson



Google’s PageRank Algorithm

0.01 0.01 0.01 0.00 0.49 0.49

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Problem with Model 4: cycles 

~π30

   ~e =  =S  n   30 1

0 1 0 0 0 0

0 0 1 0 0 0

1 3

0 0 1 3 1 3

0

1 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 1 0

30        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

B

A

C

E

F

D

Anders O. F. Hendrickson



Google’s PageRank Algorithm

0.01 0.01 0.01 0.00 0.49 0.49

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Problem with Model 4: cycles 

~π31

   ~e =  =S  n   31 1

0 1 0 0 0 0

0 0 1 0 0 0

1 3

0 0 1 3 1 3

0

1 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 1 0

31        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

B

A

C

E

F

D

Anders O. F. Hendrickson



Google’s PageRank Algorithm

0.00 0.01 0.01 0.00 0.49 0.49

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Problem with Model 4: cycles 

~π32

   ~e =  =S  n   32 1

0 1 0 0 0 0

0 0 1 0 0 0

1 3

0 0 1 3 1 3

0

1 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 1 0

32        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

B

A

C

E

F

D

Anders O. F. Hendrickson



Google’s PageRank Algorithm

0.00 0.00 0.01 0.00 0.49 0.49

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Problem with Model 4: cycles 

~π33

   ~e =  =S  n   33 1

0 1 0 0 0 0

0 0 1 0 0 0

1 3

0 0 1 3 1 3

0

1 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 1 0

33        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

B

A

C

E

F

D

Anders O. F. Hendrickson



Google’s PageRank Algorithm

0.00 0.00 0.00 0.00 0.49 0.49

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Problem with Model 4: cycles 

~π34

   ~e =  =S  n   34 1

0 1 0 0 0 0

0 0 1 0 0 0

1 3

0 0 1 3 1 3

0

1 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 1 0

34        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

B

A

C

E

F

D

Anders O. F. Hendrickson



Google’s PageRank Algorithm

0.00 0.00 0.00 0.00 0.49 0.49

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Problem with Model 4: cycles 

~π35

   ~e =  =S  n   35 1

0 1 0 0 0 0

0 0 1 0 0 0

1 3

0 0 1 3 1 3

0

1 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 1 0

35        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

B

A

C

E

F

D

Anders O. F. Hendrickson



Google’s PageRank Algorithm

0.00 0.00 0.00 0.00 0.49 0.49

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Problem with Model 4: cycles 

~π36

   ~e =  =S  n   36 1

0 1 0 0 0 0

0 0 1 0 0 0

1 3

0 0 1 3 1 3

0

1 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 1 0

36        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

B

A

C

E

F

D

Anders O. F. Hendrickson



Google’s PageRank Algorithm

0.00 0.00 0.00 0.00 0.49 0.49

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Problem with Model 4: cycles 

~π37

   ~e =  =S  n   37 1

0 1 0 0 0 0

0 0 1 0 0 0

1 3

0 0 1 3 1 3

0

1 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 1 0

37        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

B

A

C

E

F

D Bother. Anders O. F. Hendrickson



Google’s PageRank Algorithm

0.00 0.00 0.00 0.00 0.50 0.49

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Problem with Model 4: cycles 

~π38

   ~e =  =S  n   38 1

0 1 0 0 0 0

0 0 1 0 0 0

1 3

0 0 1 3 1 3

0

1 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 1 0

38        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

B

A

C

E

F

D Bother. Anders O. F. Hendrickson



Google’s PageRank Algorithm

0.00 0.00 0.00 0.00 0.50 0.50

       

The Problem Google Models Further thoughts

Five Models Markov Chains

Problem with Model 4: cycles 

~π39

   ~e =  =S  n   39 1

0 1 0 0 0 0

0 0 1 0 0 0

1 3

0 0 1 3 1 3

0

1 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 1 0

39        

      

1 6 1 6 1 6 1 6 1 6 1 6



      =      

B

A

C

E

F

D Bother. Anders O. F. Hendrickson



Google’s PageRank Algorithm

0.00 0.00 0.00 0.00 0.50 0.50

       

The Problem Google Models Further thoughts

Five Models Markov Chains

The Problem B

A

C

E

F

D In this graph, E and F form a cycle. Neither is dangling, but together they absorb all the prestige. The fix: give some probability of “teleporting” from all nodes, not just dangling ones. Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

The Problem B

A

C

E

F

D In this graph, E and F form a cycle. Neither is dangling, but together they absorb all the prestige. The fix: give some probability of “teleporting” from all nodes, not just dangling ones. Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

The Solution: The Google Matrix

At each turn, every node will share only α ≈ 85% of its prestige equally among the pages it links to, as it did before. The remaining 1 − α ≈ 15% of its prestige is shared equally among all nodes, just like with dangling nodes. This gives us a new matrix, the “Google matrix”   1 T ~e~e G = αS + (1 − α) . n

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

The Solution: The Google Matrix

At each turn, every node will share only α ≈ 85% of its prestige equally among the pages it links to, as it did before. The remaining 1 − α ≈ 15% of its prestige is shared equally among all nodes, just like with dangling nodes. This gives us a new matrix, the “Google matrix”   1 T ~e~e G = αS + (1 − α) . n

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

The Solution: The Google Matrix

At each turn, every node will share only α ≈ 85% of its prestige equally among the pages it links to, as it did before. The remaining 1 − α ≈ 15% of its prestige is shared equally among all nodes, just like with dangling nodes. This gives us a new matrix, the “Google matrix”   1 T ~e~e G = αS + (1 − α) . n

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 5: G = αS + (1 − α) n1 ~e~eT , α = 0.85     1 ~π0 = G0 ~e =   n  

0.17 0.17 0.17 0.17 0.17 0.17

       

B

A

C

E

F

D Splendid! It converged fast, too! Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 5: G = αS + (1 − α) n1 ~e~eT , α = 0.85     1 ~π1 = G1 ~e =   n  

0.21 0.17 0.17 0.07 0.21 0.17

       

B

A

C

E

F

D Splendid! It converged fast, too! Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 5: G = αS + (1 − α) n1 ~e~eT , α = 0.85     1 ~π2 = G2 ~e =   n  

0.13 0.21 0.17 0.07 0.21 0.21

       

B

A

C

E

F

D Splendid! It converged fast, too! Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 5: G = αS + (1 − α) n1 ~e~eT , α = 0.85     1 ~π3 = G3 ~e =   n  

0.13 0.14 0.20 0.07 0.25 0.21

       

B

A

C

E

F

D Splendid! It converged fast, too! Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 5: G = αS + (1 − α) n1 ~e~eT , α = 0.85     1 ~π4 = G4 ~e =   n  

0.14 0.14 0.14 0.08 0.26 0.24

       

B

A

C

E

F

D Splendid! It converged fast, too! Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 5: G = αS + (1 − α) n1 ~e~eT , α = 0.85     1 ~π5 = G5 ~e =   n  

0.14 0.15 0.14 0.07 0.27 0.24

       

B

A

C

E

F

D Splendid! It converged fast, too! Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 5: G = αS + (1 − α) n1 ~e~eT , α = 0.85     1 ~π6 = G6 ~e =   n  

0.12 0.14 0.15 0.07 0.27 0.25

       

B

A

C

E

F

D Splendid! It converged fast, too! Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 5: G = αS + (1 − α) n1 ~e~eT , α = 0.85     1 ~π7 = G7 ~e =   n  

0.12 0.13 0.14 0.07 0.28 0.26

       

B

A

C

E

F

D Splendid! It converged fast, too! Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 5: G = αS + (1 − α) n1 ~e~eT , α = 0.85     1 ~π8 = G8 ~e =   n  

0.12 0.13 0.13 0.07 0.28 0.26

       

B

A

C

E

F

D Splendid! It converged fast, too! Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 5: G = αS + (1 − α) n1 ~e~eT , α = 0.85     1 ~π9 = G9 ~e =   n  

0.12 0.13 0.14 0.06 0.29 0.27

       

B

A

C

E

F

D Splendid! It converged fast, too! Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 5: G = αS + (1 − α) n1 ~e~eT , α = 0.85 

~π10

   1 = G10 ~e =   n  

0.12 0.13 0.14 0.06 0.29 0.27

       

B

A

C

E

F

D Splendid! It converged fast, too! Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 5: G = αS + (1 − α) n1 ~e~eT , α = 0.85 

~π11

   1 = G11 ~e =   n  

0.12 0.12 0.13 0.06 0.29 0.27

       

B

A

C

E

F

D Splendid! It converged fast, too! Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 5: G = αS + (1 − α) n1 ~e~eT , α = 0.85 

~π12

   1 = G12 ~e =   n  

0.12 0.12 0.13 0.06 0.29 0.27

       

B

A

C

E

F

D Splendid! It converged fast, too! Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 5: G = αS + (1 − α) n1 ~e~eT , α = 0.85 

~π13

   1 = G13 ~e =   n  

0.12 0.12 0.13 0.06 0.29 0.27

       

B

A

C

E

F

D Splendid! It converged fast, too! Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 5: G = αS + (1 − α) n1 ~e~eT , α = 0.85 

~π14

   1 = G14 ~e =   n  

0.11 0.12 0.13 0.06 0.30 0.28

       

B

A

C

E

F

D Splendid! It converged fast, too! Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 5: G = αS + (1 − α) n1 ~e~eT , α = 0.85 

~π15

   1 = G15 ~e =   n  

0.11 0.12 0.13 0.06 0.30 0.28

       

B

A

C

E

F

D Splendid! It converged fast, too! Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 5: G = αS + (1 − α) n1 ~e~eT , α = 0.85 

~π16

   1 = G16 ~e =   n  

0.11 0.12 0.13 0.06 0.30 0.28

       

B

A

C

E

F

D Splendid! It converged fast, too! Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 5: G = αS + (1 − α) n1 ~e~eT , α = 0.85 

~π17

   1 = G17 ~e =   n  

0.11 0.12 0.13 0.06 0.30 0.28

       

B

A

C

E

F

D Splendid! It converged fast, too! Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 5: G = αS + (1 − α) n1 ~e~eT , α = 0.85 

~π18

   1 = G18 ~e =   n  

0.11 0.12 0.13 0.06 0.30 0.28

       

B

A

C

E

F

D Splendid! It converged fast, too! Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Model 5: G = αS + (1 − α) n1 ~e~eT , α = 0.85 

~π19

   1 = G19 ~e =   n  

0.11 0.12 0.13 0.06 0.30 0.28

       

B

A

C

E

F

D Splendid! It converged fast, too! Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Markov Chains 1 G = αS + (1 − α) ~e~eT . n This situation is called a Markov chain. Notice that G is square, every entry of G is nonnegative, and each column of G sums to 1. So G is stochastic. Every entry of G is greater than zero. This implies that G is regular. Regular Markov Chain Theorem If G is a regular stochastic matrix, then there exists a limiting vector ~π such that lim Gk ~x = ~π k →∞

for all starting vectors ~x . Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Markov Chains 1 G = αS + (1 − α) ~e~eT . n This situation is called a Markov chain. Notice that G is square, every entry of G is nonnegative, and each column of G sums to 1. So G is stochastic. Every entry of G is greater than zero. This implies that G is regular. Regular Markov Chain Theorem If G is a regular stochastic matrix, then there exists a limiting vector ~π such that lim Gk ~x = ~π k →∞

for all starting vectors ~x . Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Markov Chains 1 G = αS + (1 − α) ~e~eT . n This situation is called a Markov chain. Notice that G is square, every entry of G is nonnegative, and each column of G sums to 1. So G is stochastic. Every entry of G is greater than zero. This implies that G is regular. Regular Markov Chain Theorem If G is a regular stochastic matrix, then there exists a limiting vector ~π such that lim Gk ~x = ~π k →∞

for all starting vectors ~x . Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Markov Chains 1 G = αS + (1 − α) ~e~eT . n This situation is called a Markov chain. Notice that G is square, every entry of G is nonnegative, and each column of G sums to 1. So G is stochastic. Every entry of G is greater than zero. This implies that G is regular. Regular Markov Chain Theorem If G is a regular stochastic matrix, then there exists a limiting vector ~π such that lim Gk ~x = ~π k →∞

for all starting vectors ~x . Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Eigenvectors Another way to phrase this is in terms of eigenvectors. Definition Let G be a square matrix. If ~x is a vector and λ a scalar such that G~x = λ~x , we say λ is an eigenvalue and ~x an eigenvector for G. Facts In a regular stochastic matrix like G, G~π = ~π So the limiting vector ~π is an eigenvector with eigenvalue 1. In fact, 1 is the largest eigenvalue of G.

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Eigenvectors Another way to phrase this is in terms of eigenvectors. Definition Let G be a square matrix. If ~x is a vector and λ a scalar such that G~x = λ~x , we say λ is an eigenvalue and ~x an eigenvector for G. Facts In a regular stochastic matrix like G, G~π = ~π So the limiting vector ~π is an eigenvector with eigenvalue 1. In fact, 1 is the largest eigenvalue of G.

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Eigenvectors Another way to phrase this is in terms of eigenvectors. Definition Let G be a square matrix. If ~x is a vector and λ a scalar such that G~x = λ~x , we say λ is an eigenvalue and ~x an eigenvector for G. Facts In a regular stochastic matrix like G, G~π = ~π So the limiting vector ~π is an eigenvector with eigenvalue 1. In fact, 1 is the largest eigenvalue of G.

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Computation The matrix G is dense—all positive entries. Naively, multiplying G~π requires O(n2 ) operations. However, we never actually need G to compute: ~πk +1 = G~πk

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Computation The matrix G is dense—all positive entries. Naively, multiplying G~π requires O(n2 ) operations. However, we never actually need G to compute: ~πk +1 = G~πk

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Computation The matrix G is dense—all positive entries. Naively, multiplying G~π requires O(n2 ) operations. However, we never actually need G to compute: ~πk +1 = G~πk

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Computation The matrix G is dense—all positive entries. Naively, multiplying G~π requires O(n2 ) operations. However, we never actually need G to compute: ~πk +1 = G~πk   1 T = αS + (1 − α) ~e~e ~πk n

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Computation The matrix G is dense—all positive entries. Naively, multiplying G~π requires O(n2 ) operations. However, we never actually need G to compute: ~πk +1 = G~πk   1 T = αS + (1 − α) ~e~e ~πk n   1 T 1−α T ~ ~ ~ ~ = αH + ea + ee ~πk n n

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Computation The matrix G is dense—all positive entries. Naively, multiplying G~π requires O(n2 ) operations. However, we never actually need G to compute: ~πk +1 = G~πk   1 T = αS + (1 − α) ~e~e ~πk n   1 T 1−α T ~ ~ ~ ~ = αH + ea + ee ~πk n n 1 1−α T ~e~e ~πk = αH~πk + ~e~aT ~πk + n n

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Computation The matrix G is dense—all positive entries. Naively, multiplying G~π requires O(n2 ) operations. However, we never actually need G to compute: ~πk +1 = G~πk   1 T = αS + (1 − α) ~e~e ~πk n   1 T 1−α T ~ ~ ~ ~ = αH + ea + ee ~πk n n 1 1−α T ~e~e ~πk = αH~πk + ~e~aT ~πk + n n 1 = αH~πk + (~a · ~πk + 1 − α)~e n

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Five Models Markov Chains

Computation The matrix G is dense—all positive entries. Naively, multiplying G~π requires O(n2 ) operations. However, we never actually need G to compute: ~πk +1 = G~πk   1 T = αS + (1 − α) ~e~e ~πk n   1 T 1−α T ~ ~ ~ ~ = αH + ea + ee ~πk n n 1 1−α T ~e~e ~πk = αH~πk + ~e~aT ~πk + n n 1 = αH~πk + (~a · ~πk + 1 − α)~e n Since H is an extremely sparse matrix, this can be done in O(n) time! Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

The α factor

1 G = αS + (1 − α) ~e~eT . n The α factor needs to be carefully chosen! The higher α is, the more you’re using the hyperlink structure of the Web. So if α ≈ 0, your ranking will be nearly useless. But if α ≈ 1, your answer will Converge more slowly, and Fluctuate rapidly as links are created or destroyed.

Google has found α ≈ 0.85 to work well.

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

The α factor

1 G = αS + (1 − α) ~e~eT . n The α factor needs to be carefully chosen! The higher α is, the more you’re using the hyperlink structure of the Web. So if α ≈ 0, your ranking will be nearly useless. But if α ≈ 1, your answer will Converge more slowly, and Fluctuate rapidly as links are created or destroyed.

Google has found α ≈ 0.85 to work well.

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

The α factor

1 G = αS + (1 − α) ~e~eT . n The α factor needs to be carefully chosen! The higher α is, the more you’re using the hyperlink structure of the Web. So if α ≈ 0, your ranking will be nearly useless. But if α ≈ 1, your answer will Converge more slowly, and Fluctuate rapidly as links are created or destroyed.

Google has found α ≈ 0.85 to work well.

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

The α factor

1 G = αS + (1 − α) ~e~eT . n The α factor needs to be carefully chosen! The higher α is, the more you’re using the hyperlink structure of the Web. So if α ≈ 0, your ranking will be nearly useless. But if α ≈ 1, your answer will Converge more slowly, and Fluctuate rapidly as links are created or destroyed.

Google has found α ≈ 0.85 to work well.

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

The α factor

1 G = αS + (1 − α) ~e~eT . n The α factor needs to be carefully chosen! The higher α is, the more you’re using the hyperlink structure of the Web. So if α ≈ 0, your ranking will be nearly useless. But if α ≈ 1, your answer will Converge more slowly, and Fluctuate rapidly as links are created or destroyed.

Google has found α ≈ 0.85 to work well.

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

The α factor

1 G = αS + (1 − α) ~e~eT . n The α factor needs to be carefully chosen! The higher α is, the more you’re using the hyperlink structure of the Web. So if α ≈ 0, your ranking will be nearly useless. But if α ≈ 1, your answer will Converge more slowly, and Fluctuate rapidly as links are created or destroyed.

Google has found α ≈ 0.85 to work well.

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

The α factor

1 G = αS + (1 − α) ~e~eT . n The α factor needs to be carefully chosen! The higher α is, the more you’re using the hyperlink structure of the Web. So if α ≈ 0, your ranking will be nearly useless. But if α ≈ 1, your answer will Converge more slowly, and Fluctuate rapidly as links are created or destroyed.

Google has found α ≈ 0.85 to work well.

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Secrets

Everything I’ve said been made public by Google’s founders, Larry Page and Sergey Brin. But much of Google’s power comes from other factors: “Relevance” score Geographic targeting Personalization Plenty of secrets

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Problems with Google

Is popularity really the best way to rank websites? Example: Netflix search engine Example: Jim Jefson Example: Anders Hendrickson

Google bombs Example: French military victories

Link farms Google as “gatekeeper” to the web

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Problems with Google

Is popularity really the best way to rank websites? Example: Netflix search engine Example: Jim Jefson Example: Anders Hendrickson

Google bombs Example: French military victories

Link farms Google as “gatekeeper” to the web

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Problems with Google

Is popularity really the best way to rank websites? Example: Netflix search engine Example: Jim Jefson Example: Anders Hendrickson

Google bombs Example: French military victories

Link farms Google as “gatekeeper” to the web

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Problems with Google

Is popularity really the best way to rank websites? Example: Netflix search engine Example: Jim Jefson Example: Anders Hendrickson

Google bombs Example: French military victories

Link farms Google as “gatekeeper” to the web

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Problems with Google

Is popularity really the best way to rank websites? Example: Netflix search engine Example: Jim Jefson Example: Anders Hendrickson

Google bombs Example: French military victories

Link farms Google as “gatekeeper” to the web

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Problems with Google

Is popularity really the best way to rank websites? Example: Netflix search engine Example: Jim Jefson Example: Anders Hendrickson

Google bombs Example: French military victories

Link farms Google as “gatekeeper” to the web

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Problems with Google

Is popularity really the best way to rank websites? Example: Netflix search engine Example: Jim Jefson Example: Anders Hendrickson

Google bombs Example: French military victories

Link farms Google as “gatekeeper” to the web

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Problems with Google

Is popularity really the best way to rank websites? Example: Netflix search engine Example: Jim Jefson Example: Anders Hendrickson

Google bombs Example: French military victories

Link farms Google as “gatekeeper” to the web

Anders O. F. Hendrickson

Google’s PageRank Algorithm

The Problem Google Models Further thoughts

Questions?

Anders O. F. Hendrickson

Google’s PageRank Algorithm

References

Google’s PageRank and Beyond: The Science of Search Engine Rankings, by Amy N. Langville and Carl D. Meyer. (2006) Java applets by Tim Chartier http://mathdl.maa.org/images/upload_ library/23/chartier/google-opoly/index.html Sergey Brin, Lawrence Page, R. Motwami, and Terry Winograd. The PageRank citation ranking: Bringing order to the Web. Technical Report 1999-0120, Computer Science Department, Stanford University, 1999.

Anders O. F. Hendrickson

Google’s PageRank Algorithm