Embed Size (px)
Citation preview
The ProblemGoogle Models
Further thoughts
Google’s PageRank Algorithm
Anders O. F. Hendrickson
Concordia CollegeMoorhead, MN
Math/CS ColloquiumMarch 23, 2010
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Outline
1 The ProblemToo many resultsPrinciples behind a solution
2 Google ModelsFive ModelsMarkov Chains
3 Further thoughts
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Too many resultsPrinciples behind a solution
How do you organize an index?
This is two questions, really:How do you organize thetopics in the index?
(Alphabetically)
How do you organize thepages found for eachtopic?
(Sequentially)
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Too many resultsPrinciples behind a solution
How do you organize an index?
This is two questions, really:How do you organize thetopics in the index?
(Alphabetically)
How do you organize thepages found for eachtopic?
(Sequentially)
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Too many resultsPrinciples behind a solution
How do you organize an index?
This is two questions, really:How do you organize thetopics in the index?(Alphabetically)How do you organize thepages found for eachtopic?
(Sequentially)
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Too many resultsPrinciples behind a solution
How do you organize an index?
This is two questions, really:How do you organize thetopics in the index?(Alphabetically)How do you organize thepages found for eachtopic?
(Sequentially)
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Too many resultsPrinciples behind a solution
How do you organize an index?
This is two questions, really:How do you organize thetopics in the index?(Alphabetically)How do you organize thepages found for eachtopic? (Sequentially)
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Too many resultsPrinciples behind a solution
How do you organize a computer search?
For a computer search, youspecify the topic, so the firstquestion is irrelevant, but thesecond still matters.
Only one topic isdesired.How do you organize theentries found for a topic?
(Sequentially)
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Too many resultsPrinciples behind a solution
How do you organize a computer search?
For a computer search, youspecify the topic, so the firstquestion is irrelevant, but thesecond still matters.
Only one topic isdesired.How do you organize theentries found for a topic?
(Sequentially)
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Too many resultsPrinciples behind a solution
How do you organize a computer search?
For a computer search, youspecify the topic, so the firstquestion is irrelevant, but thesecond still matters.
Only one topic isdesired.How do you organize theentries found for a topic?
(Sequentially)
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Too many resultsPrinciples behind a solution
How do you organize a computer search?
For a computer search, youspecify the topic, so the firstquestion is irrelevant, but thesecond still matters.
Only one topic isdesired.How do you organize theentries found for a topic?(Sequentially)
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Too many resultsPrinciples behind a solution
Drawbacks to returning results sequentially
Sequential ordering isn’treally informative.Okay for rare topics, badfor common topics.Slight improvements:
SubtopicsMarkup
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Too many resultsPrinciples behind a solution
Drawbacks to returning results sequentially
Sequential ordering isn’treally informative.Okay for rare topics, badfor common topics.Slight improvements:
SubtopicsMarkup
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Too many resultsPrinciples behind a solution
Drawbacks to returning results sequentially
Sequential ordering isn’treally informative.Okay for rare topics, badfor common topics.Slight improvements:
SubtopicsMarkup
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Too many resultsPrinciples behind a solution
Drawbacks to returning results sequentially
Sequential ordering isn’treally informative.Okay for rare topics, badfor common topics.Slight improvements:
SubtopicsMarkup
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Too many resultsPrinciples behind a solution
Drawbacks to returning results sequentially
Sequential ordering isn’treally informative.Okay for rare topics, badfor common topics.Slight improvements:
SubtopicsMarkup
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Too many resultsPrinciples 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! DirectoryMarkup slightly possible—e.g. PDF vs. HTML, showing afew relevant lines
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Too many resultsPrinciples 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! DirectoryMarkup slightly possible—e.g. PDF vs. HTML, showing afew relevant lines
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Too many resultsPrinciples 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! DirectoryMarkup slightly possible—e.g. PDF vs. HTML, showing afew relevant lines
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Too many resultsPrinciples 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! DirectoryMarkup slightly possible—e.g. PDF vs. HTML, showing afew relevant lines
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Too many resultsPrinciples 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! DirectoryMarkup slightly possible—e.g. PDF vs. HTML, showing afew relevant lines
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Too many resultsPrinciples behind a solution
A solution: ask the librarian
An expert will know which books are particularly good orrelevant.Likewise, you could ask people for recommendations ofgood websites.Of course, the quality of the recommendation is only asgood as the quality of the recommender.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Too many resultsPrinciples behind a solution
How does this translate to the web?
Fundamental PrincipleEvery hyperlink on the Web is a recommendation to visitanother page.
So one way to figure out which pages are “important” or“good” is to trust the judgment of the other website authorsout there who link to it.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Too many resultsPrinciples behind a solution
How does this translate to the web?
Fundamental PrincipleEvery hyperlink on the Web is a recommendation to visitanother page.
So one way to figure out which pages are “important” or“good” is to trust the judgment of the other website authorsout there who link to it.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov 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 structuremathematically.
Directed Graph Matrix
A
B C
D E
F
A BFROM
C D E F
A 0 1 1 1 0 0B 1 0 0 0 0 0
TO C 0 1 0 0 0 0
D 0 1 0 0 1 0E 0 0 0 0 0 0F 0 0 0 1 0 0
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov 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 structuremathematically.
Directed Graph Matrix
A
B C
D E
F
A BFROM
C D E F
A 0 1 1 1 0 0B 1 0 0 0 0 0
TO C 0 1 0 0 0 0
D 0 1 0 0 1 0E 0 0 0 0 0 0F 0 0 0 1 0 0
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov 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 structuremathematically.
Directed Graph
Matrix
A
B C
D E
F
A BFROM
C D E F
A 0 1 1 1 0 0B 1 0 0 0 0 0
TO C 0 1 0 0 0 0
D 0 1 0 0 1 0E 0 0 0 0 0 0F 0 0 0 1 0 0
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov 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 structuremathematically.
Directed Graph Matrix
A
B C
D E
F
A BFROM
C D E F
A 0 1 1 1 0 0B 1 0 0 0 0 0
TO C 0 1 0 0 0 0
D 0 1 0 0 1 0E 0 0 0 0 0 0F 0 0 0 1 0 0
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 1: Counting hyperlinks
A BFROM
C D E F
A 0
+
1
+
1
+
1
+
0
+
0B 1
+
0
+
0
+
0
+
0
+
0
TO C 0
+
1
+
0
+
0
+
0
+
0D 0
+
1
+
0
+
0
+
1
+
0E 0
+
0
+
0
+
0
+
0
+
0F 0
+
0
+
0
+
1
+
0
+
0
= 3 A= 1 B= 1 C= 2 D= 0 E= 1 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 ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 1: Counting hyperlinks
A BFROM
C D E F
A
0 + 1 + 1 + 1 + 0 + 0
B
1 + 0 + 0 + 0 + 0 + 0
TO C
0 + 1 + 0 + 0 + 0 + 0
D
0 + 1 + 0 + 0 + 1 + 0
E
0 + 0 + 0 + 0 + 0 + 0
F
0 + 0 + 0 + 1 + 0 + 0
= 3 A= 1 B= 1 C= 2 D= 0 E= 1 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 ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 1: Counting hyperlinks
A BFROM
C D E F
A
0 + 1 + 1 + 1 + 0 + 0
B
1 + 0 + 0 + 0 + 0 + 0
TO C
0 + 1 + 0 + 0 + 0 + 0
D
0 + 1 + 0 + 0 + 1 + 0
E
0 + 0 + 0 + 0 + 0 + 0
F
0 + 0 + 0 + 1 + 0 + 0
= 3 A= 1 B= 1 C= 2 D= 0 E= 1 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 ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Matrix Multiplication
As it happens, this is exactly what matrix multiplication by
~e =
1...1
does!
0 1 1 1 0 01 0 0 0 0 00 1 0 0 0 00 1 0 0 1 00 0 0 0 0 00 0 0 1 0 0
111111
=
311201
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov 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 morethan E ’s recommendation of D?
A
B C
D E
F
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 2: Iteration
0 1 1 1 0 01 0 0 0 0 00 1 0 0 0 00 1 0 0 1 00 0 0 0 0 00 0 0 1 0 0
311201
=
431102
Let’s re-run our matrix multiplication, but multiplying eachrecommendation by the recommender’s previous score.This is M(M~e) = M2~e.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 2: Iteration
0 1 1 1 0 01 0 0 0 0 00 1 0 0 0 00 1 0 0 1 00 0 0 0 0 00 0 0 1 0 0
311201
=
431102
Let’s re-run our matrix multiplication, but multiplying eachrecommendation by the recommender’s previous score.This is M(M~e) = M2~e.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 2: Iteration
0 1 1 1 0 01 0 0 0 0 00 1 0 0 0 00 1 0 0 1 00 0 0 0 0 00 0 0 1 0 0
311201
=
431102
Let’s re-run our matrix multiplication, but multiplying eachrecommendation by the recommender’s previous score.This is M(M~e) = M2~e.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 2: Iteration
So let’s iterate! Let ~π0 = ~e and let ~πk+1 = M~πk .
M1~e =
0 1 1 1 0 01 0 0 0 0 00 1 0 0 0 00 1 0 0 1 00 0 0 0 0 00 0 0 1 0 0
111111
=
311201
= ~π1
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 2: Iteration
So let’s iterate! Let ~π0 = ~e and let ~πk+1 = M~πk .
M1~e =
0 1 1 1 0 01 0 0 0 0 00 1 0 0 0 00 1 0 0 1 00 0 0 0 0 00 0 0 1 0 0
111111
=
311201
= ~π1
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 2: Iteration
So let’s iterate! Let ~π0 = ~e and let ~πk+1 = M~πk .
M2~e =
0 1 1 1 0 01 0 0 0 0 00 1 0 0 0 00 1 0 0 1 00 0 0 0 0 00 0 0 1 0 0
311201
=
431102
= ~π2
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 2: Iteration
So let’s iterate! Let ~π0 = ~e and let ~πk+1 = M~πk .
M2~e =
0 1 1 1 0 01 0 0 0 0 00 1 0 0 0 00 1 0 0 1 00 0 0 0 0 00 0 0 1 0 0
311201
=
431102
= ~π2
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 2: Iteration
So let’s iterate! Let ~π0 = ~e and let ~πk+1 = M~πk .
M3~e =
0 1 1 1 0 01 0 0 0 0 00 1 0 0 0 00 1 0 0 1 00 0 0 0 0 00 0 0 1 0 0
431102
=
543301
= ~π3
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 2: Iteration
So let’s iterate! Let ~π0 = ~e and let ~πk+1 = M~πk .
M3~e =
0 1 1 1 0 01 0 0 0 0 00 1 0 0 0 00 1 0 0 1 00 0 0 0 0 00 0 0 1 0 0
431102
=
543301
= ~π3
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 2: Iteration
So let’s iterate! Let ~π0 = ~e and let ~πk+1 = M~πk .
M4~e =
0 1 1 1 0 01 0 0 0 0 00 1 0 0 0 00 1 0 0 1 00 0 0 0 0 00 0 0 1 0 0
543301
=
1054403
= ~π4
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 2: Iteration
So let’s iterate! Let ~π0 = ~e and let ~πk+1 = M~πk .
M4~e =
0 1 1 1 0 01 0 0 0 0 00 1 0 0 0 00 1 0 0 1 00 0 0 0 0 00 0 0 1 0 0
543301
=
1054403
= ~π4
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 2: Iteration
So let’s iterate! Let ~π0 = ~e and let ~πk+1 = M~πk .
M5~e =
0 1 1 1 0 01 0 0 0 0 00 1 0 0 0 00 1 0 0 1 00 0 0 0 0 00 0 0 1 0 0
1054403
=
13105504
= ~π5
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 2: Iteration
So let’s iterate! Let ~π0 = ~e and let ~πk+1 = M~πk .
M5~e =
0 1 1 1 0 01 0 0 0 0 00 1 0 0 0 00 1 0 0 1 00 0 0 0 0 00 0 0 1 0 0
1054403
=
13105504
= ~π5
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 2: Iteration
So let’s iterate! Let ~π0 = ~e and let ~πk+1 = M~πk .
M6~e =
0 1 1 1 0 01 0 0 0 0 00 1 0 0 0 00 1 0 0 1 00 0 0 0 0 00 0 0 1 0 0
13105504
=
2013101005
= ~π6
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 2: Iteration
So let’s iterate! Let ~π0 = ~e and let ~πk+1 = M~πk .
M6~e =
0 1 1 1 0 01 0 0 0 0 00 1 0 0 0 00 1 0 0 1 00 0 0 0 0 00 0 0 1 0 0
13105504
=
2013101005
= ~π6
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 2: Iteration
So let’s iterate! Let ~π0 = ~e and let ~πk+1 = M~πk .
M7~e =
0 1 1 1 0 01 0 0 0 0 00 1 0 0 0 00 1 0 0 1 00 0 0 0 0 00 0 0 1 0 0
2013101005
=
33201313010
= ~π7
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 2: Iteration
So let’s iterate! Let ~π0 = ~e and let ~πk+1 = M~πk .
M7~e =
0 1 1 1 0 01 0 0 0 0 00 1 0 0 0 00 1 0 0 1 00 0 0 0 0 00 0 0 1 0 0
2013101005
=
33201313010
= ~π7
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 2: Iteration
So let’s iterate! Let ~π0 = ~e and let ~πk+1 = M~πk .
M8~e =
0 1 1 1 0 01 0 0 0 0 00 1 0 0 0 00 1 0 0 1 00 0 0 0 0 00 0 0 1 0 0
33201313010
=
46332020013
= ~π8
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 2: Iteration
So let’s iterate! Let ~π0 = ~e and let ~πk+1 = M~πk .
M8~e =
0 1 1 1 0 01 0 0 0 0 00 1 0 0 0 00 1 0 0 1 00 0 0 0 0 00 0 0 1 0 0
33201313010
=
46332020013
= ~π8
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov 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 onthe results.
Solution: when B recommends 3 people (A, C, and D), let’smake that worth 1
3 each instead of 1.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov 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 onthe results.
Solution: when B recommends 3 people (A, C, and D), let’smake that worth 1
3 each instead of 1.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov 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 onthe results.
Solution: when B recommends 3 people (A, C, and D), let’smake that worth 1
3 each instead of 1.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov 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 onthe results.
Solution: when B recommends 3 people (A, C, and D), let’smake that worth 1
3 each instead of 1.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov 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 onthe results.
Solution: when B recommends 3 people (A, C, and D), let’smake that worth 1
3 each instead of 1.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov 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 onthe results.
Solution: when B recommends 3 people (A, C, and D), let’smake that worth 1
3 each instead of 1.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 3: hyperlink matrix H
Scaling the columns to sum to 1 gives us a hyperlink matrix H:
H =
0 1 1 1 0 01 0 0 0 0 00 1 0 0 0 00 1 0 0 1 00 0 0 0 0 00 0 0 1 0 0
.
So let’s let ~π0 = ~e and let ~πk+1 = H~πk .
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 3: hyperlink matrix H
Scaling the columns to sum to 1 gives us a hyperlink matrix H:
H =
0 13 1 1
2 0 01 0 0 0 0 00 1
3 0 0 0 00 1
3 0 0 1 00 0 0 0 0 00 0 0 1
2 0 0
.
So let’s let ~π0 = ~e and let ~πk+1 = H~πk .
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 3: hyperlink matrix H
Scaling the columns to sum to 1 gives us a hyperlink matrix H:
H =
0 13 1 1
2 0 01 0 0 0 0 00 1
3 0 0 0 00 1
3 0 0 1 00 0 0 0 0 00 0 0 1
2 0 0
.
So let’s let ~π0 = ~e and let ~πk+1 = H~πk .
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 3: hyperlink matrix H
~π0 = H0 1n~e =
0 13 1 1
2 0 01 0 0 0 0 00 1
3 0 0 0 00 1
3 0 0 1 00 0 0 0 0 00 0 0 1
2 0 0
0
161616161616
=
0.170.170.170.170.170.17
A
B C
D E
F
Problem: All the scores go to zero!Why? Because F is “dangling node.”
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 3: hyperlink matrix H
~π1 = H1 1n~e =
0 13 1 1
2 0 01 0 0 0 0 00 1
3 0 0 0 00 1
3 0 0 1 00 0 0 0 0 00 0 0 1
2 0 0
1
161616161616
=
0.310.170.060.220.000.08
A
B C
D E
F
Problem: All the scores go to zero!Why? Because F is “dangling node.”
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 3: hyperlink matrix H
~π2 = H2 1n~e =
0 13 1 1
2 0 01 0 0 0 0 00 1
3 0 0 0 00 1
3 0 0 1 00 0 0 0 0 00 0 0 1
2 0 0
2
161616161616
=
0.220.310.060.060.000.11
A
B C
D E
F
Problem: All the scores go to zero!Why? Because F is “dangling node.”
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 3: hyperlink matrix H
~π3 = H3 1n~e =
0 13 1 1
2 0 01 0 0 0 0 00 1
3 0 0 0 00 1
3 0 0 1 00 0 0 0 0 00 0 0 1
2 0 0
3
161616161616
=
0.190.220.100.100.000.03
A
B C
D E
F
Problem: All the scores go to zero!Why? Because F is “dangling node.”
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 3: hyperlink matrix H
~π4 = H4 1n~e =
0 13 1 1
2 0 01 0 0 0 0 00 1
3 0 0 0 00 1
3 0 0 1 00 0 0 0 0 00 0 0 1
2 0 0
4
161616161616
=
0.230.190.070.070.000.05
A
B C
D E
F
Problem: All the scores go to zero!Why? Because F is “dangling node.”
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 3: hyperlink matrix H
~π5 = H5 1n~e =
0 13 1 1
2 0 01 0 0 0 0 00 1
3 0 0 0 00 1
3 0 0 1 00 0 0 0 0 00 0 0 1
2 0 0
5
161616161616
=
0.170.230.060.060.000.04
A
B C
D E
F
Problem: All the scores go to zero!Why? Because F is “dangling node.”
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 3: hyperlink matrix H
~π6 = H6 1n~e =
0 13 1 1
2 0 01 0 0 0 0 00 1
3 0 0 0 00 1
3 0 0 1 00 0 0 0 0 00 0 0 1
2 0 0
6
161616161616
=
0.170.170.080.080.000.03
A
B C
D E
F
Problem: All the scores go to zero!Why? Because F is “dangling node.”
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 3: hyperlink matrix H
~π7 = H7 1n~e =
0 13 1 1
2 0 01 0 0 0 0 00 1
3 0 0 0 00 1
3 0 0 1 00 0 0 0 0 00 0 0 1
2 0 0
7
161616161616
=
0.170.170.060.060.000.04
A
B C
D E
F
Problem: All the scores go to zero!Why? Because F is “dangling node.”
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 3: hyperlink matrix H
~π8 = H8 1n~e =
0 13 1 1
2 0 01 0 0 0 0 00 1
3 0 0 0 00 1
3 0 0 1 00 0 0 0 0 00 0 0 1
2 0 0
8
161616161616
=
0.140.170.060.060.000.03
A
B C
D E
F
Problem: All the scores go to zero!Why? Because F is “dangling node.”
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 3: hyperlink matrix H
~π9 = H9 1n~e =
0 13 1 1
2 0 01 0 0 0 0 00 1
3 0 0 0 00 1
3 0 0 1 00 0 0 0 0 00 0 0 1
2 0 0
9
161616161616
=
0.140.140.060.060.000.03
A
B C
D E
F
Problem: All the scores go to zero!Why? Because F is “dangling node.”
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 3: hyperlink matrix H
~π10 = H10 1n~e =
0 13 1 1
2 0 01 0 0 0 0 00 1
3 0 0 0 00 1
3 0 0 1 00 0 0 0 0 00 0 0 1
2 0 0
10
161616161616
=
0.130.140.050.050.000.03
A
B C
D E
F
Problem: All the scores go to zero!Why? Because F is “dangling node.”
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 3: hyperlink matrix H
~π11 = H11 1n~e =
0 13 1 1
2 0 01 0 0 0 0 00 1
3 0 0 0 00 1
3 0 0 1 00 0 0 0 0 00 0 0 1
2 0 0
11
161616161616
=
0.120.130.050.050.000.02
A
B C
D E
F
Problem: All the scores go to zero!Why? Because F is “dangling node.”
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 3: hyperlink matrix H
~π12 = H12 1n~e =
0 13 1 1
2 0 01 0 0 0 0 00 1
3 0 0 0 00 1
3 0 0 1 00 0 0 0 0 00 0 0 1
2 0 0
12
161616161616
=
0.110.120.040.040.000.02
A
B C
D E
F
Problem: All the scores go to zero!Why? Because F is “dangling node.”
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 3: hyperlink matrix H
~π13 = H13 1n~e =
0 13 1 1
2 0 01 0 0 0 0 00 1
3 0 0 0 00 1
3 0 0 1 00 0 0 0 0 00 0 0 1
2 0 0
13
161616161616
=
0.110.110.040.040.000.02
A
B C
D E
F
Problem: All the scores go to zero!Why? Because F is “dangling node.”
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 3: hyperlink matrix H
~π14 = H14 1n~e =
0 13 1 1
2 0 01 0 0 0 0 00 1
3 0 0 0 00 1
3 0 0 1 00 0 0 0 0 00 0 0 1
2 0 0
14
161616161616
=
0.100.110.040.040.000.02
A
B C
D E
F
Problem: All the scores go to zero!Why? Because F is “dangling node.”
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 3: hyperlink matrix H
~π15 = H15 1n~e =
0 13 1 1
2 0 01 0 0 0 0 00 1
3 0 0 0 00 1
3 0 0 1 00 0 0 0 0 00 0 0 1
2 0 0
15
161616161616
=
0.090.100.040.040.000.02
A
B C
D E
F
Problem: All the scores go to zero!Why? Because F is “dangling node.”
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 3: hyperlink matrix H
~π16 = H16 1n~e =
0 13 1 1
2 0 01 0 0 0 0 00 1
3 0 0 0 00 1
3 0 0 1 00 0 0 0 0 00 0 0 1
2 0 0
16
161616161616
=
0.090.090.030.030.000.02
A
B C
D E
F
Problem: All the scores go to zero!Why? Because F is “dangling node.”
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 3: hyperlink matrix H
~π17 = H17 1n~e =
0 13 1 1
2 0 01 0 0 0 0 00 1
3 0 0 0 00 1
3 0 0 1 00 0 0 0 0 00 0 0 1
2 0 0
17
161616161616
=
0.080.090.030.030.000.02
A
B C
D E
F
Problem: All the scores go to zero!Why? Because F is “dangling node.”
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 3: hyperlink matrix H
~π18 = H18 1n~e =
0 13 1 1
2 0 01 0 0 0 0 00 1
3 0 0 0 00 1
3 0 0 1 00 0 0 0 0 00 0 0 1
2 0 0
18
161616161616
=
0.060.080.030.030.000.02
A
B C
D E
F
Problem: All the scores go to zero!Why? Because F is “dangling node.”
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 3: hyperlink matrix H
~π19 = H19 1n~e =
0 13 1 1
2 0 01 0 0 0 0 00 1
3 0 0 0 00 1
3 0 0 1 00 0 0 0 0 00 0 0 1
2 0 0
19
161616161616
=
0.060.060.030.030.000.01
A
B C
D E
F
Problem: All the scores go to zero!Why? Because F is “dangling node.”
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 3: hyperlink matrix H
~π20 = H20 1n~e =
0 13 1 1
2 0 01 0 0 0 0 00 1
3 0 0 0 00 1
3 0 0 1 00 0 0 0 0 00 0 0 1
2 0 0
20
161616161616
=
0.050.060.020.020.000.01
A
B C
D E
F
Problem: All the scores go to zero!
Why? Because F is “dangling node.”
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 3: hyperlink matrix H
~π21 = H21 1n~e =
0 13 1 1
2 0 01 0 0 0 0 00 1
3 0 0 0 00 1
3 0 0 1 00 0 0 0 0 00 0 0 1
2 0 0
21
161616161616
=
0.050.050.020.020.000.01
A
B C
D E
F
Problem: All the scores go to zero!Why? Because F is “dangling node.”
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov 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 13 1 1
2 0 01 0 0 0 0 00 1
3 0 0 0 00 1
3 0 0 1 00 0 0 0 0 00 0 0 1
2 0 0
+
0 0 0 0 0 16
0 0 0 0 0 16
0 0 0 0 0 16
0 0 0 0 0 16
0 0 0 0 0 16
0 0 0 0 0 16
= H +16
111111
(
0 0 0 0 0 1)
= H +1n~e~aT .
where ~a is a vector whose 1’s indicate dangling nodes.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov 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 13 1 1
2 0 16
1 0 0 0 0 16
0 13 0 0 0 1
60 1
3 0 0 1 16
0 0 0 0 0 16
0 0 0 12 0 1
6
+
0 0 0 0 0 16
0 0 0 0 0 16
0 0 0 0 0 16
0 0 0 0 0 16
0 0 0 0 0 16
0 0 0 0 0 16
= H +16
111111
(
0 0 0 0 0 1)
= H +1n~e~aT .
where ~a is a vector whose 1’s indicate dangling nodes.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov 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 13 1 1
2 0 01 0 0 0 0 00 1
3 0 0 0 00 1
3 0 0 1 00 0 0 0 0 00 0 0 1
2 0 0
+
0 0 0 0 0 16
0 0 0 0 0 16
0 0 0 0 0 16
0 0 0 0 0 16
0 0 0 0 0 16
0 0 0 0 0 16
= H +16
111111
(
0 0 0 0 0 1)
= H +1n~e~aT .
where ~a is a vector whose 1’s indicate dangling nodes.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov 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 13 1 1
2 0 01 0 0 0 0 00 1
3 0 0 0 00 1
3 0 0 1 00 0 0 0 0 00 0 0 1
2 0 0
+
0 0 0 0 0 16
0 0 0 0 0 16
0 0 0 0 0 16
0 0 0 0 0 16
0 0 0 0 0 16
0 0 0 0 0 16
= H +16
111111
(
0 0 0 0 0 1)
= H +1n~e~aT .
where ~a is a vector whose 1’s indicate dangling nodes.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov 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 13 1 1
2 0 01 0 0 0 0 00 1
3 0 0 0 00 1
3 0 0 1 00 0 0 0 0 00 0 0 1
2 0 0
+
0 0 0 0 0 16
0 0 0 0 0 16
0 0 0 0 0 16
0 0 0 0 0 16
0 0 0 0 0 16
0 0 0 0 0 16
= H +16
111111
(
0 0 0 0 0 1)
= H +1n~e~aT .
where ~a is a vector whose 1’s indicate dangling nodes.Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 4: S = H + 1n~e~aT
~π0 = S0 1n~e =
0 13 1 1
2 0 16
1 0 0 0 0 16
0 13 0 0 0 1
60 1
3 0 0 1 16
0 0 0 0 0 16
0 0 0 12 0 1
6
0
161616161616
=
0.170.170.170.170.170.17
A
B C
D E
F
Hooray!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 4: S = H + 1n~e~aT
~π1 = S1 1n~e =
0 13 1 1
2 0 16
1 0 0 0 0 16
0 13 0 0 0 1
60 1
3 0 0 1 16
0 0 0 0 0 16
0 0 0 12 0 1
6
1
161616161616
=
0.330.190.080.250.030.11
A
B C
D E
F
Hooray!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 4: S = H + 1n~e~aT
~π2 = S2 1n~e =
0 13 1 1
2 0 16
1 0 0 0 0 16
0 13 0 0 0 1
60 1
3 0 0 1 16
0 0 0 0 0 16
0 0 0 12 0 1
6
2
161616161616
=
0.290.350.080.110.020.14
A
B C
D E
F
Hooray!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 4: S = H + 1n~e~aT
~π3 = S3 1n~e =
0 13 1 1
2 0 16
1 0 0 0 0 16
0 13 0 0 0 1
60 1
3 0 0 1 16
0 0 0 0 0 16
0 0 0 12 0 1
6
3
161616161616
=
0.280.320.140.160.020.08
A
B C
D E
F
Hooray!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 4: S = H + 1n~e~aT
~π4 = S4 1n~e =
0 13 1 1
2 0 16
1 0 0 0 0 16
0 13 0 0 0 1
60 1
3 0 0 1 16
0 0 0 0 0 16
0 0 0 12 0 1
6
4
161616161616
=
0.340.290.120.140.010.09
A
B C
D E
F
Hooray!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 4: S = H + 1n~e~aT
~π5 = S5 1n~e =
0 13 1 1
2 0 16
1 0 0 0 0 16
0 13 0 0 0 1
60 1
3 0 0 1 16
0 0 0 0 0 16
0 0 0 12 0 1
6
5
161616161616
=
0.300.360.110.130.020.09
A
B C
D E
F
Hooray!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 4: S = H + 1n~e~aT
~π6 = S6 1n~e =
0 13 1 1
2 0 16
1 0 0 0 0 16
0 13 0 0 0 1
60 1
3 0 0 1 16
0 0 0 0 0 16
0 0 0 12 0 1
6
6
161616161616
=
0.310.320.130.150.010.08
A
B C
D E
F
Hooray!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 4: S = H + 1n~e~aT
~π7 = S7 1n~e =
0 13 1 1
2 0 16
1 0 0 0 0 16
0 13 0 0 0 1
60 1
3 0 0 1 16
0 0 0 0 0 16
0 0 0 12 0 1
6
7
161616161616
=
0.330.320.120.130.010.09
A
B C
D E
F
Hooray!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 4: S = H + 1n~e~aT
~π8 = S8 1n~e =
0 13 1 1
2 0 16
1 0 0 0 0 16
0 13 0 0 0 1
60 1
3 0 0 1 16
0 0 0 0 0 16
0 0 0 12 0 1
6
8
161616161616
=
0.310.340.120.130.010.08
A
B C
D E
F
Hooray!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 4: S = H + 1n~e~aT
~π9 = S9 1n~e =
0 13 1 1
2 0 16
1 0 0 0 0 16
0 13 0 0 0 1
60 1
3 0 0 1 16
0 0 0 0 0 16
0 0 0 12 0 1
6
9
161616161616
=
0.320.320.130.140.010.08
A
B C
D E
F
Hooray!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 4: S = H + 1n~e~aT
~π10 = S10 1n~e =
0 13 1 1
2 0 16
1 0 0 0 0 16
0 13 0 0 0 1
60 1
3 0 0 1 16
0 0 0 0 0 16
0 0 0 12 0 1
6
10
161616161616
=
0.320.330.120.130.010.08
A
B C
D E
F
Hooray!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 4: S = H + 1n~e~aT
~π11 = S11 1n~e =
0 13 1 1
2 0 16
1 0 0 0 0 16
0 13 0 0 0 1
60 1
3 0 0 1 16
0 0 0 0 0 16
0 0 0 12 0 1
6
11
161616161616
=
0.310.330.120.140.010.08
A
B C
D E
F
Hooray!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 4: S = H + 1n~e~aT
~π12 = S12 1n~e =
0 13 1 1
2 0 16
1 0 0 0 0 16
0 13 0 0 0 1
60 1
3 0 0 1 16
0 0 0 0 0 16
0 0 0 12 0 1
6
12
161616161616
=
0.320.320.120.140.010.08
A
B C
D E
F
Hooray!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 4: S = H + 1n~e~aT
~π13 = S13 1n~e =
0 13 1 1
2 0 16
1 0 0 0 0 16
0 13 0 0 0 1
60 1
3 0 0 1 16
0 0 0 0 0 16
0 0 0 12 0 1
6
13
161616161616
=
0.320.330.120.140.010.08
A
B C
D E
F
Hooray!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 4: S = H + 1n~e~aT
~π14 = S14 1n~e =
0 13 1 1
2 0 16
1 0 0 0 0 16
0 13 0 0 0 1
60 1
3 0 0 1 16
0 0 0 0 0 16
0 0 0 12 0 1
6
14
161616161616
=
0.310.330.120.140.010.08
A
B C
D E
F
Hooray!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 4: S = H + 1n~e~aT
~π15 = S15 1n~e =
0 13 1 1
2 0 16
1 0 0 0 0 16
0 13 0 0 0 1
60 1
3 0 0 1 16
0 0 0 0 0 16
0 0 0 12 0 1
6
15
161616161616
=
0.320.330.120.140.010.08
A
B C
D E
F
Hooray!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 4: S = H + 1n~e~aT
~π16 = S16 1n~e =
0 13 1 1
2 0 16
1 0 0 0 0 16
0 13 0 0 0 1
60 1
3 0 0 1 16
0 0 0 0 0 16
0 0 0 12 0 1
6
16
161616161616
=
0.310.330.120.140.010.08
A
B C
D E
F
Hooray!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 4: S = H + 1n~e~aT
~π17 = S17 1n~e =
0 13 1 1
2 0 16
1 0 0 0 0 16
0 13 0 0 0 1
60 1
3 0 0 1 16
0 0 0 0 0 16
0 0 0 12 0 1
6
17
161616161616
=
0.310.330.120.140.010.08
A
B C
D E
F
Hooray!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 4: S = H + 1n~e~aT
~π18 = S18 1n~e =
0 13 1 1
2 0 16
1 0 0 0 0 16
0 13 0 0 0 1
60 1
3 0 0 1 16
0 0 0 0 0 16
0 0 0 12 0 1
6
18
161616161616
=
0.320.330.120.140.010.08
A
B C
D E
F
Hooray!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 4: S = H + 1n~e~aT
~π19 = S19 1n~e =
0 13 1 1
2 0 16
1 0 0 0 0 16
0 13 0 0 0 1
60 1
3 0 0 1 16
0 0 0 0 0 16
0 0 0 12 0 1
6
19
161616161616
=
0.310.330.120.140.010.08
A
B C
D E
F
Hooray!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 4: S = H + 1n~e~aT
~π20 = S20 1n~e =
0 13 1 1
2 0 16
1 0 0 0 0 16
0 13 0 0 0 1
60 1
3 0 0 1 16
0 0 0 0 0 16
0 0 0 12 0 1
6
20
161616161616
=
0.320.330.120.140.010.08
A
B C
D E
F
Hooray!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 4: S = H + 1n~e~aT
~π21 = S21 1n~e =
0 13 1 1
2 0 16
1 0 0 0 0 16
0 13 0 0 0 1
60 1
3 0 0 1 16
0 0 0 0 0 16
0 0 0 12 0 1
6
21
161616161616
=
0.320.330.120.140.010.08
A
B C
D E
F
Hooray!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 4: S = H + 1n~e~aT
~π22 = S22 1n~e =
0 13 1 1
2 0 16
1 0 0 0 0 16
0 13 0 0 0 1
60 1
3 0 0 1 16
0 0 0 0 0 16
0 0 0 12 0 1
6
22
161616161616
=
0.310.330.120.140.010.08
A
B C
D E
F
Hooray!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 4: S = H + 1n~e~aT
~π23 = S23 1n~e =
0 13 1 1
2 0 16
1 0 0 0 0 16
0 13 0 0 0 1
60 1
3 0 0 1 16
0 0 0 0 0 16
0 0 0 12 0 1
6
23
161616161616
=
0.320.330.120.140.010.08
A
B C
D E
F
Hooray!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 4: S = H + 1n~e~aT
~π24 = S24 1n~e =
0 13 1 1
2 0 16
1 0 0 0 0 16
0 13 0 0 0 1
60 1
3 0 0 1 16
0 0 0 0 0 16
0 0 0 12 0 1
6
24
161616161616
=
0.320.330.120.140.010.08
A
B C
D E
F
Hooray!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 4: S = H + 1n~e~aT
~π25 = S25 1n~e =
0 13 1 1
2 0 16
1 0 0 0 0 16
0 13 0 0 0 1
60 1
3 0 0 1 16
0 0 0 0 0 16
0 0 0 12 0 1
6
25
161616161616
=
0.320.330.120.140.010.08
A
B C
D E
F
Hooray!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 4: S = H + 1n~e~aT
~π26 = S26 1n~e =
0 13 1 1
2 0 16
1 0 0 0 0 16
0 13 0 0 0 1
60 1
3 0 0 1 16
0 0 0 0 0 16
0 0 0 12 0 1
6
26
161616161616
=
0.320.330.120.140.010.08
A
B C
D E
F
Hooray!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Problem with Model 4: cycles
~π0 = S0 1n~e =
0 0 13 1 0 0
1 0 0 0 0 00 1 0 0 0 00 0 1
3 0 0 00 0 1
3 0 0 10 0 0 0 1 0
0
161616161616
=
0.170.170.170.170.170.17
A
B
C
D
E F
Bother.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Problem with Model 4: cycles
~π1 = S1 1n~e =
0 0 13 1 0 0
1 0 0 0 0 00 1 0 0 0 00 0 1
3 0 0 00 0 1
3 0 0 10 0 0 0 1 0
1
161616161616
=
0.220.170.170.060.220.17
A
B
C
D
E F
Bother.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Problem with Model 4: cycles
~π2 = S2 1n~e =
0 0 13 1 0 0
1 0 0 0 0 00 1 0 0 0 00 0 1
3 0 0 00 0 1
3 0 0 10 0 0 0 1 0
2
161616161616
=
0.110.220.170.060.220.22
A
B
C
D
E F
Bother.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Problem with Model 4: cycles
~π3 = S3 1n~e =
0 0 13 1 0 0
1 0 0 0 0 00 1 0 0 0 00 0 1
3 0 0 00 0 1
3 0 0 10 0 0 0 1 0
3
161616161616
=
0.110.110.220.060.280.22
A
B
C
D
E F
Bother.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Problem with Model 4: cycles
~π4 = S4 1n~e =
0 0 13 1 0 0
1 0 0 0 0 00 1 0 0 0 00 0 1
3 0 0 00 0 1
3 0 0 10 0 0 0 1 0
4
161616161616
=
0.130.110.110.070.300.28
A
B
C
D
E F
Bother.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Problem with Model 4: cycles
~π5 = S5 1n~e =
0 0 13 1 0 0
1 0 0 0 0 00 1 0 0 0 00 0 1
3 0 0 00 0 1
3 0 0 10 0 0 0 1 0
5
161616161616
=
0.110.130.110.040.310.30
A
B
C
D
E F
Bother.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Problem with Model 4: cycles
~π6 = S6 1n~e =
0 0 13 1 0 0
1 0 0 0 0 00 1 0 0 0 00 0 1
3 0 0 00 0 1
3 0 0 10 0 0 0 1 0
6
161616161616
=
0.070.110.130.040.330.31
A
B
C
D
E F
Bother.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Problem with Model 4: cycles
~π7 = S7 1n~e =
0 0 13 1 0 0
1 0 0 0 0 00 1 0 0 0 00 0 1
3 0 0 00 0 1
3 0 0 10 0 0 0 1 0
7
161616161616
=
0.080.070.110.040.360.33
A
B
C
D
E F
Bother.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Problem with Model 4: cycles
~π8 = S8 1n~e =
0 0 13 1 0 0
1 0 0 0 0 00 1 0 0 0 00 0 1
3 0 0 00 0 1
3 0 0 10 0 0 0 1 0
8
161616161616
=
0.080.080.070.040.370.36
A
B
C
D
E F
Bother.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Problem with Model 4: cycles
~π9 = S9 1n~e =
0 0 13 1 0 0
1 0 0 0 0 00 1 0 0 0 00 0 1
3 0 0 00 0 1
3 0 0 10 0 0 0 1 0
9
161616161616
=
0.060.080.080.020.380.37
A
B
C
D
E F
Bother.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Problem with Model 4: cycles
~π10 = S10 1n~e =
0 0 13 1 0 0
1 0 0 0 0 00 1 0 0 0 00 0 1
3 0 0 00 0 1
3 0 0 10 0 0 0 1 0
10
161616161616
=
0.050.060.080.030.400.38
A
B
C
D
E F
Bother.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Problem with Model 4: cycles
~π11 = S11 1n~e =
0 0 13 1 0 0
1 0 0 0 0 00 1 0 0 0 00 0 1
3 0 0 00 0 1
3 0 0 10 0 0 0 1 0
11
161616161616
=
0.050.050.060.030.410.40
A
B
C
D
E F
Bother.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Problem with Model 4: cycles
~π12 = S12 1n~e =
0 0 13 1 0 0
1 0 0 0 0 00 1 0 0 0 00 0 1
3 0 0 00 0 1
3 0 0 10 0 0 0 1 0
12
161616161616
=
0.050.050.050.020.420.41
A
B
C
D
E F
Bother.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Problem with Model 4: cycles
~π13 = S13 1n~e =
0 0 13 1 0 0
1 0 0 0 0 00 1 0 0 0 00 0 1
3 0 0 00 0 1
3 0 0 10 0 0 0 1 0
13
161616161616
=
0.040.050.050.020.430.42
A
B
C
D
E F
Bother.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Problem with Model 4: cycles
~π14 = S14 1n~e =
0 0 13 1 0 0
1 0 0 0 0 00 1 0 0 0 00 0 1
3 0 0 00 0 1
3 0 0 10 0 0 0 1 0
14
161616161616
=
0.030.040.050.020.440.43
A
B
C
D
E F
Bother.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Problem with Model 4: cycles
~π15 = S15 1n~e =
0 0 13 1 0 0
1 0 0 0 0 00 1 0 0 0 00 0 1
3 0 0 00 0 1
3 0 0 10 0 0 0 1 0
15
161616161616
=
0.030.030.040.020.440.44
A
B
C
D
E F
Bother.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Problem with Model 4: cycles
~π16 = S16 1n~e =
0 0 13 1 0 0
1 0 0 0 0 00 1 0 0 0 00 0 1
3 0 0 00 0 1
3 0 0 10 0 0 0 1 0
16
161616161616
=
0.030.030.030.010.450.44
A
B
C
D
E F
Bother.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Problem with Model 4: cycles
~π17 = S17 1n~e =
0 0 13 1 0 0
1 0 0 0 0 00 1 0 0 0 00 0 1
3 0 0 00 0 1
3 0 0 10 0 0 0 1 0
17
161616161616
=
0.020.030.030.010.450.45
A
B
C
D
E F
Bother.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Problem with Model 4: cycles
~π18 = S18 1n~e =
0 0 13 1 0 0
1 0 0 0 0 00 1 0 0 0 00 0 1
3 0 0 00 0 1
3 0 0 10 0 0 0 1 0
18
161616161616
=
0.020.020.030.010.460.45
A
B
C
D
E F
Bother.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Problem with Model 4: cycles
~π19 = S19 1n~e =
0 0 13 1 0 0
1 0 0 0 0 00 1 0 0 0 00 0 1
3 0 0 00 0 1
3 0 0 10 0 0 0 1 0
19
161616161616
=
0.020.020.020.010.460.46
A
B
C
D
E F
Bother.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Problem with Model 4: cycles
~π20 = S20 1n~e =
0 0 13 1 0 0
1 0 0 0 0 00 1 0 0 0 00 0 1
3 0 0 00 0 1
3 0 0 10 0 0 0 1 0
20
161616161616
=
0.020.020.020.010.470.46
A
B
C
D
E F
Bother.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Problem with Model 4: cycles
~π21 = S21 1n~e =
0 0 13 1 0 0
1 0 0 0 0 00 1 0 0 0 00 0 1
3 0 0 00 0 1
3 0 0 10 0 0 0 1 0
21
161616161616
=
0.020.020.020.010.470.47
A
B
C
D
E F
Bother.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Problem with Model 4: cycles
~π22 = S22 1n~e =
0 0 13 1 0 0
1 0 0 0 0 00 1 0 0 0 00 0 1
3 0 0 00 0 1
3 0 0 10 0 0 0 1 0
22
161616161616
=
0.010.020.020.010.470.47
A
B
C
D
E F
Bother.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Problem with Model 4: cycles
~π23 = S23 1n~e =
0 0 13 1 0 0
1 0 0 0 0 00 1 0 0 0 00 0 1
3 0 0 00 0 1
3 0 0 10 0 0 0 1 0
23
161616161616
=
0.010.010.020.010.480.47
A
B
C
D
E F
Bother.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Problem with Model 4: cycles
~π24 = S24 1n~e =
0 0 13 1 0 0
1 0 0 0 0 00 1 0 0 0 00 0 1
3 0 0 00 0 1
3 0 0 10 0 0 0 1 0
24
161616161616
=
0.010.010.010.010.480.48
A
B
C
D
E F
Bother.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Problem with Model 4: cycles
~π25 = S25 1n~e =
0 0 13 1 0 0
1 0 0 0 0 00 1 0 0 0 00 0 1
3 0 0 00 0 1
3 0 0 10 0 0 0 1 0
25
161616161616
=
0.010.010.010.000.480.48
A
B
C
D
E F
Bother.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Problem with Model 4: cycles
~π26 = S26 1n~e =
0 0 13 1 0 0
1 0 0 0 0 00 1 0 0 0 00 0 1
3 0 0 00 0 1
3 0 0 10 0 0 0 1 0
26
161616161616
=
0.010.010.010.000.480.48
A
B
C
D
E F
Bother.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Problem with Model 4: cycles
~π27 = S27 1n~e =
0 0 13 1 0 0
1 0 0 0 0 00 1 0 0 0 00 0 1
3 0 0 00 0 1
3 0 0 10 0 0 0 1 0
27
161616161616
=
0.010.010.010.000.490.48
A
B
C
D
E F
Bother.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Problem with Model 4: cycles
~π28 = S28 1n~e =
0 0 13 1 0 0
1 0 0 0 0 00 1 0 0 0 00 0 1
3 0 0 00 0 1
3 0 0 10 0 0 0 1 0
28
161616161616
=
0.010.010.010.000.490.49
A
B
C
D
E F
Bother.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Problem with Model 4: cycles
~π29 = S29 1n~e =
0 0 13 1 0 0
1 0 0 0 0 00 1 0 0 0 00 0 1
3 0 0 00 0 1
3 0 0 10 0 0 0 1 0
29
161616161616
=
0.010.010.010.000.490.49
A
B
C
D
E F
Bother.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Problem with Model 4: cycles
~π30 = S30 1n~e =
0 0 13 1 0 0
1 0 0 0 0 00 1 0 0 0 00 0 1
3 0 0 00 0 1
3 0 0 10 0 0 0 1 0
30
161616161616
=
0.010.010.010.000.490.49
A
B
C
D
E F
Bother.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Problem with Model 4: cycles
~π31 = S31 1n~e =
0 0 13 1 0 0
1 0 0 0 0 00 1 0 0 0 00 0 1
3 0 0 00 0 1
3 0 0 10 0 0 0 1 0
31
161616161616
=
0.000.010.010.000.490.49
A
B
C
D
E F
Bother.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Problem with Model 4: cycles
~π32 = S32 1n~e =
0 0 13 1 0 0
1 0 0 0 0 00 1 0 0 0 00 0 1
3 0 0 00 0 1
3 0 0 10 0 0 0 1 0
32
161616161616
=
0.000.000.010.000.490.49
A
B
C
D
E F
Bother.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Problem with Model 4: cycles
~π33 = S33 1n~e =
0 0 13 1 0 0
1 0 0 0 0 00 1 0 0 0 00 0 1
3 0 0 00 0 1
3 0 0 10 0 0 0 1 0
33
161616161616
=
0.000.000.000.000.490.49
A
B
C
D
E F
Bother.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Problem with Model 4: cycles
~π34 = S34 1n~e =
0 0 13 1 0 0
1 0 0 0 0 00 1 0 0 0 00 0 1
3 0 0 00 0 1
3 0 0 10 0 0 0 1 0
34
161616161616
=
0.000.000.000.000.490.49
A
B
C
D
E F
Bother.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Problem with Model 4: cycles
~π35 = S35 1n~e =
0 0 13 1 0 0
1 0 0 0 0 00 1 0 0 0 00 0 1
3 0 0 00 0 1
3 0 0 10 0 0 0 1 0
35
161616161616
=
0.000.000.000.000.490.49
A
B
C
D
E F
Bother.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Problem with Model 4: cycles
~π36 = S36 1n~e =
0 0 13 1 0 0
1 0 0 0 0 00 1 0 0 0 00 0 1
3 0 0 00 0 1
3 0 0 10 0 0 0 1 0
36
161616161616
=
0.000.000.000.000.490.49
A
B
C
D
E F
Bother.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Problem with Model 4: cycles
~π37 = S37 1n~e =
0 0 13 1 0 0
1 0 0 0 0 00 1 0 0 0 00 0 1
3 0 0 00 0 1
3 0 0 10 0 0 0 1 0
37
161616161616
=
0.000.000.000.000.500.49
A
B
C
D
E F
Bother.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Problem with Model 4: cycles
~π38 = S38 1n~e =
0 0 13 1 0 0
1 0 0 0 0 00 1 0 0 0 00 0 1
3 0 0 00 0 1
3 0 0 10 0 0 0 1 0
38
161616161616
=
0.000.000.000.000.500.50
A
B
C
D
E F
Bother.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Problem with Model 4: cycles
~π39 = S39 1n~e =
0 0 13 1 0 0
1 0 0 0 0 00 1 0 0 0 00 0 1
3 0 0 00 0 1
3 0 0 10 0 0 0 1 0
39
161616161616
=
0.000.000.000.000.500.50
A
B
C
D
E F
Bother.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
The Problem
A
B
C
D
E F
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 allnodes, not just dangling ones.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
The Problem
A
B
C
D
E F
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 allnodes, not just dangling ones.
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
The Solution: The Google Matrix
At each turn, every node will share only α ≈ 85% of itsprestige equally among the pages it links to, as it didbefore.The remaining 1− α ≈ 15% of its prestige is sharedequally among all nodes, just like with dangling nodes.This gives us a new matrix, the “Google matrix”
G = αS + (1− α)
(1n~e~eT
).
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
The Solution: The Google Matrix
At each turn, every node will share only α ≈ 85% of itsprestige equally among the pages it links to, as it didbefore.The remaining 1− α ≈ 15% of its prestige is sharedequally among all nodes, just like with dangling nodes.This gives us a new matrix, the “Google matrix”
G = αS + (1− α)
(1n~e~eT
).
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
The Solution: The Google Matrix
At each turn, every node will share only α ≈ 85% of itsprestige equally among the pages it links to, as it didbefore.The remaining 1− α ≈ 15% of its prestige is sharedequally among all nodes, just like with dangling nodes.This gives us a new matrix, the “Google matrix”
G = αS + (1− α)
(1n~e~eT
).
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 5: G = αS + (1− α)1n~e~eT , α = 0.85
~π0 = G0 1n~e =
0.170.170.170.170.170.17
A
B
C
D
E F
Splendid! It converged fast, too!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 5: G = αS + (1− α)1n~e~eT , α = 0.85
~π1 = G1 1n~e =
0.210.170.170.070.210.17
A
B
C
D
E F
Splendid! It converged fast, too!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 5: G = αS + (1− α)1n~e~eT , α = 0.85
~π2 = G2 1n~e =
0.130.210.170.070.210.21
A
B
C
D
E F
Splendid! It converged fast, too!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 5: G = αS + (1− α)1n~e~eT , α = 0.85
~π3 = G3 1n~e =
0.130.140.200.070.250.21
A
B
C
D
E F
Splendid! It converged fast, too!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 5: G = αS + (1− α)1n~e~eT , α = 0.85
~π4 = G4 1n~e =
0.140.140.140.080.260.24
A
B
C
D
E F
Splendid! It converged fast, too!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 5: G = αS + (1− α)1n~e~eT , α = 0.85
~π5 = G5 1n~e =
0.140.150.140.070.270.24
A
B
C
D
E F
Splendid! It converged fast, too!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 5: G = αS + (1− α)1n~e~eT , α = 0.85
~π6 = G6 1n~e =
0.120.140.150.070.270.25
A
B
C
D
E F
Splendid! It converged fast, too!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 5: G = αS + (1− α)1n~e~eT , α = 0.85
~π7 = G7 1n~e =
0.120.130.140.070.280.26
A
B
C
D
E F
Splendid! It converged fast, too!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 5: G = αS + (1− α)1n~e~eT , α = 0.85
~π8 = G8 1n~e =
0.120.130.130.070.280.26
A
B
C
D
E F
Splendid! It converged fast, too!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 5: G = αS + (1− α)1n~e~eT , α = 0.85
~π9 = G9 1n~e =
0.120.130.140.060.290.27
A
B
C
D
E F
Splendid! It converged fast, too!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 5: G = αS + (1− α)1n~e~eT , α = 0.85
~π10 = G10 1n~e =
0.120.130.140.060.290.27
A
B
C
D
E F
Splendid! It converged fast, too!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 5: G = αS + (1− α)1n~e~eT , α = 0.85
~π11 = G11 1n~e =
0.120.120.130.060.290.27
A
B
C
D
E F
Splendid! It converged fast, too!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 5: G = αS + (1− α)1n~e~eT , α = 0.85
~π12 = G12 1n~e =
0.120.120.130.060.290.27
A
B
C
D
E F
Splendid! It converged fast, too!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 5: G = αS + (1− α)1n~e~eT , α = 0.85
~π13 = G13 1n~e =
0.120.120.130.060.290.27
A
B
C
D
E F
Splendid! It converged fast, too!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 5: G = αS + (1− α)1n~e~eT , α = 0.85
~π14 = G14 1n~e =
0.110.120.130.060.300.28
A
B
C
D
E F
Splendid! It converged fast, too!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 5: G = αS + (1− α)1n~e~eT , α = 0.85
~π15 = G15 1n~e =
0.110.120.130.060.300.28
A
B
C
D
E F
Splendid! It converged fast, too!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 5: G = αS + (1− α)1n~e~eT , α = 0.85
~π16 = G16 1n~e =
0.110.120.130.060.300.28
A
B
C
D
E F
Splendid! It converged fast, too!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 5: G = αS + (1− α)1n~e~eT , α = 0.85
~π17 = G17 1n~e =
0.110.120.130.060.300.28
A
B
C
D
E F
Splendid! It converged fast, too!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 5: G = αS + (1− α)1n~e~eT , α = 0.85
~π18 = G18 1n~e =
0.110.120.130.060.300.28
A
B
C
D
E F
Splendid! It converged fast, too!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Model 5: G = αS + (1− α)1n~e~eT , α = 0.85
~π19 = G19 1n~e =
0.110.120.130.060.300.28
A
B
C
D
E F
Splendid! It converged fast, too!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Markov Chains
G = αS + (1− α)1n~e~eT .
This situation is called a Markov chain. Notice thatG is square, every entry of G is nonnegative, and eachcolumn of G sums to 1. So G is stochastic.Every entry of G is greater than zero. This implies that G isregular.
Regular Markov Chain Theorem
If G is a regular stochastic matrix, then there exists a limitingvector ~π such that
limk→∞
Gk~x = ~π
for all starting vectors ~x .
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Markov Chains
G = αS + (1− α)1n~e~eT .
This situation is called a Markov chain. Notice thatG is square, every entry of G is nonnegative, and eachcolumn of G sums to 1. So G is stochastic.Every entry of G is greater than zero. This implies that G isregular.
Regular Markov Chain Theorem
If G is a regular stochastic matrix, then there exists a limitingvector ~π such that
limk→∞
Gk~x = ~π
for all starting vectors ~x .
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Markov Chains
G = αS + (1− α)1n~e~eT .
This situation is called a Markov chain. Notice thatG is square, every entry of G is nonnegative, and eachcolumn of G sums to 1. So G is stochastic.Every entry of G is greater than zero. This implies that G isregular.
Regular Markov Chain Theorem
If G is a regular stochastic matrix, then there exists a limitingvector ~π such that
limk→∞
Gk~x = ~π
for all starting vectors ~x .
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov Chains
Markov Chains
G = αS + (1− α)1n~e~eT .
This situation is called a Markov chain. Notice thatG is square, every entry of G is nonnegative, and eachcolumn of G sums to 1. So G is stochastic.Every entry of G is greater than zero. This implies that G isregular.
Regular Markov Chain Theorem
If G is a regular stochastic matrix, then there exists a limitingvector ~π such that
limk→∞
Gk~x = ~π
for all starting vectors ~x .
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov 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 suchthat G~x = λ~x , we say λ is an eigenvalue and ~x aneigenvector for G.
FactsIn 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 ProblemGoogle Models
Further thoughts
Five ModelsMarkov 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 suchthat G~x = λ~x , we say λ is an eigenvalue and ~x aneigenvector for G.
FactsIn 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 ProblemGoogle Models
Further thoughts
Five ModelsMarkov 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 suchthat G~x = λ~x , we say λ is an eigenvalue and ~x aneigenvector for G.
FactsIn 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 ProblemGoogle Models
Further thoughts
Five ModelsMarkov 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
=
(αS + (1− α)
1n~e~eT
)~πk
=
(αH +
1n~e~aT +
1− α
n~e~eT
)~πk
= αH~πk +1n~e~aT~πk +
1− α
n~e~eT~πk
= αH~πk +1n
(~a · ~πk + 1− α)~e
Since H is an extremely sparse matrix, this can be done inO(n) time!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov 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
=
(αS + (1− α)
1n~e~eT
)~πk
=
(αH +
1n~e~aT +
1− α
n~e~eT
)~πk
= αH~πk +1n~e~aT~πk +
1− α
n~e~eT~πk
= αH~πk +1n
(~a · ~πk + 1− α)~e
Since H is an extremely sparse matrix, this can be done inO(n) time!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov 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
=
(αS + (1− α)
1n~e~eT
)~πk
=
(αH +
1n~e~aT +
1− α
n~e~eT
)~πk
= αH~πk +1n~e~aT~πk +
1− α
n~e~eT~πk
= αH~πk +1n
(~a · ~πk + 1− α)~e
Since H is an extremely sparse matrix, this can be done inO(n) time!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov 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
=
(αS + (1− α)
1n~e~eT
)~πk
=
(αH +
1n~e~aT +
1− α
n~e~eT
)~πk
= αH~πk +1n~e~aT~πk +
1− α
n~e~eT~πk
= αH~πk +1n
(~a · ~πk + 1− α)~e
Since H is an extremely sparse matrix, this can be done inO(n) time!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov 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
=
(αS + (1− α)
1n~e~eT
)~πk
=
(αH +
1n~e~aT +
1− α
n~e~eT
)~πk
= αH~πk +1n~e~aT~πk +
1− α
n~e~eT~πk
= αH~πk +1n
(~a · ~πk + 1− α)~e
Since H is an extremely sparse matrix, this can be done inO(n) time!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov 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
=
(αS + (1− α)
1n~e~eT
)~πk
=
(αH +
1n~e~aT +
1− α
n~e~eT
)~πk
= αH~πk +1n~e~aT~πk +
1− α
n~e~eT~πk
= αH~πk +1n
(~a · ~πk + 1− α)~e
Since H is an extremely sparse matrix, this can be done inO(n) time!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov 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
=
(αS + (1− α)
1n~e~eT
)~πk
=
(αH +
1n~e~aT +
1− α
n~e~eT
)~πk
= αH~πk +1n~e~aT~πk +
1− α
n~e~eT~πk
= αH~πk +1n
(~a · ~πk + 1− α)~e
Since H is an extremely sparse matrix, this can be done inO(n) time!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Five ModelsMarkov 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
=
(αS + (1− α)
1n~e~eT
)~πk
=
(αH +
1n~e~aT +
1− α
n~e~eT
)~πk
= αH~πk +1n~e~aT~πk +
1− α
n~e~eT~πk
= αH~πk +1n
(~a · ~πk + 1− α)~e
Since H is an extremely sparse matrix, this can be done inO(n) time!
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
The α factor
G = αS + (1− α)1n~e~eT .
The α factor needs to be carefully chosen!The higher α is, the more you’re using the hyperlinkstructure of the Web.So if α ≈ 0, your ranking will be nearly useless.But if α ≈ 1, your answer will
Converge more slowly, andFluctuate 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 ProblemGoogle Models
Further thoughts
The α factor
G = αS + (1− α)1n~e~eT .
The α factor needs to be carefully chosen!The higher α is, the more you’re using the hyperlinkstructure of the Web.So if α ≈ 0, your ranking will be nearly useless.But if α ≈ 1, your answer will
Converge more slowly, andFluctuate 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 ProblemGoogle Models
Further thoughts
The α factor
G = αS + (1− α)1n~e~eT .
The α factor needs to be carefully chosen!The higher α is, the more you’re using the hyperlinkstructure of the Web.So if α ≈ 0, your ranking will be nearly useless.But if α ≈ 1, your answer will
Converge more slowly, andFluctuate 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 ProblemGoogle Models
Further thoughts
The α factor
G = αS + (1− α)1n~e~eT .
The α factor needs to be carefully chosen!The higher α is, the more you’re using the hyperlinkstructure of the Web.So if α ≈ 0, your ranking will be nearly useless.But if α ≈ 1, your answer will
Converge more slowly, andFluctuate 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 ProblemGoogle Models
Further thoughts
The α factor
G = αS + (1− α)1n~e~eT .
The α factor needs to be carefully chosen!The higher α is, the more you’re using the hyperlinkstructure of the Web.So if α ≈ 0, your ranking will be nearly useless.But if α ≈ 1, your answer will
Converge more slowly, andFluctuate 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 ProblemGoogle Models
Further thoughts
The α factor
G = αS + (1− α)1n~e~eT .
The α factor needs to be carefully chosen!The higher α is, the more you’re using the hyperlinkstructure of the Web.So if α ≈ 0, your ranking will be nearly useless.But if α ≈ 1, your answer will
Converge more slowly, andFluctuate 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 ProblemGoogle Models
Further thoughts
The α factor
G = αS + (1− α)1n~e~eT .
The α factor needs to be carefully chosen!The higher α is, the more you’re using the hyperlinkstructure of the Web.So if α ≈ 0, your ranking will be nearly useless.But if α ≈ 1, your answer will
Converge more slowly, andFluctuate 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 ProblemGoogle 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 powercomes from other factors:
“Relevance” scoreGeographic targetingPersonalizationPlenty of secrets
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Problems with Google
Is popularity really the best way to rank websites?Example: Netflix search engineExample: Jim JefsonExample: Anders Hendrickson
Google bombsExample: French military victories
Link farmsGoogle as “gatekeeper” to the web
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Problems with Google
Is popularity really the best way to rank websites?Example: Netflix search engineExample: Jim JefsonExample: Anders Hendrickson
Google bombsExample: French military victories
Link farmsGoogle as “gatekeeper” to the web
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Problems with Google
Is popularity really the best way to rank websites?Example: Netflix search engineExample: Jim JefsonExample: Anders Hendrickson
Google bombsExample: French military victories
Link farmsGoogle as “gatekeeper” to the web
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Problems with Google
Is popularity really the best way to rank websites?Example: Netflix search engineExample: Jim JefsonExample: Anders Hendrickson
Google bombsExample: French military victories
Link farmsGoogle as “gatekeeper” to the web
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Problems with Google
Is popularity really the best way to rank websites?Example: Netflix search engineExample: Jim JefsonExample: Anders Hendrickson
Google bombsExample: French military victories
Link farmsGoogle as “gatekeeper” to the web
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Problems with Google
Is popularity really the best way to rank websites?Example: Netflix search engineExample: Jim JefsonExample: Anders Hendrickson
Google bombsExample: French military victories
Link farmsGoogle as “gatekeeper” to the web
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Problems with Google
Is popularity really the best way to rank websites?Example: Netflix search engineExample: Jim JefsonExample: Anders Hendrickson
Google bombsExample: French military victories
Link farmsGoogle as “gatekeeper” to the web
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Problems with Google
Is popularity really the best way to rank websites?Example: Netflix search engineExample: Jim JefsonExample: Anders Hendrickson
Google bombsExample: French military victories
Link farmsGoogle as “gatekeeper” to the web
Anders O. F. Hendrickson Google’s PageRank Algorithm
The ProblemGoogle Models
Further thoughts
Questions?
Anders O. F. Hendrickson Google’s PageRank Algorithm
References
Google’s PageRank and Beyond: The Science of SearchEngine Rankings, by Amy N. Langville and Carl D. Meyer.(2006)Java applets by Tim Chartierhttp://mathdl.maa.org/images/upload_library/23/chartier/google-opoly/index.html
Sergey Brin, Lawrence Page, R. Motwami, and TerryWinograd. The PageRank citation ranking: Bringing orderto the Web. Technical Report 1999-0120, ComputerScience Department, Stanford University, 1999.
Anders O. F. Hendrickson Google’s PageRank Algorithm
