The small world problem: Kevin Bacon, Paul Erdos, William˝...
Transcript of The small world problem: Kevin Bacon, Paul Erdos, William˝...
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The small world problem: KevinBacon, Paul Erdős, William
McKinley, and me – 5 degrees ofgraph theory
Ryan Martin
Mathematics Department
Iowa State University
mailto:[email protected]://www.math.iastate.edu/http://www.iastate.edu/
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Joint Work
This talk is based on joint work with
Tom Bohman,Carnegie Mellon University
Alan Frieze,Carnegie Mellon University
Michael Krivelevich,Tel Aviv University
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.1/64
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Six? degrees of separation
In the Kevin Bacon Game, it is postulated that the centerof the Hollywood universe is
Kevin Bacon.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.2/64
http://www.cs.virginia.edu/oraclehttp://oracleofbacon.org/cgi-bin/oracle/movielinks?firstname=Bacon%2C+Kevin&game=1&secondname=kevin+baconhttp://www.cs.virginia.edu/cgi-bin/oracle/center-cgi?who=Hopper,+Dennishttp://www.cs.virginia.edu/cgi-bin/oracle/center-cgi?who=Hopper,+Dennishttp://www.imdb.com
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Six? degrees of separation
In the Kevin Bacon Game, it is postulated that the centerof the Hollywood universe is
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.2/64
http://www.cs.virginia.edu/oraclehttp://oracleofbacon.org/cgi-bin/oracle/movielinks?firstname=Bacon%2C+Kevin&game=1&secondname=kevin+baconhttp://www.cs.virginia.edu/cgi-bin/oracle/center-cgi?who=Hopper,+Dennishttp://www.cs.virginia.edu/cgi-bin/oracle/center-cgi?who=Hopper,+Dennishttp://www.imdb.com
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Six? degrees of separation
In the Kevin Bacon Game, it is postulated that the centerof the Hollywood universe is
Kevin Bacon.
This is false. It is Dennis Hopper.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.2/64
http://www.cs.virginia.edu/oraclehttp://oracleofbacon.org/cgi-bin/oracle/movielinks?firstname=Bacon%2C+Kevin&game=1&secondname=kevin+baconhttp://www.cs.virginia.edu/cgi-bin/oracle/center-cgi?who=Hopper,+Dennishttp://www.cs.virginia.edu/cgi-bin/oracle/center-cgi?who=Hopper,+Dennishttp://www.imdb.com
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Six? degrees of separation
In the Kevin Bacon Game, it is postulated that the centerof the Hollywood universe is
Kevin Bacon.
This is false. It is Dennis Hopper.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.2/64
http://www.cs.virginia.edu/oraclehttp://oracleofbacon.org/cgi-bin/oracle/movielinks?firstname=Bacon%2C+Kevin&game=1&secondname=kevin+baconhttp://www.cs.virginia.edu/cgi-bin/oracle/center-cgi?who=Hopper,+Dennishttp://www.cs.virginia.edu/cgi-bin/oracle/center-cgi?who=Hopper,+Dennishttp://www.imdb.com
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Six? degrees of separation
In the Kevin Bacon Game, it is postulated that the centerof the Hollywood universe is
Kevin Bacon.
We link two actors together if they appeared together inthe same movie.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.2/64
http://www.cs.virginia.edu/oraclehttp://oracleofbacon.org/cgi-bin/oracle/movielinks?firstname=Bacon%2C+Kevin&game=1&secondname=kevin+baconhttp://www.imdb.com
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Six? degrees of separation
In the Kevin Bacon Game, it is postulated that the centerof the Hollywood universe is
Kevin Bacon.
We link two actors together if they appeared together inthe same movie.
(They must be together on a cast list at the IMDb.)
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.2/64
http://www.cs.virginia.edu/oraclehttp://oracleofbacon.org/cgi-bin/oracle/movielinks?firstname=Bacon%2C+Kevin&game=1&secondname=kevin+baconhttp://www.imdb.com
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Bacon number
The actor’s Bacon number is the fewest number of stepsit takes to connect that actor to
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.3/64
http://www.cs.virginia.edu/oracle/http://www.imdb.com/name/nm0000102/
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Bacon number
The actor’s
#is the fewest number of steps
it takes to connect that actor to
Kevin Bacon.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.3/64
http://www.imdb.com/name/nm0000102/http://www.imdb.com/name/nm0000102/
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Bacon number
The actor’s Bacon number is the fewest number of stepsit takes to connect that actor to
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.3/64
http://www.cs.virginia.edu/oracle/http://www.imdb.com/name/nm0000102/
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Bacon number
An actor can have infinite Kevin Bacon number.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.3/64
http://www.imdb.com/name/nm0000102/
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Bacon number
An actor can have infinite
#.
(For example, a TV actor who appears in no moviecredits.)
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.3/64
http://www.imdb.com/name/nm0000102/
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Example: Kevin Costner
Kevin Costner is linked to Kevin Bacon because bothappeared in
JFK (1991).
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.4/64
http://www.imdb.com/name/nm0000126/http://www.imdb.com/name/nm0000102/http://www.imdb.com/title/tt0102138/http://www.imdb.com/name/nm0000126/http://oracleofbacon.org/cgi-bin/oracle/movielinks?firstname=Bacon%2C+Kevin&game=1&secondname=kevin+bacon
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Example: Kevin Costner
is linked to because bothappeared in
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.4/64
http://www.imdb.com/name/nm0000126/http://www.imdb.com/name/nm0000102/http://www.imdb.com/title/tt0102138/http://www.imdb.com/name/nm0000126/http://oracleofbacon.org/cgi-bin/oracle/movielinks?firstname=Bacon%2C+Kevin&game=1&secondname=kevin+bacon
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Example: Kevin Costner
Kevin Costner is linked to Kevin Bacon because bothappeared in
JFK (1991).
So, Kevin Costner’s Kevin Bacon number is
???
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.4/64
http://www.imdb.com/name/nm0000126/http://www.imdb.com/name/nm0000102/http://www.imdb.com/title/tt0102138/http://www.imdb.com/name/nm0000126/http://oracleofbacon.org/cgi-bin/oracle/movielinks?firstname=Bacon%2C+Kevin&game=1&secondname=kevin+bacon
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Example: Kevin Costner
Kevin Costner is linked to Kevin Bacon because bothappeared in
JFK (1991).
So, Kevin Costner’s Kevin Bacon number is
1▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.4/64
http://www.imdb.com/name/nm0000126/http://www.imdb.com/name/nm0000102/http://www.imdb.com/title/tt0102138/http://www.imdb.com/name/nm0000126/http://oracleofbacon.org/cgi-bin/oracle/movielinks?firstname=Bacon%2C+Kevin&game=1&secondname=kevin+bacon
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Example: Kevin Costner
Kevin Costner is linked to Kevin Bacon because bothappeared in
JFK (1991).
So, ’s
#is
1The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.4/64
http://www.imdb.com/name/nm0000126/http://www.imdb.com/name/nm0000102/http://www.imdb.com/title/tt0102138/http://www.imdb.com/name/nm0000126/http://oracleofbacon.org/cgi-bin/oracle/movielinks?firstname=Bacon%2C+Kevin&game=1&secondname=kevin+bacon
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Illustration: Kevin Costner
x x
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.5/64
http://www.imdb.com/name/nm0000126/http://www.imdb.com/title/tt0102138/http://www.imdb.com/name/nm0000102/
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Illustration: Kevin Costner
x x
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.5/64
http://www.imdb.com/name/nm0000126/http://www.imdb.com/title/tt0102138/http://www.imdb.com/name/nm0000102/
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Illustration: Kevin Costner
x x
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.5/64
http://www.imdb.com/name/nm0000126/http://www.imdb.com/title/tt0102138/http://www.imdb.com/name/nm0000102/
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Illustration: Kevin Costner
x x
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.5/64
http://www.imdb.com/name/nm0000126/http://www.imdb.com/title/tt0102138/http://www.imdb.com/name/nm0000102/
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E.g.: Henry “Fonz” Winkler
We know that
appeared with in
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.6/64
http://www.imdb.com/name/nm0001857/http://www.imdb.com/name/nm0000474/http://www.imdb.com/title/tt0084412/
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E.g.: Henry “Fonz” Winkler
We know that
Henry Winkler appeared with Michael Keaton inNight Shift (1982)
appeared with in
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.6/64
http://www.imdb.com/name/nm0001857/http://www.imdb.com/name/nm0000474/http://www.imdb.com/title/tt0084412/http://www.imdb.com/name/nm0000474/http://www.imdb.com/name/nm0000107/http://www.imdb.com/title/tt0096895/
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E.g.: Henry “Fonz” Winkler
We know that
Henry Winkler appeared with Michael Keaton inNight Shift (1982)
Michael Keaton appeared with Kim Basinger inBatman (1989)
appeared with in
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.6/64
http://www.imdb.com/name/nm0001857/http://www.imdb.com/name/nm0000474/http://www.imdb.com/title/tt0084412/http://www.imdb.com/name/nm0000474/http://www.imdb.com/name/nm0000107/http://www.imdb.com/title/tt0096895/http://www.imdb.com/name/nm0000107/http://www.imdb.com/name/nm0000620/http://www.imdb.com/title/tt0091635/
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E.g.: Henry “Fonz” Winkler
We know that
Henry Winkler appeared with Michael Keaton inNight Shift (1982)
Michael Keaton appeared with Kim Basinger inBatman (1989)
Kim Basinger appeared with Mickey Rourke in9 1/2 Weeks (1986)
appeared with in
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.6/64
http://www.imdb.com/name/nm0001857/http://www.imdb.com/name/nm0000474/http://www.imdb.com/title/tt0084412/http://www.imdb.com/name/nm0000474/http://www.imdb.com/name/nm0000107/http://www.imdb.com/title/tt0096895/http://www.imdb.com/name/nm0000107/http://www.imdb.com/name/nm0000620/http://www.imdb.com/title/tt0091635/http://www.imdb.com/name/nm0000620/http://www.imdb.com/name/nm0000102/http://www.imdb.com/title/tt0083833/
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E.g.: Henry “Fonz” Winkler
We know that
Henry Winkler appeared with Michael Keaton inNight Shift (1982)
Michael Keaton appeared with Kim Basinger inBatman (1989)
Kim Basinger appeared with Mickey Rourke in9 1/2 Weeks (1986)
Mickey Rourke appeared with Kevin Bacon inDiner (1982)
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.6/64
http://www.imdb.com/name/nm0001857/http://www.imdb.com/name/nm0000474/http://www.imdb.com/title/tt0084412/http://www.imdb.com/name/nm0000474/http://www.imdb.com/name/nm0000107/http://www.imdb.com/title/tt0096895/http://www.imdb.com/name/nm0000107/http://www.imdb.com/name/nm0000620/http://www.imdb.com/title/tt0091635/http://www.imdb.com/name/nm0000620/http://www.imdb.com/name/nm0000102/http://www.imdb.com/title/tt0083833/
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A chain: Henry “Fonz” Winkler
x x x x x
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.7/64
http://www.imdb.com/name/nm0001857/http://www.imdb.com/title/tt0084412/http://www.imdb.com/name/nm0000474/http://www.imdb.com/title/tt0096895/http://www.imdb.com/name/nm0000107/http://www.imdb.com/title/tt0091635/http://www.imdb.com/name/nm0000620/http://www.imdb.com/title/tt0083833/http://www.imdb.com/name/nm0000102/
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A chain: Henry “Fonz” Winkler
x x x x x
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.7/64
http://www.imdb.com/name/nm0001857/http://www.imdb.com/title/tt0084412/http://www.imdb.com/name/nm0000474/http://www.imdb.com/title/tt0096895/http://www.imdb.com/name/nm0000107/http://www.imdb.com/title/tt0091635/http://www.imdb.com/name/nm0000620/http://www.imdb.com/title/tt0083833/http://www.imdb.com/name/nm0000102/
-
A chain: Henry “Fonz” Winkler
x x x x x
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.7/64
http://www.imdb.com/name/nm0001857/http://www.imdb.com/title/tt0084412/http://www.imdb.com/name/nm0000474/http://www.imdb.com/title/tt0096895/http://www.imdb.com/name/nm0000107/http://www.imdb.com/title/tt0091635/http://www.imdb.com/name/nm0000620/http://www.imdb.com/title/tt0083833/http://www.imdb.com/name/nm0000102/
-
A chain: Henry “Fonz” Winkler
x x x x x
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.7/64
http://www.imdb.com/name/nm0001857/http://www.imdb.com/title/tt0084412/http://www.imdb.com/name/nm0000474/http://www.imdb.com/title/tt0096895/http://www.imdb.com/name/nm0000107/http://www.imdb.com/title/tt0091635/http://www.imdb.com/name/nm0000620/http://www.imdb.com/title/tt0083833/http://www.imdb.com/name/nm0000102/
-
A chain: Henry “Fonz” Winkler
x x x x x
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.7/64
http://www.imdb.com/name/nm0001857/http://www.imdb.com/title/tt0084412/http://www.imdb.com/name/nm0000474/http://www.imdb.com/title/tt0096895/http://www.imdb.com/name/nm0000107/http://www.imdb.com/title/tt0091635/http://www.imdb.com/name/nm0000620/http://www.imdb.com/title/tt0083833/http://www.imdb.com/name/nm0000102/
-
A chain: Henry “Fonz” Winkler
x x x x x
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.7/64
http://www.imdb.com/name/nm0001857/http://www.imdb.com/title/tt0084412/http://www.imdb.com/name/nm0000474/http://www.imdb.com/title/tt0096895/http://www.imdb.com/name/nm0000107/http://www.imdb.com/title/tt0091635/http://www.imdb.com/name/nm0000620/http://www.imdb.com/title/tt0083833/http://www.imdb.com/name/nm0000102/
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A chain: Henry “Fonz” Winkler
x x x x x
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.7/64
http://www.imdb.com/name/nm0001857/http://www.imdb.com/title/tt0084412/http://www.imdb.com/name/nm0000474/http://www.imdb.com/title/tt0096895/http://www.imdb.com/name/nm0000107/http://www.imdb.com/title/tt0091635/http://www.imdb.com/name/nm0000620/http://www.imdb.com/title/tt0083833/http://www.imdb.com/name/nm0000102/
-
A chain: Henry “Fonz” Winkler
x x x x x
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.7/64
http://www.imdb.com/name/nm0001857/http://www.imdb.com/title/tt0084412/http://www.imdb.com/name/nm0000474/http://www.imdb.com/title/tt0096895/http://www.imdb.com/name/nm0000107/http://www.imdb.com/title/tt0091635/http://www.imdb.com/name/nm0000620/http://www.imdb.com/title/tt0083833/http://www.imdb.com/name/nm0000102/
-
A chain: Henry “Fonz” Winkler
x x x x x
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.7/64
http://www.imdb.com/name/nm0001857/http://www.imdb.com/title/tt0084412/http://www.imdb.com/name/nm0000474/http://www.imdb.com/title/tt0096895/http://www.imdb.com/name/nm0000107/http://www.imdb.com/title/tt0091635/http://www.imdb.com/name/nm0000620/http://www.imdb.com/title/tt0083833/http://www.imdb.com/name/nm0000102/
-
A chain: Henry “Fonz” Winkler
x x x x x
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.7/64
http://www.imdb.com/name/nm0001857/http://www.imdb.com/title/tt0084412/http://www.imdb.com/name/nm0000474/http://www.imdb.com/title/tt0096895/http://www.imdb.com/name/nm0000107/http://www.imdb.com/title/tt0091635/http://www.imdb.com/name/nm0000620/http://www.imdb.com/title/tt0083833/http://www.imdb.com/name/nm0000102/
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More: Henry “Fonz” Winkler
But it is also true that
appeared with ??? in .
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.8/64
http://www.imdb.com/name/nm0001857/http://www.imdb.com/title/tt0185431/
-
More: Henry “Fonz” Winkler
But it is also true that
appeared with in .
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.8/64
http://www.imdb.com/name/nm0001857/http://www.imdb.com/name/nm0397212/http://www.imdb.com/title/tt0185431/
-
More: Henry “Fonz” Winkler
But it is also true that
Henry Winkler appeared with Clint Howard inLittle Nicky (2000).
appeared with in
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.8/64
http://www.imdb.com/name/nm0001857/http://www.imdb.com/name/nm0397212/http://www.imdb.com/title/tt0185431/http://www.imdb.com/name/nm0397212/http://www.imdb.com/name/nm0000102/http://posters.imdb.com/Covers/15/68/12.jpg
-
More: Henry “Fonz” Winkler
But it is also true that
Henry Winkler appeared with Clint Howard inLittle Nicky (2000).
Clint Howard appeared with Kevin Bacon inMy Dog Skip (2000).
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.8/64
http://www.imdb.com/name/nm0001857/http://www.imdb.com/name/nm0397212/http://www.imdb.com/title/tt0185431/http://www.imdb.com/name/nm0397212/http://www.imdb.com/name/nm0000102/http://www.imdb.com/title/tt0156812/
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Better: Henry “Fonz” Winkler
x x x
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.9/64
http://www.imdb.com/name/nm0001857/http://www.imdb.com/title/tt0185431/http://www.imdb.com/name/nm0397212/http://posters.imdb.com/Covers/15/68/12.jpghttp://www.imdb.com/name/nm0000102/
-
Better: Henry “Fonz” Winkler
x x x
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.9/64
http://www.imdb.com/name/nm0001857/http://www.imdb.com/title/tt0185431/http://www.imdb.com/name/nm0397212/http://posters.imdb.com/Covers/15/68/12.jpghttp://www.imdb.com/name/nm0000102/
-
Better: Henry “Fonz” Winkler
x x x
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.9/64
http://www.imdb.com/name/nm0001857/http://www.imdb.com/title/tt0185431/http://www.imdb.com/name/nm0397212/http://posters.imdb.com/Covers/15/68/12.jpghttp://www.imdb.com/name/nm0000102/
-
Better: Henry “Fonz” Winkler
x x x
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.9/64
http://www.imdb.com/name/nm0001857/http://www.imdb.com/title/tt0185431/http://www.imdb.com/name/nm0397212/http://posters.imdb.com/Covers/15/68/12.jpghttp://www.imdb.com/name/nm0000102/
-
Better: Henry “Fonz” Winkler
x x x
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.9/64
http://www.imdb.com/name/nm0001857/http://www.imdb.com/title/tt0185431/http://www.imdb.com/name/nm0397212/http://posters.imdb.com/Covers/15/68/12.jpghttp://www.imdb.com/name/nm0000102/
-
Better: Henry “Fonz” Winkler
x x x
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.9/64
http://www.imdb.com/name/nm0001857/http://www.imdb.com/title/tt0185431/http://www.imdb.com/name/nm0397212/http://posters.imdb.com/Covers/15/68/12.jpghttp://www.imdb.com/name/nm0000102/
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Can we do even better?
has never appeared in a film with .
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.10/64
http://www.imdb.com/name/nm0000102/http://www.imdb.com/name/nm0001857/
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Can we do even better?
Kevin Bacon has never appeared in a film withHenry Winkler.
So, ’s
#is
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.10/64
http://www.imdb.com/name/nm0000102/http://www.imdb.com/name/nm0001857/http://www.imdb.com/name/nm0001857/http://oracleofbacon.org/cgi-bin/oracle/movielinks?firstname=Bacon%2C+Kevin&game=1&secondname=kevin+bacon
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Can we do even better?
Kevin Bacon has never appeared in a film withHenry Winkler.
So, Henry Winkler’s Kevin Bacon number is
???
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.10/64
http://www.imdb.com/name/nm0000102/http://www.imdb.com/name/nm0001857/http://www.imdb.com/name/nm0001857/http://oracleofbacon.org/cgi-bin/oracle/movielinks?firstname=Bacon%2C+Kevin&game=1&secondname=kevin+bacon
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Can we do even better?
Kevin Bacon has never appeared in a film withHenry Winkler.
So, Henry Winkler’s Kevin Bacon number is
2
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.10/64
http://www.imdb.com/name/nm0000102/http://www.imdb.com/name/nm0001857/http://www.imdb.com/name/nm0001857/http://oracleofbacon.org/cgi-bin/oracle/movielinks?firstname=Bacon%2C+Kevin&game=1&secondname=kevin+bacon
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In sum: Henry “Fonz” Winkler
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(((
hhhhhhhh
hhhx xx xxx
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.11/64
http://www.imdb.com/name/nm0001857/http://www.imdb.com/name/nm0000474/http://www.imdb.com/name/nm0000107/http://www.imdb.com/name/nm0000620/http://www.imdb.com/name/nm0000102/http://www.imdb.com/name/nm0397212/
-
In sum: Henry “Fonz” Winkler
PPPPPPhhhhhh((((((
������
((((((((
(((
hhhhhhhh
hhhx xx xxx
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.11/64
http://www.imdb.com/name/nm0001857/http://www.imdb.com/name/nm0000474/http://www.imdb.com/name/nm0000107/http://www.imdb.com/name/nm0000620/http://www.imdb.com/name/nm0000102/http://www.imdb.com/name/nm0397212/
-
In sum: Henry “Fonz” Winkler
PPPPPPhhhhhh((((((
������
((((((((
(((
hhhhhhhh
hhhx xx xxx
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.11/64
http://www.imdb.com/name/nm0001857/http://www.imdb.com/name/nm0000474/http://www.imdb.com/name/nm0000107/http://www.imdb.com/name/nm0000620/http://www.imdb.com/name/nm0000102/http://www.imdb.com/name/nm0397212/
-
Experimental data
# of actors
0 11 18062 1450243 3951264 954975 7451
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.12/64
http://oracleofbacon.org/cgi-bin/oracle/movielinks?firstname=Bacon%2C+Kevin&game=1&secondname=kevin+baconhttp://oracleofbacon.org/cgi-bin/oracle/movielinks?firstname=Bacon%2C+Kevin&game=1&secondname=kevin+baconhttp://www.cs.virginia.edu/cgi-bin/oracle/bynumber-cgi?who=Kevin+Bacon&howmany=1http://oracleofbacon.org/cgi-bin/oracle/movielinks?firstname=Bacon%2C+Kevin&game=1&secondname=kevin+bacon
-
Experimental data
# of actors
0 11 18062 1450243 3951264 954975 7451
# of actors
6 9337 1068 13
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.12/64
http://oracleofbacon.org/cgi-bin/oracle/movielinks?firstname=Bacon%2C+Kevin&game=1&secondname=kevin+baconhttp://oracleofbacon.org/cgi-bin/oracle/movielinks?firstname=Bacon%2C+Kevin&game=1&secondname=kevin+baconhttp://www.cs.virginia.edu/cgi-bin/oracle/bynumber-cgi?who=Kevin+Bacon&howmany=1http://oracleofbacon.org/cgi-bin/oracle/movielinks?firstname=Bacon%2C+Kevin&game=1&secondname=kevin+bacon
-
What about the high numbers?
As we said before, there are actors with infinite
#.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.13/64
http://www.cs.virginia.edu/oracle/http://www.cs.virginia.edu/oracle/
-
What about the high numbers?
As we said before, there are actors with infinite
#.
The actors with large
#are obscure and the
reason why is fairly obvious.
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.13/64
http://www.cs.virginia.edu/oracle/http://www.cs.virginia.edu/oracle/
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Kevin Bacon not so special
Most successful actors follow the same pattern:
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.14/64
-
Kevin Bacon not so special
Most successful actors follow the same pattern:
For every pair of successful actors, they areconnected by a path of length ≤ 5
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.14/64
-
Kevin Bacon not so special
Most successful actors follow the same pattern:
For every pair of successful actors, they areconnected by a path of length ≤ 5
Why?
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.14/64
-
Our model
We will represent actors by vertices
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.15/64
-
Our model
We will represent actors by vertices
x
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.15/64
-
Our model
We will represent actors by vertices
xand connect them with edges
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.15/64
-
Our model
We will represent actors by vertices
xand connect them with edges
x xif they appeared in the same film.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.15/64
-
Our model
We will represent actors by vertices
xand connect them with edges
x xif they appeared in the same film.
This is a graph.
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.15/64
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Model parameters
There are n actors.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.16/64
-
Model parameters
There are n actors.
Fix a constant d.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.16/64
-
Model parameters
There are n actors.
Fix a constant d.
We will begin with an arbitrary graph H.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.16/64
-
Model parameters
There are n actors.
Fix a constant d.
We will begin with an arbitrary graph H.In H, each actor is connected to at least dn otheractors.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.16/64
-
Model parameters
There are n actors.
Fix a constant d.
We will begin with an arbitrary graph H.In H, each actor is connected to at least dn otheractors.
The constant d can be extremely tiny:
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.16/64
-
Model parameters
There are n actors.
Fix a constant d.
We will begin with an arbitrary graph H.In H, each actor is connected to at least dn otheractors.
The constant d can be extremely tiny:
0.1
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.16/64
-
Model parameters
There are n actors.
Fix a constant d.
We will begin with an arbitrary graph H.In H, each actor is connected to at least dn otheractors.
The constant d can be extremely tiny:
0.1, 0.01
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.16/64
-
Model parameters
There are n actors.
Fix a constant d.
We will begin with an arbitrary graph H.In H, each actor is connected to at least dn otheractors.
The constant d can be extremely tiny:
0.1, 0.01, 0.000001
It just needs to be independent of n.
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.16/64
-
Random casting
We add f(n) random casting connections.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.17/64
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Random casting
We add f(n) random casting connections.
What does random mean?
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.17/64
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Random edges
Let N be the number of pairs with no connectionbetween them (non-edges). We can create m newrandom edges in two ways:
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.18/64
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Random edges
Let N be the number of pairs with no connectionbetween them (non-edges). We can create m newrandom edges in two ways:
For every set of m unconnected pairs, choose oneset uniformly at random.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.18/64
-
Random edges
Let N be the number of pairs with no connectionbetween them (non-edges). We can create m newrandom edges in two ways:
For every set of m unconnected pairs, choose oneset uniformly at random.
Connect a previously unconnected pair,independently, with probability m/N .
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.18/64
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Random edges
Let N be the number of pairs with no connectionbetween them (non-edges). We can create m newrandom edges in two ways:
For every set of m unconnected pairs, choose oneset uniformly at random.
Connect a previously unconnected pair,independently, with probability m/N (coin flips).
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.18/64
-
Random edges
Let N be the number of pairs with no connectionbetween them (non-edges). We can create m newrandom edges in two ways:
For every set of m unconnected pairs, choose oneset uniformly at random.
Connect a previously unconnected pair,independently, with probability m/N (coin flips).The average number of new connections is m.
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.18/64
-
The models
For our purposes, these produce the same results.
The question:
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.19/64
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The models
For our purposes, these produce the same results.
The question:
What is the longest distance between any pair ofactors ?
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.19/64
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The models
For our purposes, these produce the same results.
The question:
What is the longest distance between any pair ofactors (diameter)?
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.19/64
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The theorem
Theorem. If f(n) → ∞ as n → ∞, thenPr (diam ≤ 5) → 1
Important points:
Recall f(n) is the number of random connections.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.20/64
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The theorem
Theorem. If f(n) → ∞ as n → ∞, thenPr (diam ≤ 5) → 1
Important points:
Recall f(n) is the number of random connections.
“5” doesn’t depend on d at all.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.20/64
-
The theorem
Theorem. If f(n) → ∞ as n → ∞, thenPr (diam ≤ 5) → 1
Important points:
Recall f(n) is the number of random connections.
“5” doesn’t depend on d at all.
f(n) can be very small:
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.20/64
-
The theorem
Theorem. If f(n) → ∞ as n → ∞, thenPr (diam ≤ 5) → 1
Important points:
Recall f(n) is the number of random connections.
“5” doesn’t depend on d at all.
f(n) can be very small:√
n
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.20/64
-
The theorem
Theorem. If f(n) → ∞ as n → ∞, thenPr (diam ≤ 5) → 1
Important points:
Recall f(n) is the number of random connections.
“5” doesn’t depend on d at all.
f(n) can be very small:√
n, log n
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.20/64
-
The theorem
Theorem. If f(n) → ∞ as n → ∞, thenPr (diam ≤ 5) → 1
Important points:
Recall f(n) is the number of random connections.
“5” doesn’t depend on d at all.
f(n) can be very small:√
n, log n,√
log log log n
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.20/64
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Qualifying the model
This is a nice result on a pretty good model.
The model only assumes some density conditions and alittle bit of randomness.
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.21/64
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The proof
To prove the theorem, you need the Regularity Lemma.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.22/64
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The proof
To prove the theorem, you need the Regularity Lemma.
The Regularity Lemma is ’s powerfuland complicated graph theoretic tool.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.22/64
http://www.renyi.hu/~szemered/
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The proof
To prove the theorem, you need the Regularity Lemma.
The Regularity Lemma is Endre Szemerédi’s powerfuland complicated graph theoretic tool.
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.22/64
http://www.renyi.hu/~szemered/
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Best possible?
The theorem is “tight”:
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.23/64
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Best possible?
The theorem is “tight”:
If there aren’t an infinite number of edges added,then some H ’s will be disconnected.
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.23/64
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What about closer connections?
To get diam ≤ 4, you need randomconnections.
To get diam ≤ 3, you need randomconnections.
To get diam ≤ 2, you need randomconnections.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.24/64
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What about closer connections?
To get diam ≤ 4, you need c1 log n randomconnections.
To get diam ≤ 3, you need c1 log n randomconnections.
To get diam ≤ 2, you need randomconnections.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.24/64
-
What about closer connections?
To get diam ≤ 4, you need c1 log n randomconnections.
To get diam ≤ 3, you need c1 log n randomconnections.
To get diam ≤ 2, you need c2n log n randomconnections.
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.24/64
-
Origins of the problem
The original question posed by Stanley Milgram
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.25/64
-
Origins of the problem
The original question posed by Stanley Milgram
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.25/64
-
Origins of the problem
The original question posed by Stanley Milgramasked what the average distance was among peoplein a network.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.25/64
-
Origins of the problem
The original question posed by Stanley Milgramasked what the average distance was among peoplein a network.
His methodology was very flawed
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.25/64
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Origins of the problem
The original question posed by Stanley Milgramasked what the average distance was among peoplein a network.
His methodology was very flawed – he sent out lettersfrom a single source and waited for a return message,many didn’t come back.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.25/64
-
Origins of the problem
The original question posed by Stanley Milgramasked what the average distance was among peoplein a network.
His methodology was very flawed – he sent out lettersfrom a single source and waited for a return message,many didn’t come back.
But his assertion that the average distance was around 6stuck.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.25/64
-
Origins of the problem
But his assertion that the average distance was around 6stuck.
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.25/64
-
Advantages and disadvantages
Our model has some advantages and somedisadvantages.
Disadvantages:
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.26/64
-
Advantages and disadvantages
Our model has some advantages and somedisadvantages.
Disadvantages:Each person must beconnected to at leastdn other people.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.26/64
-
Advantages and disadvantages
Our model has some advantages and somedisadvantages.
Disadvantages:Each person must beconnected to at leastdn other people.
Not good for modellingthe Internet.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.26/64
-
Advantages and disadvantages
Our model has some advantages and somedisadvantages.
Disadvantages:Each person must beconnected to at leastdn other people.
Not good for modellingthe Internet.
Advantages:
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.26/64
-
Advantages and disadvantages
Our model has some advantages and somedisadvantages.
Disadvantages:Each person must beconnected to at leastdn other people.
Not good for modellingthe Internet.
Advantages:Very weak restrictionon structure.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.26/64
-
Advantages and disadvantages
Our model has some advantages and somedisadvantages.
Disadvantages:Each person must beconnected to at leastdn other people.
Not good for modellingthe Internet.
Advantages:Very weak restrictionon structure.
The number ofrandom connectionsis tiny.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.26/64
-
Advantages and disadvantages
Our model has some advantages and somedisadvantages.
Disadvantages:Each person must beconnected to at leastdn other people.
Not good for modellingthe Internet.
Advantages:Very weak restrictionon structure.
The number ofrandom connectionsis tiny.
An upper bound of 5on the diameter.
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.26/64
-
Another model
There is another model due to Fan Chung
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.27/64
-
Another model
There is another model due to Fan Chung
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.27/64
-
Another model
There is another model due to Fan Chungand others
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.27/64
-
Another model
There is another model due to Fan Chungand others (no pictures)
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.27/64
-
Another model
There is another model due to Fan Chungand others (no pictures)which eliminates the problems of not being applicable tothe Internet.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.27/64
-
Another model
There is another model due to Fan Chungand others (no pictures)which eliminates the problems of not being applicable tothe Internet.
However . . .
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.27/64
-
The Chung model
Advantages:
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.28/64
-
The Chung model
Advantages:Easy to use, easy todo computations.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.28/64
-
The Chung model
Advantages:Easy to use, easy todo computations.
Models the Internetquite well.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.28/64
-
The Chung model
Advantages:Easy to use, easy todo computations.
Models the Internetquite well.
Disdvantages:
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.28/64
-
The Chung model
Advantages:Easy to use, easy todo computations.
Models the Internetquite well.
Disdvantages:The number of randomconnections is huge.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.28/64
-
The Chung model
Advantages:Easy to use, easy todo computations.
Models the Internetquite well.
Disdvantages:The number of randomconnections is huge.
Weak result: The av-erage distance is ≈log2 n/ log2 d̃.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.28/64
-
The Chung model
Advantages:Easy to use, easy todo computations.
Models the Internetquite well.
Disdvantages:The number of randomconnections is huge.
Weak result: The av-erage distance is ≈log2 n/ log2 d̃, where d̃relates to the aver-age degree (numberof connections).
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.28/64
-
Applying our model
To have it be very likely that everyone is con-nected by a path of no more than 5 acquain-tances, just arrange a few random meetings.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.29/64
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Applying our model
To have it be very likely that everyone is con-nected by a path of no more than 5 acquain-tances, just arrange a few random meetings.
Think about people at Iowa State.
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.29/64
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Iowa State cliques
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▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.30/64
-
Iowa State cliques
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The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.30/64
-
More Iowa State cliques
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▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.31/64
-
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Computer Scientists
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.31/64
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The cliché
The cliché states that every pair of people is separatedby at most
six degrees of separation.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.32/64
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The cliché
The cliché states that every pair of people is separatedby at most
six degrees of separation.
In fact, it is
FIVE degrees of separationand there’s an actual proof!
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.32/64
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Erdős number
One of the most prolific mathematicians of the 20thcentury was
Paul (Pál) ErdősMarch 26, 1913-September 20, 1996
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.33/64
http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Erdos.html
-
Erdős number
One of the most prolific mathematicians of the 20thcentury was
Paul (Pál) ErdősMarch 26, 1913-September 20, 1996
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.33/64
http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Erdos.html
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Erdős number project
The Erdős number project is concerned with the distanceof mathematicians from Paul Erdős.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.34/64
http://www.oakland.edu/~grossman/erdoshp.htmlhttp://www.ams.org/mathscinet
-
Erdős number project
The
#project is concerned with the distance
of mathematicians from Paul Erdős.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.34/64
http://www.oakland.edu/~grossman/erdoshp.htmlhttp://www.oakland.edu/~grossman/erdoshp.htmlhttp://www.ams.org/mathscinet
-
Erdős number project
The
#project is concerned with the distance
of mathematicians from Paul Erdős.
Two mathematicians are connected if they co-authored apaper together and that paper appears in MathematicalReviews, accessible by MathSciNet.
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.34/64
http://www.oakland.edu/~grossman/erdoshp.htmlhttp://www.oakland.edu/~grossman/erdoshp.htmlhttp://www.ams.org/mathscinet
-
Most prolific authors
: 1401 papers (Erdős number )
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.35/64
http://www.oakland.edu/~grossman/erdoshp.html
-
Most prolific authors
: 1401 papers (Erdős number )
Drumi Bainov: 782 (Erdős number )
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.35/64
http://www.oakland.edu/~grossman/erdoshp.html
-
Most prolific authors
: 1401 papers (Erdős number )
Drumi Bainov: 782 (Erdős number )
Leonard Carlitz: 730 (Erdős number )
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.35/64
http://www.oakland.edu/~grossman/erdoshp.html
-
Most prolific authors
: 1401 papers (Erdős number )
Drumi Bainov: 782 (Erdős number )
Leonard Carlitz: 730 (Erdős number )
Lucien Godeaux: 644 (Erdős number )
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.35/64
http://www.oakland.edu/~grossman/erdoshp.html
-
Most prolific authors
: 1401 papers (Erdős number )
Drumi Bainov: 782 (Erdős number )
Leonard Carlitz: 730 (Erdős number )
Lucien Godeaux: 644 (Erdős number )
Saharon Shelah: 600 (Erdős number )
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.35/64
http://www.oakland.edu/~grossman/erdoshp.html
-
Most prolific authors
: 1401 papers (Erdős number 0)
Drumi Bainov: 782 (Erdős number 4)
Leonard Carlitz: 730 (Erdős number 2)
Lucien Godeaux: 644 (Erdős number ∞)
Saharon Shelah: 600 (Erdős number 1)
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.35/64
http://www.oakland.edu/~grossman/erdoshp.html
-
Erdős number statistics
wrote 1401 papers in Math Reviews.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.36/64
http://www.oakland.edu/~grossman/erdoshp.htmlhttp://www.oakland.edu/~grossman/papers.html
-
Erdős number statistics
wrote 1401 papers in Math Reviews.
There are 337, 000 vertices (authors) in the graph.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.36/64
http://www.oakland.edu/~grossman/erdoshp.htmlhttp://www.oakland.edu/~grossman/papers.html
-
Erdős number statistics
wrote 1401 papers in Math Reviews.
There are 337, 000 vertices (authors) in the graph.
There are about 496, 000 edges.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.36/64
http://www.oakland.edu/~grossman/erdoshp.htmlhttp://www.oakland.edu/~grossman/papers.html
-
Erdős number statistics
wrote 1401 papers in Math Reviews.
There are 337, 000 vertices (authors) in the graph.
There are about 496, 000 edges.
Average number of authors per paper: 1.45
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.36/64
http://www.oakland.edu/~grossman/erdoshp.htmlhttp://www.oakland.edu/~grossman/papers.html
-
Erdős number statistics
wrote 1401 papers in Math Reviews.
There are 337, 000 vertices (authors) in the graph.
There are about 496, 000 edges.
Average number of authors per paper: 1.45
Average number of papers per author: 6.87
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.36/64
http://www.oakland.edu/~grossman/erdoshp.htmlhttp://www.oakland.edu/~grossman/papers.html
-
Experimental data
#
0 1
1 502
2 5713
3 26422
4 62136
5 66157
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.37/64
http://www.oakland.edu/~grossman/erdoshp.htmlhttp://www.oakland.edu/enp/thedata.htmlhttp://www.oakland.edu/enp/Erdos0http://www.oakland.edu/enp/Erdos2http://www.oakland.edu/~grossman/erdoshp.htmlhttp://www.oakland.edu/~grossman/erdoshp.html
-
Experimental data
#
0 1
1 509
2 6984
3 26422
4 62136
5 66157
#
6 32280
7 10431
8 3214
9 953
10 262
11 94(Most recent data)
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.37/64
http://www.oakland.edu/~grossman/erdoshp.htmlhttp://www.oakland.edu/enp/thedata.htmlhttp://www.oakland.edu/enp/Erdos0http://www.oakland.edu/enp/Erdos2http://www.oakland.edu/~grossman/erdoshp.htmlhttp://www.oakland.edu/~grossman/erdoshp.html
-
Experimental data
#
0 1
1 509
2 6984
3 26422
4 62136
5 66157
#
6 32280
7 10431
8 3214
9 953
10 262
11 94
#
12 23
13 4
14 7
15 1
(Most recent data)
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.37/64
http://www.oakland.edu/~grossman/erdoshp.htmlhttp://www.oakland.edu/enp/thedata.htmlhttp://www.oakland.edu/enp/Erdos0http://www.oakland.edu/enp/Erdos2http://www.oakland.edu/~grossman/erdoshp.htmlhttp://www.oakland.edu/~grossman/erdoshp.html
-
Experimental data
#
0 1
1 509
2 6984
3 26422
4 62136
5 66157
#
6 32280
7 10431
8 3214
9 953
10 262
11 94
#
12 23
13 4
14 7
15 1
(R. G. Kamalov)
(Most recent data)
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.37/64
http://www.oakland.edu/~grossman/erdoshp.htmlhttp://www.oakland.edu/enp/thedata.htmlhttp://www.oakland.edu/enp/Erdos0http://www.oakland.edu/enp/Erdos2http://www.oakland.edu/~grossman/erdoshp.htmlhttp://www.oakland.edu/~grossman/erdoshp.html
-
The unknown mathematician
xxx
x
xHHHHHHHH���������
��������
\\\\\\\\\
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.38/64
http://www.oakland.edu/~grossman/erdoshp.html
-
The unknown mathematician
xxx
x
xHHHHHHHH���������
��������
\\\\\\\\\
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.38/64
http://www.oakland.edu/~grossman/erdoshp.html
-
The unknown mathematician
xxx
x
xHHHHHHHH���������
��������
\\\\\\\\\
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.38/64
http://www.oakland.edu/~grossman/erdoshp.html
-
The unknown mathematician
xxx
x
xHHHHHHHH���������
��������
\\\\\\\\\
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.38/64
http://www.oakland.edu/~grossman/erdoshp.html
-
The unknown mathematician
xxx
x
xHHHHHHHH���������
��������
\\\\\\\\\
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.38/64
http://www.oakland.edu/~grossman/erdoshp.html
-
The unknown mathematician
xxx
x
xHHHHHHHH���������
��������
\\\\\\\\\
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.38/64
http://www.oakland.edu/~grossman/erdoshp.html
-
Computer networks
Graphs model much more serious stuff.
I.e.,
computer networks,
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.39/64
-
Computer networks
Graphs model much more serious stuff.
I.e.,
computer networks,
shipping routes,
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.39/64
-
Computer networks
Graphs model much more serious stuff.
I.e.,
computer networks,
shipping routes,
distribution networks.
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.39/64
-
Network question
In networks we are concerned with one particularquantity:
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.40/64
-
Network question
In networks we are concerned with one particularquantity:
connectivity: A connected graph is k-connected ifremoving any set of k − 1 vertices (and all relevantedges) leaves the graph connected.
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.40/64
-
Same model
n computers
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.41/64
-
Same model
n computers
in H, each computer is connected to ≥ dn others
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.41/64
-
Same model
n computers
in H, each computer is connected to ≥ dn others
add f(n) random connections
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.41/64
-
Same model
n computers
in H, each computer is connected to ≥ dn others
add f(n) random connections
Of course, we want high connectivity with as littlerandomness as possible.
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.41/64
-
Connectivity theorem
Theorem. Let k be a function of n that is ≪ n.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.42/64
-
Connectivity theorem
Theorem. Let k be a function of n that is ≪ n. (That is,k grows more slowly than n as n → ∞.)
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.42/64
-
Connectivity theorem
Theorem. Let k be a function of n that is ≪ n. (That is,k grows more slowly than n as n → ∞.) Let H have theproperty that each vertex is connected to at least dnother vertices.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.42/64
-
Connectivity theorem
Theorem. Let k be a function of n that is ≪ n. (That is,k grows more slowly than n as n → ∞.) Let H have theproperty that each vertex is connected to at least dnother vertices.
If f(n) ≫ k, then the graph becomes k-connected,with high probability.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.42/64
-
Connectivity theorem
Theorem. Let k be a function of n that is ≪ n. (That is,k grows more slowly than n as n → ∞.) Let H have theproperty that each vertex is connected to at least dnother vertices.
If f(n) ≫ k, then the graph becomes k-connected,with high probability.
If d < 1/2, there is an H0 such that for every k ≪ n,f(n) = k − 1 ensures that the graph fails to bek-connected, with high probability.
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.42/64
-
Bottom line
A way to interpret this theorem is:
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.43/64
-
Bottom line
A way to interpret this theorem is:
If you need k-connectivity,
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.43/64
-
Bottom line
A way to interpret this theorem is:
If you need k-connectivity,
then you need to add a little more(asymptotically) random edges than k.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.43/64
-
Bottom line
A way to interpret this theorem is:
If you need k-connectivity,
then you need to add a little more(asymptotically) random edges than k.
If fewer than k random edges are added, k-connectivitydoes not necessarily occur.
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.43/64
-
Worst case
What is that H0?
H0 =
SS��S
S""""b
bbb
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S��S
S""""b
bbb
�� S
S��S
S""""bb
bb�� S
S��S
S""""b
bbb
AAAA
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SSb
bbb
��""""
AAAA����
����
""""
AAAA
����
""""
AAAA
bb
bb
��
SS b
bbbAAAA""""
����
��
SS
""""
AAAA����
����
""""
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��
""""
""""b
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SS �
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SS
bb
bb""""
bb
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SS""""
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SS""""
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bb
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SS
bb
bb
xx x x xx
xxx x xx x
x x xxxxx x x xx xx x x x
xxxx x x x
x x x xxxxxx x x
xDisjoint cliques give the worst case.
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.44/64
-
Other properties
We’ve used this model to investigate other properties:
Hamilton cycle
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.45/64
-
Other properties
We’ve used this model to investigate other properties:
Hamilton cycle
Small cliques as subgraphs
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.45/64
-
Other properties
We’ve used this model to investigate other properties:
Hamilton cycle
Small cliques as subgraphs
Chromatic number
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.45/64
-
What next?
Shall we conclude with
more mathematics,
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.46/64
-
What next?
Shall we conclude with
more mathematics,
people with high Bacon numbers,
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.46/64
-
What next?
Shall we conclude with
more mathematics,
people with high Bacon numbers,
mathematicians and dead presidents, or
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.46/64
-
What next?
Shall we conclude with
more mathematics,
people with high Bacon numbers,
mathematicians and dead presidents, or
connections between Bacon numbers and Erdősnumbers?
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.46/64
-
It’s just killing you, isn’t it?
Let us return to the Kevin Bacon question.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.47/64
http://oracleofbacon.org/cgi-bin/oracle/movielinks?firstname=Bacon%2C+Kevin&game=1&secondname=kevin+baconhttp://oracleofbacon.org/cgi-bin/oracle/movielinks?firstname=Bacon%2C+Kevin&game=1&secondname=kevin+bacon
-
It’s just killing you, isn’t it?
Let us return to the Kevin Bacon question.
We want to find actors with an
infinite
#and
with
#=8 .
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.47/64
http://oracleofbacon.org/cgi-bin/oracle/movielinks?firstname=Bacon%2C+Kevin&game=1&secondname=kevin+baconhttp://oracleofbacon.org/cgi-bin/oracle/movielinks?firstname=Bacon%2C+Kevin&game=1&secondname=kevin+bacon
-
Infinite Kevin Bacon number
is someone with infinite
#.
Thomas Alva Edison only appeared in one movie (abrief documentary) and was the only actor.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.48/64
http://oracleofbacon.org/cgi-bin/oracle/movielinks?firstname=Bacon%2C+Kevin&game=1&secondname=kevin+baconhttp://us.imdb.com/Name?Edison,+Thomas+A.http://us.imdb.com/Name?Edison,+Thomas+A.http://us.imdb.com/Name?Edison,+Thomas+A.http://us.imdb.com/Title?0216924http://us.imdb.com/Title?0216924http://us.imdb.com/Title?0216924http://us.imdb.com/Title?0216924http://us.imdb.com/Title?0216924http://us.imdb.com/Title?0216924http://us.imdb.com/Title?0216924http://us.imdb.com/Title?0216924http://us.imdb.com/Title?0216924http://us.imdb.com/Title?0216924
-
Infinite Kevin Bacon number
is someone with infinite
#.
Thomas Alva Edison only appeared in one movie (abrief documentary) and was the only actor.
Not soon coming to DVD:
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.48/64
http://oracleofbacon.org/cgi-bin/oracle/movielinks?firstname=Bacon%2C+Kevin&game=1&secondname=kevin+baconhttp://us.imdb.com/Name?Edison,+Thomas+A.http://us.imdb.com/Name?Edison,+Thomas+A.http://us.imdb.com/Name?Edison,+Thomas+A.http://us.imdb.com/Title?0216924http://us.imdb.com/Title?0216924http://us.imdb.com/Title?0216924http://us.imdb.com/Title?0216924http://us.imdb.com/Title?0216924http://us.imdb.com/Title?0216924http://us.imdb.com/Title?0216924http://us.imdb.com/Title?0216924http://us.imdb.com/Title?0216924http://us.imdb.com/Title?0216924
-
Infinite Kevin Bacon number
is someone with infinite
#.
Thomas Alva Edison only appeared in one movie (abrief documentary) and was the only actor.
Not soon coming to DVD: Mr. Edison at Work inHis Chemical Laboratory (1897).
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.48/64
http://oracleofbacon.org/cgi-bin/oracle/movielinks?firstname=Bacon%2C+Kevin&game=1&secondname=kevin+baconhttp://us.imdb.com/Name?Edison,+Thomas+A.http://us.imdb.com/Name?Edison,+Thomas+A.http://us.imdb.com/Name?Edison,+Thomas+A.http://us.imdb.com/Title?0216924http://us.imdb.com/Title?0216924http://us.imdb.com/Title?0216924http://us.imdb.com/Title?0216924http://us.imdb.com/Title?0216924http://us.imdb.com/Title?0216924http://us.imdb.com/Title?0216924http://us.imdb.com/Title?0216924http://us.imdb.com/Title?0216924http://us.imdb.com/Title?0216924
-
Kevin Bacon number 7
Joseph Wheeler appeared in two films:
General Wheeler and Secretary ofWar Alger at Camp Wikoff (1898), a shortdocumentary, in which he appeared with RussellAlexander Alger (Kevin Bacon number 6)
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.49/64
http://us.imdb.com/Name?Wheeler,+Joseph+(IV)http://us.imdb.com/Name?Wheeler,+Joseph+(IV)http://us.imdb.com/Title?0239423http://us.imdb.com/Title?0239423http://us.imdb.com/Title?0239423http://us.imdb.com/Title?0239423http://us.imdb.com/Title?0239423http://us.imdb.com/Title?0239423http://us.imdb.com/Title?0239423http://us.imdb.com/Title?0239423http://us.imdb.com/Title?0239423http://us.imdb.com/Title?0239423http://us.imdb.com/Name?Alger,+Russell+Alexanderhttp://us.imdb.com/Name?Alger,+Russell+Alexanderhttp://us.imdb.com/Name?Alger,+Russell+Alexanderhttp://us.imdb.com/Title?0220793http://us.imdb.com/Title?0220793http://us.imdb.com/Title?0220793http://us.imdb.com/Title?0220793http://us.imdb.com/Name?Shafter,+William+Rufushttp://us.imdb.com/Name?Shafter,+William+Rufushttp://us.imdb.com/Name?Shafter,+William+Rufus
-
Kevin Bacon number 7
Joseph Wheeler appeared in two films:
General Wheeler and Secretary ofWar Alger at Camp Wikoff (1898), a shortdocumentary, in which he appeared with RussellAlexander Alger (Kevin Bacon number 6)
Surrender of General Toral (1898), again,a short documentary, with William Rufus Shafter.
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.49/64
http://us.imdb.com/Name?Wheeler,+Joseph+(IV)http://us.imdb.com/Name?Wheeler,+Joseph+(IV)http://us.imdb.com/Title?0239423http://us.imdb.com/Title?0239423http://us.imdb.com/Title?0239423http://us.imdb.com/Title?0239423http://us.imdb.com/Title?0239423http://us.imdb.com/Title?0239423http://us.imdb.com/Title?0239423http://us.imdb.com/Title?0239423http://us.imdb.com/Title?0239423http://us.imdb.com/Title?0239423http://us.imdb.com/Name?Alger,+Russell+Alexanderhttp://us.imdb.com/Name?Alger,+Russell+Alexanderhttp://us.imdb.com/Name?Alger,+Russell+Alexanderhttp://us.imdb.com/Title?0220793http://us.imdb.com/Title?0220793http://us.imdb.com/Title?0220793http://us.imdb.com/Title?0220793http://us.imdb.com/Name?Shafter,+William+Rufushttp://us.imdb.com/Name?Shafter,+William+Rufushttp://us.imdb.com/Name?Shafter,+William+Rufus
-
Kevin Bacon number 8
William Rufus Shafter also appeared in two films:
Surrender of General Toral (1898) withJoseph Wheeler.
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.50/64
http://us.imdb.com/Name?Shafter,+William+Rufushttp://us.imdb.com/Name?Shafter,+William+Rufushttp://us.imdb.com/Name?Shafter,+William+Rufushttp://us.imdb.com/Title?0220793http://us.imdb.com/Title?0220793http://us.imdb.com/Title?0220793http://us.imdb.com/Title?0220793http://us.imdb.com/Name?Wheeler,+Joseph+(IV)http://us.imdb.com/Name?Wheeler,+Joseph+(IV)http://us.imdb.com/Title?0239582http://us.imdb.com/Title?0239582http://us.imdb.com/Title?0239582
-
Kevin Bacon number 8
William Rufus Shafter also appeared in two films:
Surrender of General Toral (1898) withJoseph Wheeler.
Major General Shafter (1898) as the onlycredited cast member.
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.50/64
http://us.imdb.com/Name?Shafter,+William+Rufushttp://us.imdb.com/Name?Shafter,+William+Rufushttp://us.imdb.com/Name?Shafter,+William+Rufushttp://us.imdb.com/Title?0220793http://us.imdb.com/Title?0220793http://us.imdb.com/Title?0220793http://us.imdb.com/Title?0220793http://us.imdb.com/Name?Wheeler,+Joseph+(IV)http://us.imdb.com/Name?Wheeler,+Joseph+(IV)http://us.imdb.com/Title?0239582http://us.imdb.com/Title?0239582http://us.imdb.com/Title?0239582
-
Conclusion
Russell Alexander Alger has
#=6,
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.51/64
http://us.imdb.com/Name?Alger,+Russell+Alexanderhttp://us.imdb.com/Name?Alger,+Russell+Alexanderhttp://us.imdb.com/Name?Alger,+Russell+Alexanderhttp://oracleofbacon.org/cgi-bin/oracle/movielinks?firstname=Bacon%2C+Kevin&game=1&secondname=kevin+baconhttp://us.imdb.com/Name?Wheeler,+Joseph+(IV)http://us.imdb.com/Name?Wheeler,+Joseph+(IV)http://oracleofbacon.org/cgi-bin/oracle/movielinks?firstname=Bacon%2C+Kevin&game=1&secondname=kevin+baconhttp://us.imdb.com/Name?Shafter,+William+Rufushttp://us.imdb.com/Name?Shafter,+William+Rufushttp://us.imdb.com/Name?Shafter,+William+Rufushttp://oracleofbacon.org/cgi-bin/oracle/movielinks?firstname=Bacon%2C+Kevin&game=1&secondname=kevin+bacon
-
Conclusion
Russell Alexander Alger has
#=6,
Joseph Wheeler has
#=7, and
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.51/64
http://us.imdb.com/Name?Alger,+Russell+Alexanderhttp://us.imdb.com/Name?Alger,+Russell+Alexanderhttp://us.imdb.com/Name?Alger,+Russell+Alexanderhttp://oracleofbacon.org/cgi-bin/oracle/movielinks?firstname=Bacon%2C+Kevin&game=1&secondname=kevin+baconhttp://us.imdb.com/Name?Wheeler,+Joseph+(IV)http://us.imdb.com/Name?Wheeler,+Joseph+(IV)http://oracleofbacon.org/cgi-bin/oracle/movielinks?firstname=Bacon%2C+Kevin&game=1&secondname=kevin+baconhttp://us.imdb.com/Name?Shafter,+William+Rufushttp://us.imdb.com/Name?Shafter,+William+Rufushttp://us.imdb.com/Name?Shafter,+William+Rufushttp://oracleofbacon.org/cgi-bin/oracle/movielinks?firstname=Bacon%2C+Kevin&game=1&secondname=kevin+bacon
-
Conclusion
Russell Alexander Alger has
#=6,
Joseph Wheeler has
#=7, and
William Rufus Shafter has
#=8.
The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.51/64
http://us.imdb.com/Name?Alger,+Russell+Alexanderhttp://us.imdb.com/Name?Alger,+Russell+Alexanderhttp://us.imdb.com/Name?Alger,+Russell+Alexanderhttp://oracleofbacon.org/cgi-bin/oracle/movielinks?firstname=Bacon%2C+Kevin&game=1&secondname=kevin+baconhttp://us.imdb.com/Name?Wheeler,+Joseph+(IV)http://us.imdb.com/Name?Wheeler,+Joseph+(IV)http://oracleofbacon.org/cgi-bin/oracle/movielinks?firstname=Bacon%2C+Kevin&game=1&secondname=kevin+baconhttp://us.imdb.com/Name?Shafter,+William+Rufushttp://us.imdb.com/Name?Shafter,+William+Rufushttp://us.imdb.com/Name?Shafter,+William+Rufushttp://oracleofbacon.org/cgi-bin/oracle/movielinks?firstname=Bacon%2C+Kevin&game=1&secondname=kevin+bacon
-
The chain
8 William Rufus Shafter was in Surrender ofGeneral Toral (1898) with Joseph Wheeler
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.52/64
http://us.imdb.com/Name?Shafter,+William+Rufushttp://us.imdb.com/Name?Shafter,+William+Rufushttp://us.imdb.com/Name?Shafter,+William+Rufushttp://us.imdb.com/Title?0220793http://us.imdb.com/Title?0220793http://us.imdb.com/Title?0220793http://us.imdb.com/Title?0220793http://us.imdb.com/Name?Wheeler,+Joseph+(IV)http://us.imdb.com/Name?Wheeler,+Joseph+(IV)http://us.imdb.com/Name?Wheeler,+Joseph+(IV)http://us.imdb.com/Name?Wheeler,+Joseph+(IV)http://us.imdb.com/Title?0239582http://us.imdb.com/Title?0239582http://us.imdb.com/Title?0239582http://us.imdb.com/Title?0239582http://us.imdb.com/Title?0239582http://us.imdb.com/Title?0239582http://us.imdb.com/Title?0239582http://us.imdb.com/Title?0239582http://us.imdb.com/Title?0239582http://us.imdb.com/Title?0239582http://us.imdb.com/Name?Alger,+Russell+Alexanderhttp://us.imdb.com/Name?Alger,+Russell+Alexanderhttp://us.imdb.com/Name?Alger,+Russell+Alexanderhttp://us.imdb.com/Name?Alger,+Russell+Alexanderhttp://us.imdb.com/Name?Alger,+Russell+Alexanderhttp://us.imdb.com/Name?Alger,+Russell+Alexanderhttp://www.imdb.com/title/tt0272253/http://www.imdb.com/title/tt0272253/http://www.imdb.com/title/tt0272253/http://www.imdb.com/title/tt0272253/http://www.imdb.com/title/tt0272253/http://www.imdb.com/title/tt0272253/http://us.imdb.com/Name?McKinley,+William+(I)http://us.imdb.com/Name?McKinley,+William+(I)http://us.imdb.com/Name?McKinley,+William+(I)http://www.imdb.com/name/nm1299773/http://www.imdb.com/name/nm1299773/http://www.imdb.com/name/nm1299773/http://www.imdb.com/name/nm1299773/http://www.imdb.com/title/tt0232229/http://www.imdb.com/title/tt0232229/http://www.imdb.com/title/tt0232229/http://www.imdb.com/title/tt0232229/http://www.imdb.com/title/tt0232229/http://www.imdb.com/title/tt0232229/http://www.imdb.com/title/tt0232229/http://www.imdb.com/title/tt0232229/http://www.imdb.com/title/tt0232229/http://www.imdb.com/title/tt0232229/http://www.imdb.com/title/tt0232229/http://www.imdb.com/title/tt0232229/http://www.imdb.com/name/nm0740485/http://www.imdb.com/name/nm0740485/http://www.imdb.com/name/nm0740485/http://www.imdb.com/name/nm0740485/http://www.imdb.com/name/nm0740485/http://www.imdb.com/name/nm0740485/http://www.imdb.com/name/nm0740485/http://www.imdb.com/name/nm0740485/http://www.imdb.com/title/tt0008787/http://www.imdb.com/title/tt0008787/http://www.imdb.com/title/tt0008787/http://www.imdb.com/title/tt0008787/http://www.imdb.com/title/tt0008787/http://www.imdb.com/title/tt0008787/http://www.imdb.com/name/nm0569747/http://www.imdb.com/name/nm0569747/http://www.imdb.com/name/nm0569747/http://www.imdb.com/name/nm0569747/http://www.imdb.com/title/tt0033535/http://www.imdb.com/title/tt0033535/http://www.imdb.com/title/tt0033535/http://www.imdb.com/title/tt0033535/http://www.imdb.com/title/tt0033535/http://www.imdb.com/name/nm0741731/http://www.imdb.com/name/nm0741731/http://www.imdb.com/name/nm0741731/http://www.imdb.com/name/nm0741731/http://www.imdb.com/title/tt0113870/http://www.imdb.com/title/tt0113870/http://www.imdb.com/title/tt0113870/http://www.imdb.com/title/tt0113870/
-
The chain
8 William Rufus Shafter was in Surrender ofGeneral Toral (1898) with Joseph Wheeler
7 Joseph Wheeler was in General Wheelerand Secretary of War Alger at CampWikoff (1898) with Russell Alexander Alger
▽The small world problem: Kevin Bacon, Paul Erdős, William McKinley, and me – 5 degrees of graph theory – p.52/64
http://us.imdb.com/Name?Shafter,+William+Rufushttp://us.imdb.com/Name?Shafter,+William+Rufushttp://us.imdb.com/Name?Shafter,+William+Rufushttp://us.imdb.com/Title?0220793http://us.imdb.com/Title?0220793http://us.imdb.com/Title?0220793http://us.imdb.com/Title?0220793http://us.imdb.com/Name?Wheeler,+Joseph+(IV)http://us.imdb.com/Name?Wheeler,+Joseph+(IV)http://us.imdb.com/Name?Wheeler,+Joseph+(IV)http://us.imdb.com/Name?Wheeler,+Joseph+(IV)http://us.imdb.com/Title?0239582http://us.imdb.com/Title?0239582http://us.imdb.com/Title?0239582http://us.imdb.com/Title?0239582http://us.imdb.com/Title?0239582http://us.imdb.com/Title?0239582http://us.imdb.com/Title?0239582http://us.imdb.com/Title?0239582http://us.imdb.com/Title?0239582http://us.imdb.com/Title?0239582http://us.imdb.com/Name?Alger,+Russell+Alexanderhttp://us.imdb.com/Name?Alger,+Russell+Alexanderhttp://us.imdb.com/Name?Alger,+Russell+Alexanderhttp://us.imdb.com/Name?Alger,+Russell+Alexanderhttp://us.imdb.com/Name?Alger,+Russell+Alexanderhttp://us.imdb.com/Name?Alger,+Russell+Alexanderhttp://www.imdb.com/title/tt0272253/http://www.imdb.com/title/tt0272253/http://www.imdb.com/title/tt0272253/http://www.imdb.com/title/tt0272253/http://www.imdb.com/title/tt0272253/http://www.imdb.com/title/tt0272253/http://us.imdb.com/Name?McKinley,+William+(I)http://us.imdb.com/Name?McKinley,+William+(I)http://us.imdb.com/Name?McKinley,+William+(I)http://www.imdb.com/name/nm1299773/http://www.imdb.com/name/nm1299773/http://www.imdb.com/name/nm1299773/http://www.imdb.com/name/nm1299773/http://www.imdb.com/title/tt0232229/http://www.imdb.com/title/tt0232229/http://www.imdb.com/title/tt0232229/http://www.imdb.com/title/tt0232