The small world problem: Kevin Bacon, Paul Erdos, William˝ …orion.math.iastate.edu/rymartin/talks/SixDegrees/recruit... · 2005. 9. 20. · The small world problem: Kevin Bacon,

The small world problem: KevinBacon, Paul Erdős, William

McKinley, and me – 5 degrees ofgraph theory

Ryan Martin

[email protected]

Mathematics Department

Iowa State University

mailto:[email protected]://www.math.iastate.edu/http://www.iastate.edu/

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

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



Kevin Bacon.

This is false. It is Dennis Hopper.





Kevin Bacon.

We link two actors together if they appeared together inthe same movie.


http://www.cs.virginia.edu/oraclehttp://oracleofbacon.org/cgi-bin/oracle/movielinks?firstname=Bacon%2C+Kevin&game=1&secondname=kevin+baconhttp://www.imdb.com



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.)


http://www.cs.virginia.edu/oraclehttp://oracleofbacon.org/cgi-bin/oracle/movielinks?firstname=Bacon%2C+Kevin&game=1&secondname=kevin+baconhttp://www.imdb.com

Bacon number

The actor’s Bacon number is the fewest number of stepsit takes to connect that actor to


http://www.cs.virginia.edu/oracle/http://www.imdb.com/name/nm0000102/

Bacon number

The actor’s

#is the fewest number of steps

it takes to connect that actor to

Kevin Bacon.


http://www.imdb.com/name/nm0000102/http://www.imdb.com/name/nm0000102/

Bacon number

The actor’s Bacon number is the fewest number of stepsit takes to connect that actor to


http://www.cs.virginia.edu/oracle/http://www.imdb.com/name/nm0000102/

Bacon number

An actor can have infinite Kevin Bacon number.


http://www.imdb.com/name/nm0000102/

Bacon number

An actor can have infinite

#.

(For example, a TV actor who appears in no moviecredits.)


http://www.imdb.com/name/nm0000102/

Example: Kevin Costner

Kevin Costner is linked to Kevin Bacon because bothappeared in

JFK (1991).


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


is linked to because bothappeared in





JFK (1991).

So, Kevin Costner’s Kevin Bacon number is

???





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




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


Illustration: Kevin Costner

x x


http://www.imdb.com/name/nm0000126/http://www.imdb.com/title/tt0102138/http://www.imdb.com/name/nm0000102/

E.g.: Henry “Fonz” Winkler

We know that

appeared with in


http://www.imdb.com/name/nm0001857/http://www.imdb.com/name/nm0000474/http://www.imdb.com/title/tt0084412/


We know that

Henry Winkler appeared with Michael Keaton inNight Shift (1982)

appeared with in


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/


We know that


Michael Keaton appeared with Kim Basinger inBatman (1989)

appeared with in


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/


We know that



Kim Basinger appeared with Mickey Rourke in9 1/2 Weeks (1986)

appeared with in


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/


We know that



Kim Basinger appeared with Mickey Rourke in9 1/2 Weeks (1986)

Mickey Rourke appeared with Kevin Bacon inDiner (1982)


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/

A chain: Henry “Fonz” Winkler

x x x x x


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/

More: Henry “Fonz” Winkler

But it is also true that

appeared with ??? in .


http://www.imdb.com/name/nm0001857/http://www.imdb.com/title/tt0185431/



appeared with in .


http://www.imdb.com/name/nm0001857/http://www.imdb.com/name/nm0397212/http://www.imdb.com/title/tt0185431/



Henry Winkler appeared with Clint Howard inLittle Nicky (2000).

appeared with in


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



Henry Winkler appeared with Clint Howard inLittle Nicky (2000).

Clint Howard appeared with Kevin Bacon inMy Dog Skip (2000).


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/

Better: Henry “Fonz” Winkler

x x x


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/

Can we do even better?

has never appeared in a film with .


http://www.imdb.com/name/nm0000102/http://www.imdb.com/name/nm0001857/


Kevin Bacon has never appeared in a film withHenry Winkler.

So, ’s

#is


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



So, Henry Winkler’s Kevin Bacon number is

???





So, Henry Winkler’s Kevin Bacon number is

2



In sum: Henry “Fonz” Winkler

PPPPPPhhhhhh((((((

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((((((((

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hhhx xx xxx


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


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


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

#.


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.


http://www.cs.virginia.edu/oracle/http://www.cs.virginia.edu/oracle/

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




For every pair of successful actors, they areconnected by a path of length ≤ 5

Why?


Our model

We will represent actors by vertices


Our model


x


Our model


xand connect them with edges


Our model



x xif they appeared in the same film.


Our model



x xif they appeared in the same film.

This is a graph.


Model parameters

There are n actors.


Model parameters

There are n actors.

Fix a constant d.


Model parameters

There are n actors.

Fix a constant d.

We will begin with an arbitrary graph H.


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.


Model parameters

There are n actors.

Fix a constant d.


The constant d can be extremely tiny:


Model parameters

There are n actors.

Fix a constant d.



0.1


Model parameters

There are n actors.

Fix a constant d.



0.1, 0.01


Model parameters

There are n actors.

Fix a constant d.



0.1, 0.01, 0.000001

It just needs to be independent of n.


Random casting

We add f(n) random casting connections.


Random casting

We add f(n) random casting connections.

What does random mean?


Random edges

Let N be the number of pairs with no connectionbetween them (non-edges). We can create m newrandom edges in two ways:


Random edges


For every set of m unconnected pairs, choose oneset uniformly at random.


Random edges



Connect a previously unconnected pair,independently, with probability m/N .


Random edges



Connect a previously unconnected pair,independently, with probability m/N (coin flips).


Random edges



Connect a previously unconnected pair,independently, with probability m/N (coin flips).The average number of new connections is m.


The models

For our purposes, these produce the same results.

The question:


The models


The question:

What is the longest distance between any pair ofactors ?


The models


The question:

What is the longest distance between any pair ofactors (diameter)?


The theorem

Theorem. If f(n) → ∞ as n → ∞, thenPr (diam ≤ 5) → 1

Important points:

Recall f(n) is the number of random connections.


The theorem


Important points:


“5” doesn’t depend on d at all.


The theorem


Important points:



f(n) can be very small:


The theorem


Important points:



f(n) can be very small:√

n


The theorem


Important points:




n, log n


The theorem


Important points:




n, log n,√

log log log n


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 proof

To prove the theorem, you need the Regularity Lemma.


The proof


The Regularity Lemma is ’s powerfuland complicated graph theoretic tool.


http://www.renyi.hu/~szemered/

The proof


The Regularity Lemma is Endre Szemerédi’s powerfuland complicated graph theoretic tool.


http://www.renyi.hu/~szemered/

Best possible?

The theorem is “tight”:


Best possible?

The theorem is “tight”:

If there aren’t an infinite number of edges added,then some H ’s will be disconnected.


What about closer connections?

To get diam ≤ 4, you need randomconnections.





To get diam ≤ 4, you need c1 log n randomconnections.







To get diam ≤ 2, you need c2n log n randomconnections.


Origins of the problem

The original question posed by Stanley Milgram



The original question posed by Stanley Milgramasked what the average distance was among peoplein a network.




His methodology was very flawed




His methodology was very flawed – he sent out lettersfrom a single source and waited for a return message,many didn’t come back.




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.



But his assertion that the average distance was around 6stuck.


Advantages and disadvantages

Our model has some advantages and somedisadvantages.

Disadvantages:




Disadvantages:Each person must beconnected to at leastdn other people.





Not good for modellingthe Internet.






Advantages:






Advantages:Very weak restrictionon structure.







The number ofrandom connectionsis tiny.







The number ofrandom connectionsis tiny.

An upper bound of 5on the diameter.


Another model

There is another model due to Fan Chung


Another model

There is another model due to Fan Chungand others


Another model

There is another model due to Fan Chungand others (no pictures)


Another model

There is another model due to Fan Chungand others (no pictures)which eliminates the problems of not being applicable tothe Internet.


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 Chung model

Advantages:


The Chung model

Advantages:Easy to use, easy todo computations.


The Chung model


Models the Internetquite well.


The Chung model



Disdvantages:


The Chung model



Disdvantages:The number of randomconnections is huge.


The Chung model




Weak result: The av-erage distance is ≈log2 n/ log2 d̃.


The Chung model




Weak result: The av-erage distance is ≈log2 n/ log2 d̃, where d̃relates to the aver-age degree (numberof connections).


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.


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.


Iowa State cliques

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Computer Scientists


The cliché

The cliché states that every pair of people is separatedby at most

six degrees of separation.


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!


Erdős number

One of the most prolific mathematicians of the 20thcentury was

Paul (Pál) ErdősMarch 26, 1913-September 20, 1996


http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Erdos.html

Erdős number project

The Erdős number project is concerned with the distanceof mathematicians from Paul Erdős.


http://www.oakland.edu/~grossman/erdoshp.htmlhttp://www.ams.org/mathscinet


The

#project is concerned with the distance

of mathematicians from Paul Erdős.


http://www.oakland.edu/~grossman/erdoshp.htmlhttp://www.oakland.edu/~grossman/erdoshp.htmlhttp://www.ams.org/mathscinet


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.


http://www.oakland.edu/~grossman/erdoshp.htmlhttp://www.oakland.edu/~grossman/erdoshp.htmlhttp://www.ams.org/mathscinet

Most prolific authors

: 1401 papers (Erdős number )


http://www.oakland.edu/~grossman/erdoshp.html



Drumi Bainov: 782 (Erdős number )






Leonard Carlitz: 730 (Erdős number )







Lucien Godeaux: 644 (Erdős number )







Lucien Godeaux: 644 (Erdős number )

Saharon Shelah: 600 (Erdős number )




: 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)



Erdős number statistics

wrote 1401 papers in Math Reviews.


http://www.oakland.edu/~grossman/erdoshp.htmlhttp://www.oakland.edu/~grossman/papers.html



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 authors per paper: 1.45

Average number of papers per author: 6.87



Experimental data

#

0 1

1 502

2 5713

3 26422

4 62136

5 66157


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)



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)



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)



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Computer networks

Graphs model much more serious stuff.

I.e.,

computer networks,


Computer networks


I.e.,

computer networks,

shipping routes,


Computer networks


I.e.,

computer networks,

shipping routes,

distribution networks.


Network question

In networks we are concerned with one particularquantity:


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.


Same model

n computers


Same model

n computers

in H, each computer is connected to ≥ dn others


Same model

n computers


add f(n) random connections


Same model

n computers


add f(n) random connections

Of course, we want high connectivity with as littlerandomness as possible.


Connectivity theorem

Theorem. Let k be a function of n that is ≪ n.



Theorem. Let k be a function of n that is ≪ n. (That is,k grows more slowly than n as n → ∞.)



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 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.


Bottom line

A way to interpret this theorem is:


Bottom line


If you need k-connectivity,


Bottom line



then you need to add a little more(asymptotically) random edges than k.


Bottom line



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.


Worst case

What is that H0?

H0 =

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Other properties

We’ve used this model to investigate other properties:

Hamilton cycle


Other properties


Hamilton cycle

Small cliques as subgraphs


Other properties


Hamilton cycle

Small cliques as subgraphs

Chromatic number


What next?

Shall we conclude with

more mathematics,


What next?


more mathematics,

people with high Bacon numbers,


What next?


more mathematics,


mathematicians and dead presidents, or


What next?


more mathematics,


mathematicians and dead presidents, or

connections between Bacon numbers and Erdősnumbers?


It’s just killing you, isn’t it?

Let us return to the Kevin Bacon question.


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 .


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.


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



#.


Not soon coming to DVD:





#.


Not soon coming to DVD: Mr. Edison at Work inHis Chemical Laboratory (1897).



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)


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


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.


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


William Rufus Shafter also appeared in two films:

Surrender of General Toral (1898) withJoseph Wheeler.


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


William Rufus Shafter also appeared in two films:

Surrender of General Toral (1898) withJoseph Wheeler.

Major General Shafter (1898) as the onlycredited cast member.


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,


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


#=6,

Joseph Wheeler has

#=7, and



Conclusion


#=6,

Joseph Wheeler has

#=7, and

William Rufus Shafter has

#=8.



The chain

8 William Rufus Shafter was in Surrender ofGeneral Toral (1898) with Joseph Wheeler


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


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

The small world problem: Kevin Bacon, Paul Erdos, William˝ …orion.math.iastate.edu/rymartin/talks/SixDegrees/recruit... · 2005. 9. 20. · The small world problem: Kevin Bacon,

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