six steps to kevin bacon

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					  Six degrees of graph theory:
                      ˝
Kevin Bacon, Paul Erdos, William
       McKinley and me
               Ryan Martin
           rymartin@iastate.edu


            Assistant Professor
          Mathematics Department
           Iowa State University




                                                                  o
                 Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 1/4
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




                                                                         o
                        Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 2/4
Six? degrees of separation
In the Kevin Bacon Game, it is postulated that the
center of the Hollywood universe is

                    Kevin Bacon.




                                                                        o
                       Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 3/4
Six? degrees of separation
In the Kevin Bacon Game, it is postulated that the
center of the Hollywood universe is


                                        .




                                                                        o
                       Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 3/4
Six? degrees of separation
In the Kevin Bacon Game, it is postulated that the
center of the Hollywood universe is

                      Kevin Bacon.


This is false. It is Rod Steiger.




                                                                           o
                          Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 3/4
Six? degrees of separation
In the Kevin Bacon Game, it is postulated that the
center of the Hollywood universe is

                      Kevin Bacon.


This is false. It is Rod Steiger.




                                                                           o
                          Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 3/4
Six? degrees of separation
In the Kevin Bacon Game, it is postulated that the
center of the Hollywood universe is

                    Kevin Bacon.


We link two actors together if they appeared together
in the same movie.




                                                                        o
                       Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 3/4
Six? degrees of separation
In the Kevin Bacon Game, it is postulated that the
center of the Hollywood universe is

                    Kevin Bacon.


We link two actors together if they appeared together
in the same movie.

(They must be together on a cast list at the IMDb.)




                                                                         o
                        Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 3/4
Bacon number
The actor’s

                    Bacon number

is the fewest number of steps it takes to connect that
actor to

                     Kevin Bacon.




                                                                         o
                        Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 4/4
Bacon number
The actor’s
                                        #
is the fewest number of steps it takes to connect that
actor to


                                         .




                                                                         o
                        Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 4/4
Bacon number
The actor’s

                    Bacon number

is the fewest number of steps it takes to connect that
actor to

                     Kevin Bacon.




                                                                         o
                        Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 4/4
Bacon number
An actor can have infinite
                                      #
                                            .




                                                                        o
                       Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 4/4
Bacon number
An actor can have infinite
                                      #
                                            .


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




                                                                        o
                       Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 4/4
Example: Kevin Costner


Kevin Costner is linked to Kevin Bacon because both
appeared in

                    JFK (1991).




                                                                       o
                      Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 5/4
Example: Kevin Costner


              is linked to                                          because both
appeared in




                                                                        o
                       Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 5/4
Example: Kevin Costner
is linked to because both appeared in

                    JFK (1991).




So, Kevin Costner’s Kevin Bacon number is

                         ???


                                                                        o
                       Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 5/4
Example: Kevin Costner
is linked to because both appeared in

                    JFK (1991).




So, Kevin Costner’s Kevin Bacon number is

                             1

                                                                        o
                       Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 5/4
Example: Kevin Costner
is linked to because both appeared in

                       JFK (1991).


                                   #
So,               ’s                                                        is

                               1

                                                                          o
                         Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 5/4
Illustration: Kevin Costner



      x                                                                      x




                                                                 o
                Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 6/4
Illustration: Kevin Costner



      x                                                                      x




                                                                 o
                Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 6/4
Illustration: Kevin Costner



      x                                                                      x




                                                                 o
                Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 6/4
Illustration: Kevin Costner



      x                                                                      x




                                                                 o
                Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 6/4
E.g.: Henry “Fonz” Winkler
We know that



  •       appeared with                        in




                                                                      o
                     Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 7/4
E.g.: Henry “Fonz” Winkler
We know that
 • Henry Winkler appeared with Michael Keaton in
   Night Shift (1982)



  •       appeared with                        in




                                                                      o
                     Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 7/4
E.g.: Henry “Fonz” Winkler
We know that
 • Henry Winkler appeared with Michael Keaton in
   Night Shift (1982)
 • Michael Keaton appeared with Kim Basinger in
   Batman (1989)



  •       appeared with                        in




                                                                      o
                     Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 7/4
E.g.: Henry “Fonz” Winkler
We know that
 • Henry Winkler appeared with Michael Keaton in
   Night Shift (1982)
 • Michael Keaton appeared with Kim Basinger in
   Batman (1989)
 • Kim Basinger appeared with Mickey Rourke in
   9 1/2 Weeks (1986)



  •       appeared with                        in


                                                                      o
                     Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 7/4
E.g.: Henry “Fonz” Winkler
We know that
 • Henry Winkler appeared with Michael Keaton in
   Night Shift (1982)
 • Michael Keaton appeared with Kim Basinger in
   Batman (1989)
 • Kim Basinger appeared with Mickey Rourke in
   9 1/2 Weeks (1986)
 • Mickey Rourke appeared with Kevin Bacon in
   Diner (1982)




                                                                      o
                     Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 7/4
A chain: Henry “Fonz” Winkler



     x     x            x                         x                         x




                                                                o
               Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 8/4
A chain: Henry “Fonz” Winkler



     x     x            x                         x                         x




                                                                o
               Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 8/4
A chain: Henry “Fonz” Winkler



     x     x            x                         x                         x




                                                                o
               Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 8/4
A chain: Henry “Fonz” Winkler



     x     x            x                         x                         x




                                                                o
               Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 8/4
A chain: Henry “Fonz” Winkler



     x     x            x                         x                         x




                                                                o
               Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 8/4
A chain: Henry “Fonz” Winkler



     x     x            x                         x                         x




                                                                o
               Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 8/4
A chain: Henry “Fonz” Winkler



     x     x            x                         x                         x




                                                                o
               Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 8/4
A chain: Henry “Fonz” Winkler



     x     x            x                         x                         x




                                                                o
               Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 8/4
A chain: Henry “Fonz” Winkler



     x     x            x                         x                         x




                                                                o
               Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 8/4
A chain: Henry “Fonz” Winkler



     x     x            x                         x                         x




                                                                o
               Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 8/4
More: Henry “Fonz” Winkler
But it is also true that




  •             appeared with            ???                       in
      (2000).




                                                                            o
                           Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 9/4
More: Henry “Fonz” Winkler
But it is also true that




  •             appeared with                                       in
      (2000).




                                                                            o
                           Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 9/4
More: Henry “Fonz” Winkler
But it is also true that

  •   Henry Winkler appeared with Clint Howard
      in Little Nicky (2000).




  •              appeared with                                      in
      (2000)



                                                                            o
                           Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 9/4
More: Henry “Fonz” Winkler
But it is also true that

  •   Henry Winkler appeared with Clint Howard
      in Little Nicky (2000).
  •   Clint Howard appeared with Kevin Bacon in
      My Dog Skip (2000).




                                                                            o
                           Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 9/4
Better: Henry “Fonz” Winkler



     x                  x                                                   x




                                                               o
              Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 10/4
Better: Henry “Fonz” Winkler



     x                  x                                                   x




                                                               o
              Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 10/4
Better: Henry “Fonz” Winkler



     x                  x                                                   x




                                                               o
              Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 10/4
Better: Henry “Fonz” Winkler



     x                  x                                                   x




                                                               o
              Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 10/4
Better: Henry “Fonz” Winkler



     x                  x                                                   x




                                                               o
              Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 10/4
Better: Henry “Fonz” Winkler



     x                  x                                                   x




                                                               o
              Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 10/4
Can we do even better?

        has never appeared in a film with


    .




                                                                  o
                 Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 11/4
Can we do even better?

Kevin Bacon has never appeared in a film with
Henry Winkler.

                                  #
So,              ’s                                                       is




                                                                       o
                      Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 11/4
Can we do even better?

Kevin Bacon has never appeared in a film with
Henry Winkler.



So, Henry Winkler’s Kevin Bacon number is

                         ???



                                                                       o
                      Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 11/4
Can we do even better?

Kevin Bacon has never appeared in a film with
Henry Winkler.



So, Henry Winkler’s Kevin Bacon number is

                             2



                                                                       o
                      Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 11/4
In sum: Henry “Fonz” Winkler



                     x
               @@@@hhhhhh
     x                           hx
            @@
     @@ @ @@               h hhh
     €€                          
                               
           € xh            x 
        €€
                           
                hh h x @@@
           €h             @
                   h@@




                                                                 o
                Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 12/4
In sum: Henry “Fonz” Winkler



                     x
               @@@@hhhhhh
     x                           hx
            @@
     @@ @ @@               h hhh
     €€                          
                               
           € xh            x 
        €€
                           
                hh h x @@@
           €h             @
                   h@@




                                                                 o
                Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 12/4
In sum: Henry “Fonz” Winkler



                     x
               @@@@hhhhhh
     x                           hx
            @@
     @@ @ @@               h hhh
     €€                          
                               
           € xh            x 
        €€
                           
                hh h x @@@
           €h             @
                   h@@




                                                                 o
                Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 12/4
Experimental data

       # of actors
   0        1
   1      1806
   2    145024
   3    395126
   4     95497
   5      7451



                                                                      o
                     Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 13/4
Experimental data

       # of actors
   0        1                                       # of actors
   1      1806
                                    6                  933
   2    145024
                                    7                  106
   3    395126
                                    8                   13
   4     95497
   5      7451



                                                                      o
                     Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 13/4
What about the high numbers?
As we said before, there are actors with infinite
       #
         .




                                                                        o
                       Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 14/4
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.




                                                                        o
                       Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 14/4
Kevin Bacon not so special
Most successful actors follow the same pattern:




                                                                        o
                       Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 15/4
Kevin Bacon not so special
Most successful actors follow the same pattern:

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




                                                                        o
                       Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 15/4
Kevin Bacon not so special
Most successful actors follow the same pattern:

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



                       Why?




                                                                        o
                       Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 15/4
Our model
We will represent actors by vertices




                                                                        o
                       Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 16/4
Our model
We will represent actors by vertices
                                 x




                                                                        o
                       Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 16/4
Our model
We will represent actors by vertices
                                 x

and connect them with edges




                                                                        o
                       Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 16/4
Our model
We will represent actors by vertices
                                 x

and connect them with edges
                      x                       x

if they appeared in the same film.




                                                                        o
                       Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 16/4
Our model
We will represent actors by vertices
                                 x

and connect them with edges
                      x                       x

if they appeared in the same film.

                   This is a graph.




                                                                        o
                       Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 16/4
Model parameters
 •   There are n actors.




                                                                            o
                           Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 17/4
Model parameters
 •   There are n actors.
 •   Fix a constant d.




                                                                            o
                           Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 17/4
Model parameters
 •   There are n actors.
 •   Fix a constant d.
 •   We will begin with an arbitrary graph H.




                                                                       o
                      Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 17/4
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 other
     actors.




                                                                         o
                        Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 17/4
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 other
     actors.
 •   The constant d can be extremely tiny:




                                                                         o
                        Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 17/4
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 other
     actors.
 •   The constant d can be extremely tiny:
                  0.1




                                                                         o
                        Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 17/4
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 other
     actors.
 •   The constant d can be extremely tiny:
                  0.1,     0.01




                                                                          o
                         Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 17/4
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 other
     actors.
 •   The constant d can be extremely tiny:
                  0.1,     0.01,                 0.000001
     It just needs to be independent of n.



                                                                          o
                         Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 17/4
Random casting
We add f (n) random casting connections.




                                                                       o
                      Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 18/4
Random casting
We add f (n) random casting connections.




What does random mean?




                                                                       o
                      Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 18/4
Random edges
Let N be the number of pairs with no connection
between them (non-edges). We can create m new
random edges in two ways:




                                                                       o
                      Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 19/4
Random edges
Let N be the number of pairs with no connection
between them (non-edges). We can create m new
random edges in two ways:

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




                                                                         o
                        Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 19/4
Random edges
Let N be the number of pairs with no connection
between them (non-edges). We can create m new
random edges in two ways:

  •   For every set of m unconnected pairs, choose one
      set uniformly at random.
  •   Connect a previously unconnected pair,
      independently, with probability m/N .




                                                                         o
                        Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 19/4
Random edges
Let N be the number of pairs with no connection
between them (non-edges). We can create m new
random edges in two ways:

  •   For every set of m unconnected pairs, choose one
      set uniformly at random.
  •   Connect a previously unconnected pair,
      independently, with probability m/N (coin flips).




                                                                         o
                        Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 19/4
Random edges
Let N be the number of pairs with no connection
between them (non-edges). We can create m new
random edges in two ways:

  •   For every set of m unconnected pairs, choose one
      set uniformly at random.
  •   Connect a previously unconnected pair,
      independently, with probability m/N (coin flips).
      The average number of new connections is m.




                                                                         o
                        Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 19/4
The models
For our purposes, these produce the same results.


The question:




                                                                        o
                       Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 20/4
The models
For our purposes, these produce the same results.


The question:
    What is the longest distance between any
    pair of actors           ?




                                                                        o
                       Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 20/4
The models
For our purposes, these produce the same results.


The question:
    What is the longest distance between any
    pair of actors (diameter)?




                                                                        o
                       Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 20/4
The theorem
Theorem. If f (n) → ∞ as n → ∞, then
                Pr (diam ≤ 5) → 1


Important points:
  • Recall f (n) is the number of random
    connections.




                                                                       o
                      Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 21/4
The theorem
Theorem. If f (n) → ∞ as n → ∞, then
                Pr (diam ≤ 5) → 1


Important points:
  • Recall f (n) is the number of random
    connections.
  • “5” doesn’t depend on d at all.




                                                                       o
                      Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 21/4
The theorem
Theorem. If f (n) → ∞ as n → ∞, then
                Pr (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:




                                                                       o
                      Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 21/4
The theorem
Theorem. If f (n) → ∞ as n → ∞, then
                Pr (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


                                                                       o
                      Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 21/4
The theorem
Theorem. If f (n) → ∞ as n → ∞, then
                Pr (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


                                                                       o
                      Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 21/4
The theorem
Theorem. If f (n) → ∞ as n → ∞, then
                Pr (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


                                                                        o
                       Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 21/4
Qualifying the model
This is a nice result on a pretty good model.



The model only assumes some density conditions and
a little bit of randomness.




                                                                         o
                        Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 22/4
The proof
To prove the theorem, you need the Regularity
Lemma.




                                                                       o
                      Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 23/4
The proof
To prove the theorem, you need the Regularity
Lemma.


The Regularity Lemma is                   ’s
powerful and complicated graph theoretic tool.




                                                                       o
                      Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 23/4
The proof
To prove the theorem, you need the Regularity
Lemma.


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




                                                                       o
                      Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 23/4
Best possible?

The theorem is “tight”:




                                                                           o
                          Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 24/4
Best possible?

The theorem is “tight”:




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




                                                                           o
                          Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 24/4
What about closer connections?
  •   To get diam ≤ 4, you need                                      random
      connections.
  •   To get diam ≤ 3, you need                                      random
      connections.
  •   To get diam ≤ 2, you need                                          random
      connections.




                                                                        o
                       Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 25/4
What about closer connections?
  •   To get diam ≤ 4, you need c1 log n random
      connections.
  •   To get diam ≤ 3, you need c1 log n random
      connections.
  •   To get diam ≤ 2, you need                                          random
      connections.




                                                                        o
                       Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 25/4
What about closer connections?
  •   To get diam ≤ 4, you need c1 log n random
      connections.
  •   To get diam ≤ 3, you need c1 log n random
      connections.
  •   To get diam ≤ 2, you need c2 n log n random
      connections.




                                                                         o
                        Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 25/4
Applying this knowledge

   To have it be very likely that everyone is
   connected by a path of no more than 5
   acquaintances, just arrange a few random
   meetings.




                                                                     o
                    Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 26/4
Applying this knowledge

     To have it be very likely that everyone is
     connected by a path of no more than 5
     acquaintances, just arrange a few random
     meetings.

Think about people at Central College.




                                                                       o
                      Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 26/4
Central College cliques
 '                         $
         xhhx
                   2x
         h
         t
    x                  ˜x
               22 ˜
               e  ¥e
           t ¡¡ e
        222 
      €x
    2                ˜
    € €t   €€  ¥ e   x
 x „ i ¡  & x  hx
  ¤¤„ l ¡ t€ e ¥ $ hh
       l i             e
                   e
                  €$
            l t ˜ g 
           x &        ˜x¤
 ¤                   ˜
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            l
            
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                       3
    v      &@ x 3 x f¤
      v x @ @3 l
                   t
                   t
        @ e
        &          ¨
             x
         e ¨¨i
            e i
             ¨
               eix
 &                         %
    Mathematicians




                                                                      o
                     Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 27/4
Central College cliques
 '                        '
                          $                        $
        xhhx
                    x                 3x – x
        h                               –
                                x
        t         2                3 d –
    x   222  ¥e ˜ x                             x
                 2˜
                                h h @x
              2
              e                 3 h  44
          t ¡¡ e                3
      €x                              x
    2              ˜                    ¢
    € €t                                hd
                                        @h ¢
                              x
                                      4
                                      
                                      @
                     x
          €€  ¥ e                    4
 x „ i ¡  & x  hx
                     e
  ¤¤„ l ¡ t€ e ¥ $ hh                r
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                  e
                 €$                      r e  ¥
           l t ˜ g 
          x–
        „i¡  & ˜˜x¤       x
 ¤ 
  v        
           l
          – –t        g ¤  
                           @ @e˜
                               @             e¥
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                 – 33 f    e ˜ e v ˜˜ x
                           ˜
                           ˜
          &@ x l 3 x f¤
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        &
        @                        — lt
                              e v   —x¨
                                   ——
            x                       v x$
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                                      $ g¨ x
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           e i
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              eix
                                   2
                                  –  t
 &                        &
                          %                        %
    Mathematicians                   Social Scientists




                                                                       o
                      Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 27/4
More Central College cliques
 '                      $
        x xx – x
        €
        €
           ––

        x       x   x
                       But for us to get
    4
  xh x
   4                   diameter ≤ 5, we
  4
  4 h
  h h               x  do need each per-
    x
           xx        x
                       son to know at
     x
        —22 x
                       least dn others be-
         x2
     ——
     —
                       fore we add few
 &                     %
                       random edges.




                                                                         o
                        Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 28/4
More Central College cliques
 '                     $
        x xx – x
        €
        €
           ––

       x   x       x
                     But for us to get
     4
 xh x
   4                 diameter ≤ 5, we
 4
 4 h
 h h             x   do need each per-
   x
         x   x    x
                     son to know at
      x
         —22 x
                     least dn others be-
           x2
      ——
      —
                     fore we add few
 &                   %
                     random edges.
 Computer Scientists




                                                                        o
                       Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 28/4
The cliché
The cliché states that every pair of people is separated
by at most
               six degrees of separation.




                                                                         o
                        Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 29/4
The cliché
The cliché states that every pair of people is separated
by at most
               six degrees of separation.


In fact, it is
                 FIVE degrees of separation
and there’s an actual proof!




                                                                           o
                          Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 29/4
   ˝
Erdos number
One of the most prolific mathematicians of the 20th
century was




                              o
                Paul (Pál) Erd˝ s
        March 26, 1913-September 20, 1996


                                                                       o
                      Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 30/4
   ˝
Erdos number
One of the most prolific mathematicians of the 20th
century was




                              o
                Paul (Pál) Erd˝ s
        March 26, 1913-September 20, 1996


                                                                       o
                      Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 30/4
   ˝
Erdos number project

The Erd˝ s number project is concerned with the
        o
                                        o
distance of mathematicians from Paul Erd˝ s.




                                                                       o
                      Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 31/4
   ˝
Erdos number project
          #
The               project is concerned with the
                                        o
distance of mathematicians from Paul Erd˝ s.




                                                                       o
                      Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 31/4
   ˝
Erdos number project
          #
The               project is concerned with the
                                        o
distance of mathematicians from Paul Erd˝ s.


Two mathematicians are connected if they
co-authored a paper together and that paper appears in
Mathematical Reviews, accessible by MathSciNet.




                                                                        o
                       Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 31/4
Most prolific authors


  •   : 1401 papers (Erd˝ s number )
                        o




                                                                   o
                  Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 32/4
Most prolific authors


  •        : 1401 papers (Erd˝ s number )
                             o
  •   Drumi Bainov: 782 (Erd˝ s number )
                            o




                                                                        o
                       Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 32/4
Most prolific authors


  •        : 1401 papers (Erd˝ s number )
                             o
  •   Drumi Bainov: 782 (Erd˝ s number )
                            o
  •   Leonard Carlitz: 730 (Erd˝ s number )
                               o




                                                                        o
                       Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 32/4
Most prolific authors


  •        : 1401 papers (Erd˝ s number )
                             o
  •   Drumi Bainov: 782 (Erd˝ s number )
                            o
  •   Leonard Carlitz: 730 (Erd˝ s number )
                               o
  •   Lucien Godeaux: 644 (Erd˝ s number
                              o                                                   )




                                                                        o
                       Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 32/4
Most prolific authors


  •        : 1401 papers (Erd˝ s number )
                             o
  •   Drumi Bainov: 782 (Erd˝ s number )
                            o
  •   Leonard Carlitz: 730 (Erd˝ s number )
                               o
  •   Lucien Godeaux: 644 (Erd˝ s number
                              o                                                   )
  •   Saharon Shelah: 600 (Erd˝ s number )
                              o


                                                                        o
                       Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 32/4
Most prolific authors


  •        : 1401 papers (Erd˝ s number 0)
                             o
  •   Drumi Bainov: 782 (Erd˝ s number 4)
                            o
  •   Leonard Carlitz: 730 (Erd˝ s number 2)
                               o
  •   Lucien Godeaux: 644 (Erd˝ s number ∞)
                              o
  •   Saharon Shelah: 600 (Erd˝ s number 1)
                              o


                                                                         o
                        Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 32/4
   ˝
Erdos number statistics


  •    wrote 1401 papers in Math Reviews.




                                                                   o
                  Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 33/4
   ˝
Erdos number statistics


  •         wrote 1401 papers in Math Reviews.
  •   There are 337, 000 vertices (authors) in the graph.




                                                                          o
                         Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 33/4
   ˝
Erdos number statistics


  •         wrote 1401 papers in Math Reviews.
  •   There are 337, 000 vertices (authors) in the graph.

  •   There are about 496, 000 edges.




                                                                          o
                         Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 33/4
   ˝
Erdos 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



                                                                          o
                         Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 33/4
   ˝
Erdos 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

                                                                          o
                         Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 33/4
Experimental data

         #
   0     1
   1    502
   2   5713
   3   26422
   4   62136
   5   66157
   6   32280
   7   10431

                                                                o
               Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 34/4
Experimental data

            #                        #
   0        1          8            3214
   1       509         9            953
   2      6984        10            262
   3      26422       11             94
   4      62136       12             23
   5      66157       13             4
   6      32280       14             7
   7      10431       15             1
 (Most recent data)

                                                                       o
                      Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 34/4
Experimental data

            #                        #
   0        1          8            3214
   1       509         9            953
   2      6984        10            262
   3      26422       11             94
   4      62136       12             23
   5      66157       13             4
   6      32280       14             7
   7      10431       15             1   (R. G. Kamalov)
 (Most recent data)

                                                                       o
                      Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 34/4
The unknown mathematician
                                     x
                                    Dl
                                   D  l
                               D                  l
                           D                          l
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              D                                                           l
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      D                                                                            l
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     €€
          €€                                                                      
                  €€                                             
                     €                    
                                     x
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                                   €€




                                                                                o
                               Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 35/4
The unknown mathematician
                                     x
                                    Dl
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                     €                    
                                     x
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                                   €€




                                                                                o
                               Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 35/4
The unknown mathematician
                                     x
                                    Dl
                                   D  l
                               D                  l
                           D                          l
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                  D                                                  l
              D                                                           l
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                  €€                                             
                     €                    
                                     x
                           €€           
                                   €€




                                                                                o
                               Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 35/4
The unknown mathematician
                                     x
                                    Dl
                                   D  l
                               D                  l
                           D                          l
                       D                               l
                   D                                            l
                  D                                                  l
              D                                                           l
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      D                                                                            l
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          €€                                                                      
                  €€                                             
                     €                    
                                     x
                           €€           
                                   €€




                                                                                o
                               Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 35/4
The unknown mathematician
                                     x
                                    Dl
                                   D  l
                               D                  l
                           D                          l
                       D                               l
                   D                                            l
                  D                                                  l
              D                                                           l
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     x                                   x                                          lx
      D                                                                            l
     D
     €€
          €€                                                                      
                  €€                                             
                     €                    
                                     x
                           €€           
                                   €€




                                                                                o
                               Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 35/4
The unknown mathematician
                                     x
                                    Dl
                                   D  l
                               D                  l
                           D                          l
                       D                               l
                   D                                            l
                  D                                                  l
              D                                                           l
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     x                                   x                                          lx
      D                                                                            l
     D
     €€
          €€                                                                      
                  €€                                             
                     €                    
                                     x
                           €€           
                                   €€




                                                                                o
                               Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 35/4
Computer networks
Graphs model much more serious stuff.

I.e.,
   •    computer networks,
   •    shipping routes,
   •    distribution networks.




                                                                            o
                           Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 36/4
Network question
In networks we are concerned with one particular
quantity:




                                                                       o
                      Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 37/4
Network question
In networks we are concerned with one particular
quantity:


connectivity: A connected graph is k-connected if
    removing any set of k − 1 vertices (and all
    relevant edges) leaves the graph connected.




                                                                       o
                      Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 37/4
Same model
 •   n computers




                                                                    o
                   Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 38/4
Same model
 •   n computers

 •   in H, each computer is connected to ≥ dn others




                                                                        o
                       Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 38/4
Same model
 •   n computers

 •   in H, each computer is connected to ≥ dn others

 •   add f (n) random connections




                                                                        o
                       Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 38/4
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 little
randomness as possible.




                                                                         o
                        Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 38/4
Connectivity theorem
Theorem. Let k be a function of n that is ≪ n. Let H
have the property that each vertex is connected to at
least dn other vertices.

  •   If f (n) ≫ k, then the graph becomes
      k-connected, with high probability.




                                                                         o
                        Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 39/4
Connectivity theorem
Theorem. Let k be a function of n that is ≪ n. Let H
have the property that each vertex is connected to at
least dn other 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 be k-connected, with high probability.



                                                                         o
                        Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 39/4
Bottom line
A way to interpret this theorem is:




                                                                        o
                       Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 40/4
Bottom line
A way to interpret this theorem is:

    If you need k-connectivity,




                                                                        o
                       Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 40/4
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.




                                                                        o
                       Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 40/4
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-connectivity does not necessarily occur.




                                                                        o
                       Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 40/4
Worst case
What is that H0 ?
         x x          x x            x x                         x x
         ˜˜4
        e 4 ƒ        ˜˜4
                     e 4 ƒ          ˜˜4
                                    e 4 ƒ                       ˜˜4
                                                                e 4 ƒ
       x e ˜x       x
                    4 e ˜x         x
                                   4 e ˜x                      x
                                                               4 e ˜x
         4 ¡¡
            ˜         4 ¡¡
                         ˜           4 ¡¡
                                        ˜                        4 ¡¡
                                                                    ˜
       
       4      ƒ           ƒ             ƒ                          ƒ
       ˜ ¡ 4
       ˜      4     ˜ ¡ 4
                    ˜      4       ˜ ¡ 4
                                   ˜      4                    ˜ ¡ 4
                                                               ˜      4
        ƒx˜x          x
                      4 ex           x
                                     4 ex                        x
                                                                 4 ex
       ƒ ˜e 
           4        ƒ ˜e 
                        4          ƒ ˜e 
                                       4                       ƒ ˜e 
                                                                   4
         ¡ e
         4           ƒ¡ ˜          ƒ¡ ˜                       ƒ¡ ˜
         x x          x x            x x                         x x
H0 =
         ˜¡4
        e 4 ƒ
          ˜           ˜¡
                       ˜4
                     e 4 ƒ          ˜¡4
                                    e 4 ƒ
                                      ˜                          ˜¡
                                                                e 4 ƒ
                                                                  ˜4
       x
        e ¡ ˜x     x
                    4 e¡ ˜ x       x
                                   4 e¡ ˜ x                    x
                                                               4 e¡ ˜x
         4 ˜          4 ˜            4 ˜                         4 ˜
       4      ƒ           ƒ             ƒ                          ƒ
       ˜ ¡ 4
       ˜      4     ˜ ¡ 4
                    ˜      4       ˜ ¡ 4
                                   ˜      4                    ˜ ¡ 4
                                                               ˜      4
        ƒx˜x          x
                      4 ex           x
                                     4 ex                        x
                                                                 4 ex
       ƒ ˜e 
           4        ƒ ˜e 
                        4          ƒ ˜e 
                                       4                       ƒ ˜e 
                                                                   4
         ¡ e
         4           ƒ¡ ˜          ƒ¡ ˜                       ƒ¡ ˜


Disjoint cliques give the worst case.



                                                                          o
                         Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 41/4
Other properties
We’ve used this model to investigate other properties:

  •   Hamilton cycle




                                                                        o
                       Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 42/4
Other properties
We’ve used this model to investigate other properties:

  •   Hamilton cycle
  •   Small cliques as subgraphs




                                                                         o
                        Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 42/4
Other properties
We’ve used this model to investigate other properties:

  •   Hamilton cycle
  •   Small cliques as subgraphs
  •   Chromatic number




                                                                          o
                         Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 42/4
It’s just killing you, isn’t it?
Let us return to the Kevin Bacon question.




                                                                        o
                       Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 43/4
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 .


                                                                         o
                        Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 43/4
Infinite Kevin Bacon number
                                                               #
      is someone with infinite                                        .

Thomas Alva Edison only appeared in one movie (a
brief documentary) and was the only actor.




                                                                      o
                     Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 44/4
Infinite Kevin Bacon number
                                                               #
      is someone with infinite                                        .

Thomas Alva Edison only appeared in one movie (a
brief documentary) and was the only actor.

Not soon coming to DVD:




                                                                      o
                     Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 44/4
Infinite Kevin Bacon number
                                                               #
      is someone with infinite                                        .

Thomas Alva Edison only appeared in one movie (a
brief documentary) and was the only actor.

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



                                                                      o
                     Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 44/4
Kevin Bacon number 7
Joseph Wheeler appeared in two films:

  •   General Wheeler and Secretary of War
      Alger at Camp Wikoff (1898), a short
      documentary, in which he appeared with Russell
      Alexander Alger (Kevin Bacon number 6)




                                                                        o
                       Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 45/4
Kevin Bacon number 7
Joseph Wheeler appeared in two films:

  •   General Wheeler and Secretary of War
      Alger at Camp Wikoff (1898), a short
      documentary, in which he appeared with Russell
      Alexander Alger (Kevin Bacon number 6)

  •   Surrender of General Toral (1898), again, a
      short documentary, with William Rufus Shafter.




                                                                         o
                        Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 45/4
Kevin Bacon number 8
William Rufus Shafter also appeared in two films:

  •   Surrender of General Toral (1898) with
      Joseph Wheeler.




                                                                         o
                        Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 46/4
Kevin Bacon number 8
William Rufus Shafter also appeared in two films:

  •   Surrender of General Toral (1898) with
      Joseph Wheeler.

  •   Major General Shafter (1898) as the only
      credited cast member.




                                                                         o
                        Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 46/4
Conclusion

                                                                      #
 •   Russell Alexander Alger has                                             =6,




                                                                       o
                      Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 47/4
Conclusion

                                                                      #
 •   Russell Alexander Alger has                                             =6,
                                             #
 •   Joseph Wheeler has                              =7, and



                                                                       o
                      Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 47/4
Conclusion

                                                                      #
 •   Russell Alexander Alger has                                             =6,
                                             #
 •   Joseph Wheeler has                              =7, and
                                                                #
 •   William Rufus Shafter has                                         =8.

                                                                       o
                      Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 47/4
The chain
 8 William Rufus Shafter was in Surrender of
   General Toral (1898) with Joseph Wheeler




                                                                     o
                    Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 48/4
The chain
 8 William Rufus Shafter was in Surrender of
   General Toral (1898) with Joseph Wheeler

 7 Joseph Wheeler was in General Wheeler and
   Secretary of War Alger at Camp Wikoff
   (1898) with Russell Alexander Alger




                                                                     o
                    Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 48/4
The chain
 6 Russell Alexander Alger was in President
   McKinley’s Inspection of Camp Wikoff
   (1898) with President William McKinley




                                                                     o
                    Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 48/4
The chain
 6 Russell Alexander Alger was in President
   McKinley’s Inspection of Camp Wikoff
   (1898) with President William McKinley

 5 President William McKinley was in President
   McKinley Taking the Oath (1901) with
   U. S. Senator Marcus Hanna




                                                                     o
                    Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 48/4
The chain
 6 Russell Alexander Alger was in President
   McKinley’s Inspection of Camp Wikoff
   (1898) with President William McKinley

 5 President William McKinley was in President
   McKinley Taking the Oath (1901) with
   U. S. Senator Marcus Hanna (R-OH)




                                                                     o
                    Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 48/4
The chain
 4 U. S. Senator Marcus Hanna (R-OH) was in
   Opening of the Pan-American Exposition
   Showing Vice President Roosevelt
   Leading the Procession (1901) with
   President Theodore Roosevelt




                                                                     o
                    Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 48/4
The chain
 4 U. S. Senator Marcus Hanna (R-OH) was in
   Opening of the Pan-American Exposition
   Showing Vice President Roosevelt
   Leading the Procession (1901) with
   President Theodore Roosevelt

 3 President Theodore Roosevelt was in
   Womanhood, the Glory of the Nation
   (1917) with Walter McGrail



                                                                     o
                    Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 48/4
The chain
 2 Walter McGrail was in Dick Tracy vs. Crime
   Inc. (1941) with Wally Rose




                                                                    o
                   Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 48/4
The chain
 2 Walter McGrail was in Dick Tracy vs. Crime
   Inc. (1941) with Wally Rose

 1 Wally Rose was in Murder in the First (1995)
   with




                                                                     o
                    Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 48/4
The chain
 2 Walter McGrail was in Dick Tracy vs. Crime
   Inc. (1941) with Wally Rose

 1 Wally Rose was in Murder in the First (1995)
   with




                                                                     o
                    Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 48/4
Thanks
   Thank you for allowing me to speak today.


      rymartin@iastate.edu


 http://orion.math.iastate.edu/rymartin/




                                                                     o
                    Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 49/4

				
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