The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory
Ryan Martin [email protected]
Mathematics Department Iowa State University 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 Erdos,˝ 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 center of the Hollywood universe is
Kevin Bacon.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.2/64 Six? degrees of separation
In the Kevin Bacon Game, it is postulated that the center of the Hollywood universe is
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.2/64 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 Dennis Hopper.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.2/64 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 Dennis Hopper.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.2/64 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.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.2/64 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.)
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.2/64 Bacon number
The actor’s Bacon number is the fewest number of steps it takes to connect that actor to
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.3/64 Bacon number
# The actor’s is the fewest number of steps it takes to connect that actor to
Kevin Bacon.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.3/64 Bacon number
The actor’s Bacon number is the fewest number of steps it takes to connect that actor to
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.3/64 Bacon number
An actor can have infinite Kevin Bacon number.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.3/64 Bacon number
# An actor can have infinite .
(For example, a TV actor who appears in no movie credits.)
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.3/64 Example: Kevin Costner
Kevin Costner is linked to Kevin Bacon because both appeared in
JFK (1991).
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.4/64 Example: Kevin Costner
is linked to because both appeared in
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.4/64 Example: Kevin Costner
Kevin Costner is linked to Kevin Bacon because both appeared in
JFK (1991).
So, Kevin Costner’s Kevin Bacon number is ???
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.4/64 Example: Kevin Costner
Kevin Costner is linked to Kevin Bacon because both appeared in
JFK (1991).
So, Kevin Costner’s Kevin Bacon number is 1
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.4/64 Example: Kevin Costner
Kevin Costner is linked to Kevin Bacon because both appeared in
JFK (1991).
# So, ’s is 1
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.4/64 Illustration: Kevin Costner
x x
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.5/64 Illustration: Kevin Costner
x x
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.5/64 Illustration: Kevin Costner
x x
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.5/64 Illustration: Kevin Costner
x x
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.5/64 E.g.: Henry “Fonz” Winkler
We know that
appeared with in
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.6/64 E.g.: Henry “Fonz” Winkler
We know that Henry Winkler appeared with Michael Keaton in Night Shift (1982)
appeared with in
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.6/64 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
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.6/64 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
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.6/64 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)
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.6/64 A chain: Henry “Fonz” Winkler
x x x x x
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.7/64 A chain: Henry “Fonz” Winkler
x x x x x
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.7/64 A chain: Henry “Fonz” Winkler
x x x x x
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.7/64 A chain: Henry “Fonz” Winkler
x x x x x
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.7/64 A chain: Henry “Fonz” Winkler
x x x x x
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.7/64 A chain: Henry “Fonz” Winkler
x x x x x
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.7/64 A chain: Henry “Fonz” Winkler
x x x x x
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.7/64 A chain: Henry “Fonz” Winkler
x x x x x
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.7/64 A chain: Henry “Fonz” Winkler
x x x x x
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.7/64 A chain: Henry “Fonz” Winkler
x x x x x
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.7/64 More: Henry “Fonz” Winkler
But it is also true that
appeared with ??? in .
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.8/64 More: Henry “Fonz” Winkler
But it is also true that
appeared with in .
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.8/64 More: Henry “Fonz” Winkler
But it is also true that
Henry Winkler appeared with Clint Howard in Little Nicky (2000).
appeared with in
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.8/64 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).
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.8/64 Better: Henry “Fonz” Winkler
x x x
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.9/64 Better: Henry “Fonz” Winkler
x x x
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.9/64 Better: Henry “Fonz” Winkler
x x x
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.9/64 Better: Henry “Fonz” Winkler
x x x
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.9/64 Better: Henry “Fonz” Winkler
x x x
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.9/64 Better: Henry “Fonz” Winkler
x x x
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.9/64 Can we do even better?
has never appeared in a film with .
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.10/64 Can we do even better?
Kevin Bacon has never appeared in a film with Henry Winkler. # So, ’s is
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.10/64 Can we do even better?
Kevin Bacon has never appeared in a film with Henry Winkler.
So, Henry Winkler’s Kevin Bacon number is ???
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.10/64 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
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.10/64 In sum: Henry “Fonz” Winkler
((hh ((( hhh ((( hhh ((( hhh PP x PP x PPhh (( x hhhh(((( x x x
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.11/64 In sum: Henry “Fonz” Winkler
((hh ((( hhh ((( hhh ((( hhh PP x PP x PPhh (( x hhhh(((( x x x
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.11/64 In sum: Henry “Fonz” Winkler
((hh ((( hhh ((( hhh ((( hhh PP x PP x PPhh (( x hhhh(((( x x x
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.11/64 Experimental data
# of actors 0 1 1 1806 2 145024 3 395126 4 95497 5 7451
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.12/64 Experimental data
# of actors
0 1 # of actors 1 1806 6 933 2 145024 7 106 3 395126 8 13 4 95497 5 7451
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.12/64 What about the high numbers?
As we said before, there are actors with infinite # .
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.13/64 What about the high numbers?
As we said before, there are actors with infinite # .
# The actors with large are obscure and the reason why is fairly obvious.
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.13/64 Kevin Bacon not so special
Most successful actors follow the same pattern:
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.14/64 Kevin Bacon not so special
Most successful actors follow the same pattern:
For every pair of successful actors, they are connected by a path of length 5 ≤
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.14/64 Kevin Bacon not so special
Most successful actors follow the same pattern:
For every pair of successful actors, they are connected by a path of length 5 ≤
Why?
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.14/64 Our model
We will represent actors by vertices
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.15/64 Our model
We will represent actors by vertices
x
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.15/64 Our model
We will represent actors by vertices
x and connect them with edges
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.15/64 Our model
We will represent actors by vertices
x and connect them with edges
x x if they appeared in the same film.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.15/64 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.
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.15/64 Model parameters
There are n actors.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.16/64 Model parameters
There are n actors. Fix a constant d.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.16/64 Model parameters
There are n actors. Fix a constant d. We will begin with an arbitrary graph H.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.16/64 Model parameters
There are n actors. Fix a constant d. We will begin with an arbitrary graph H. In H, each actor is connected to at least dn other actors.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.16/64 Model parameters
There are n actors. Fix a constant d. We will begin with an arbitrary graph H. In H, each actor is connected to at least dn other actors. The constant d can be extremely tiny:
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.16/64 Model parameters
There are n actors. Fix a constant d. We will begin with an arbitrary graph H. In H, each actor is connected to at least dn other actors. The constant d can be extremely tiny:
0.1
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.16/64 Model parameters
There are n actors. Fix a constant d. We will begin with an arbitrary graph H. In H, each actor is connected to at least dn other actors. The constant d can be extremely tiny:
0.1, 0.01
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.16/64 Model parameters
There are n actors. Fix a constant d. We will begin with an arbitrary graph H. In H, each actor is connected to at least dn other actors. The constant d can be extremely tiny:
0.1, 0.01, 0.000001
It just needs to be independent of n.
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.16/64 Random casting
We add f(n) random casting connections.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.17/64 Random casting
We add f(n) random casting connections.
What does random mean?
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.17/64 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:
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.18/64 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.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.18/64 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.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.18/64 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 small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.18/64 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.
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.18/64 The models
For our purposes, these produce the same results.
The question:
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.19/64 The models
For our purposes, these produce the same results.
The question:
What is the longest distance between any pair of actors ?
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.19/64 The models
For our purposes, these produce the same results.
The question:
What is the longest distance between any pair of actors (diameter)?
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.19/64 The theorem
Theorem. If f(n) as n , then → ∞ → ∞ Pr (diam 5) 1 ≤ →
Important points: Recall f(n) is the number of random connections.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.20/64 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.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.20/64 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:
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.20/64 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
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.20/64 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
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.20/64 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
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.20/64 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.
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.21/64 The proof
To prove the theorem, you need the Regularity Lemma.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.22/64 The proof
To prove the theorem, you need the Regularity Lemma.
The Regularity Lemma is ’s powerful and complicated graph theoretic tool.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.22/64 The proof
To prove the theorem, you need the Regularity Lemma.
The Regularity Lemma is Endre Szemerédi’s powerful and complicated graph theoretic tool.
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.22/64 Best possible?
The theorem is “tight”:
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.23/64 Best possible?
The theorem is “tight”:
If there aren’t an infinite number of edges added, then some H’s will be disconnected.
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.23/64 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.≤
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.24/64 What about closer connections?
To get diam 4, you need c1 log n random connections.≤
To get diam 3, you need c1 log n random connections.≤
To get diam 2, you need random connections.≤
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.24/64 What about closer connections?
To get diam 4, you need c1 log n random connections.≤
To get diam 3, you need c1 log n random connections.≤
To get diam 2, you need c2n log n random connections.≤
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.24/64 Origins of the problem
The original question posed by Stanley Milgram
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.25/64 Origins of the problem
The original question posed by Stanley Milgram
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.25/64 Origins of the problem
The original question posed by Stanley Milgram asked what the average distance was among people in a network.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.25/64 Origins of the problem
The original question posed by Stanley Milgram asked what the average distance was among people in a network. His methodology was very flawed
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.25/64 Origins of the problem
The original question posed by Stanley Milgram asked what the average distance was among people in a network. His methodology was very flawed – he sent out letters from a single source and waited for a return message, many didn’t come back.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.25/64 Origins of the problem
The original question posed by Stanley Milgram asked what the average distance was among people in a network. His methodology was very flawed – he sent out letters from a single source and waited for a return message, many didn’t come back. But his assertion that the average distance was around 6 stuck.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.25/64 Origins of the problem
But his assertion that the average distance was around 6 stuck.
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.25/64 Advantages and disadvantages
Our model has some advantages and some disadvantages. Disadvantages:
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.26/64 Advantages and disadvantages
Our model has some advantages and some disadvantages. Disadvantages: Each person must be connected to at least dn other people.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.26/64 Advantages and disadvantages
Our model has some advantages and some disadvantages. Disadvantages: Each person must be connected to at least dn other people. Not good for modelling the Internet.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.26/64 Advantages and disadvantages
Our model has some advantages and some disadvantages. Disadvantages: Advantages: Each person must be connected to at least dn other people. Not good for modelling the Internet.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.26/64 Advantages and disadvantages
Our model has some advantages and some disadvantages. Disadvantages: Advantages: Each person must be Very weak restriction connected to at least on structure. dn other people. Not good for modelling the Internet.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.26/64 Advantages and disadvantages
Our model has some advantages and some disadvantages. Disadvantages: Advantages: Each person must be Very weak restriction connected to at least on structure. dn other people. The number of Not good for modelling random connections the Internet. is tiny.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.26/64 Advantages and disadvantages
Our model has some advantages and some disadvantages. Disadvantages: Advantages: Each person must be Very weak restriction connected to at least on structure. dn other people. The number of Not good for modelling random connections the Internet. is tiny. An upper bound of 5 on the diameter.
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.26/64 Another model
There is another model due to Fan Chung
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.27/64 Another model
There is another model due to Fan Chung
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.27/64 Another model
There is another model due to Fan Chung and others
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.27/64 Another model
There is another model due to Fan Chung and others (no pictures)
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.27/64 Another model
There is another model due to Fan Chung and others (no pictures) which eliminates the problems of not being applicable to the Internet.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.27/64 Another model
There is another model due to Fan Chung and others (no pictures) which eliminates the problems of not being applicable to the Internet.
However . . .
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.27/64 The Chung model
Advantages:
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.28/64 The Chung model
Advantages: Easy to use, easy to do computations.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.28/64 The Chung model
Advantages: Easy to use, easy to do computations. Models the Internet quite well.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.28/64 The Chung model
Advantages: Disdvantages: Easy to use, easy to do computations. Models the Internet quite well.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.28/64 The Chung model
Advantages: Disdvantages: Easy to use, easy to The number of random do computations. connections is huge. Models the Internet quite well.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.28/64 The Chung model
Advantages: Disdvantages: Easy to use, easy to The number of random do computations. connections is huge. Models the Internet Weak result: The av- quite well. erage distance is ≈ log2 n/ log2 d˜.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.28/64 The Chung model
Advantages: Disdvantages: Easy to use, easy to The number of random do computations. connections is huge. Models the Internet Weak result: The av- quite well. erage distance is ≈ log2 n/ log2 d˜, where d˜ relates to the aver- age degree (number of connections).
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.28/64 Applying our model
To have it be very likely that everyone is con- nected by a path of no more than 5 acquain- tances, just arrange a few random meetings.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.29/64 Applying our model
To have it be very likely that everyone is con- nected by a path of no more than 5 acquain- tances, just arrange a few random meetings.
Think about people at Iowa State.
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.29/64 Iowa State cliques
'hhh $ J .. .. ¡ A ..b .. xJ .. A b . .. b.. .. ¡ xA ...... PP. ...... P .. xA ...... PJ ...... T .. ¡P .. ...... A .. A...... ll.. P .. ...... J PA.. ...... xT .. Q.. x...... ¡ ... Q...... l b ...... x.. b ...... J ...... TT.. ...... ¡` l b...... ` x...... `J x !!...... ` ...... A l `! .. .. x... ` ..... x ... ! `.. ... J ... xA((.! lJ x ... ( .. ..( .. ¨ .. ¨ .. Z A ¨.. .. x Z¨ .. x.. x A .. x .. .. A.. x . x &Mathematicians %
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.30/64 Iowa State cliques
hh ' h $' ..` $ !... `` J ...... ! .. ... ¡ A ..b ... @ ".. .. !! ...... x J .. A h .. .. .. b h ... " ... . ¡ x .. b.... h... x(@. . ..P A ...... (hh .. .. P. ...... ("" ...... PPJ .. xA ...... x .. .. T ...... H e ...... ¡P A .. ...... P .. A...... H .. .. ll.. .. ... .. x ...... J PA. ...... xT .. Q... x.. Z Hxe .. x .. .. ¡ .. Q.. A .. .. x.. l bb ...... x H ...... J ...... Z e.. .. T.. ...... ((( H ... T..¡ b...... ( ... ` l .. .. xA b .. ` x...... `J x !!.. .. bb L .. A ` .. .. lb x... A l `! ..... x e ... ``..... b A b ... !J . xL b(...hh x ... ! l bA l( .. hh .. xA((.. J x xe (( ...... (( .. a .. . .. ¨ a J.. ¨ .. ¨ L l ... .. a ... ¨ Z A ¨.. e al...¨ .. x x..... J Z¨ .. L ..... x.. x x L ...... x A .. e ...... Q x .. e ...... QJ .. ``...... `...... A. x x x x x x x &Mathematicians%& Social Scientists %
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.30/64 More Iowa State cliques
`` ' PP ` $ x x x " x But for us to get di- " "" ameter 5, we do hhhhx x x need each≤ person x x x to know at least dn x aa others before we aax x x x add few random x x edges. & %
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.31/64 More Iowa State cliques
`` ' PP ` $ x x x " x But for us to get di- " "" ameter 5, we do hhhhx x x need each≤ person x x x to know at least dn x aa others before we aax x x x add few random x x edges. &Computer Scientists %
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.31/64 The cliché
The cliché states that every pair of people is separated by at most six degrees of separation.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.32/64 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!
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.32/64 Erdos˝ number
One of the most prolific mathematicians of the 20th century was
Paul (Pál) Erdos˝ March 26, 1913-September 20, 1996
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.33/64 Erdos˝ number
One of the most prolific mathematicians of the 20th century was
Paul (Pál) Erdos˝ March 26, 1913-September 20, 1996
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.33/64 Erdos˝ number project
The Erdos˝ number project is concerned with the distance of mathematicians from Paul Erdos.˝
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.34/64 Erdos˝ number project
# The project is concerned with the distance of mathematicians from Paul Erdos.˝
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.34/64 Erdos˝ number project
# The project is concerned with the distance of mathematicians from Paul Erdos.˝
Two mathematicians are connected if they co-authored a paper together and that paper appears in Mathematical Reviews, accessible by MathSciNet.
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.34/64 Most prolific authors
: 1401 papers (Erdos˝ number )
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.35/64 Most prolific authors
: 1401 papers (Erdos˝ number )
Drumi Bainov: 782 (Erdos˝ number )
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.35/64 Most prolific authors
: 1401 papers (Erdos˝ number )
Drumi Bainov: 782 (Erdos˝ number )
Leonard Carlitz: 730 (Erdos˝ number )
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.35/64 Most prolific authors
: 1401 papers (Erdos˝ number )
Drumi Bainov: 782 (Erdos˝ number )
Leonard Carlitz: 730 (Erdos˝ number )
Lucien Godeaux: 644 (Erdos˝ number )
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.35/64 Most prolific authors
: 1401 papers (Erdos˝ number )
Drumi Bainov: 782 (Erdos˝ number )
Leonard Carlitz: 730 (Erdos˝ number )
Lucien Godeaux: 644 (Erdos˝ number )
Saharon Shelah: 600 (Erdos˝ number )
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.35/64 Most prolific authors
: 1401 papers (Erdos˝ number 0)
Drumi Bainov: 782 (Erdos˝ number 4)
Leonard Carlitz: 730 (Erdos˝ number 2)
Lucien Godeaux: 644 (Erdos˝ number ) ∞ Saharon Shelah: 600 (Erdos˝ number 1)
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.35/64 Erdos˝ number statistics
wrote 1401 papers in Math Reviews.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.36/64 Erdos˝ number statistics
wrote 1401 papers in Math Reviews.
There are 337, 000 vertices (authors) in the graph.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.36/64 Erdos˝ number statistics
wrote 1401 papers in Math Reviews.
There are 337, 000 vertices (authors) in the graph.
There are about 496, 000 edges.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.36/64 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
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.36/64 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
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.36/64 Experimental data
# 0 1 1 502 2 5713 3 26422 4 62136 5 66157
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.37/64 Experimental data
# # 0 1 6 32280 1 509 7 10431 2 6984 8 3214 3 26422 9 953 4 62136 10 262 5 66157 11 94 (Most recent data)
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.37/64 Experimental data
# # # 0 1 6 32280 12 23 1 509 7 10431 13 4 2 6984 8 3214 14 7 3 26422 9 953 15 1 4 62136 10 262 5 66157 11 94 (Most recent data)
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.37/64 Experimental data
# # # 0 1 6 32280 12 23 1 509 7 10431 13 4 2 6984 8 3214 14 7 3 26422 9 953 15 1 4 62136 10 262 (R. G. Kamalov) 5 66157 11 94 (Most recent data)
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.37/64 The unknown mathematician
\ x\ \ \ \ \ \ \ \ H ¨ H ¨ H ¨ x H x ¨ x H ¨ H ¨ HH¨¨ x
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.38/64 The unknown mathematician
\ x\ \ \ \ \ \ \ \ H ¨ H ¨ H ¨ x H x ¨ x H ¨ H ¨ HH¨¨ x
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.38/64 The unknown mathematician
\ x\ \ \ \ \ \ \ \ H ¨ H ¨ H ¨ x H x ¨ x H ¨ H ¨ HH¨¨ x
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.38/64 The unknown mathematician
\ x\ \ \ \ \ \ \ \ H ¨ H ¨ H ¨ x H x ¨ x H ¨ H ¨ HH¨¨ x
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.38/64 The unknown mathematician
\ x\ \ \ \ \ \ \ \ H ¨ H ¨ H ¨ x H x ¨ x H ¨ H ¨ HH¨¨ x
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.38/64 The unknown mathematician
\ x\ \ \ \ \ \ \ \ H ¨ H ¨ H ¨ x H x ¨ x H ¨ H ¨ HH¨¨ x
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.38/64 Computer networks
Graphs model much more serious stuff.
I.e.,
computer networks,
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.39/64 Computer networks
Graphs model much more serious stuff.
I.e.,
computer networks,
shipping routes,
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.39/64 Computer networks
Graphs model much more serious stuff.
I.e.,
computer networks,
shipping routes,
distribution networks.
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.39/64 Network question
In networks we are concerned with one particular quantity:
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.40/64 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.
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.40/64 Same model
n computers
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.41/64 Same model
n computers
in H, each computer is connected to dn others ≥
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.41/64 Same model
n computers
in H, each computer is connected to dn others ≥
add f(n) random connections
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.41/64 Same model
n computers
in H, each computer is connected to dn others ≥
add f(n) random connections
Of course, we want high connectivity with as little randomness as possible.
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.41/64 Connectivity theorem
Theorem. Let k be a function of n that is n. ≪
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.42/64 Connectivity theorem
Theorem. Let k be a function of n that is n. (That is, k grows more slowly than n as n .) ≪ → ∞
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.42/64 Connectivity theorem
Theorem. Let k be a function of n that is n. (That is, k grows more slowly than n as n .) Let≪H have the property that each vertex is connected→ ∞ to at least dn other vertices.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.42/64 Connectivity theorem
Theorem. Let k be a function of n that is n. (That is, k grows more slowly than n as n .) Let≪H have 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.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.42/64 Connectivity theorem
Theorem. Let k be a function of n that is n. (That is, k grows more slowly than n as n .) Let≪H have 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.
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.42/64 Bottom line
A way to interpret this theorem is:
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.43/64 Bottom line
A way to interpret this theorem is:
If you need k-connectivity,
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.43/64 Bottom line
A way to interpret this theorem is:
If you need k-connectivity, then you need to add a little more (asymptotically) random edges than k.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.43/64 Bottom line
A way to interpret this theorem is:
If you need k-connectivity, then you need to add a little more (asymptotically) random edges than k.
If fewer than k random edges are added, k-connectivity does not necessarily occur.
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.43/64 Worst case
What is that H0?
bb"" bb"" bb"" bb"" "A b¡S "A b¡S "A b¡S "A b¡S " ¡ bS " ¡ bS " ¡ bS " ¡ bS bb xA x"" bb xA x"" bb xA x"" bb xA x"" S b¡"A S b¡"A S b¡A" S b¡"A xS¡"bA x xS"¡ bA x xS"¡ bA x xS¡"bA x H0 = xbb""x xbb""x xbb""x xbb""x "A b¡S "A b¡S "A b¡S "A b¡S " ¡ bS " ¡ bS " ¡ bS " ¡ bS bb xA x"" bb xA x"" bb xA x"" bb xA x"" S b¡"A S b¡"A S b¡"A S b¡"A xS¡"bA x xS¡"bA x xS"¡ bA x xS¡"bA x x x x x x x x x Disjoint cliques give the worst case.
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.44/64 Other properties
We’ve used this model to investigate other properties:
Hamilton cycle
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.45/64 Other properties
We’ve used this model to investigate other properties:
Hamilton cycle
Small cliques as subgraphs
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.45/64 Other properties
We’ve used this model to investigate other properties:
Hamilton cycle
Small cliques as subgraphs
Chromatic number
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.45/64 What next?
Shall we conclude with
more mathematics,
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.46/64 What next?
Shall we conclude with
more mathematics,
people with high Bacon numbers,
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.46/64 What next?
Shall we conclude with
more mathematics,
people with high Bacon numbers,
mathematicians and dead presidents, or
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.46/64 What next?
Shall we conclude with
more mathematics,
people with high Bacon numbers,
mathematicians and dead presidents, or
connections between Bacon numbers and Erdos˝ numbers?
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.46/64 It’s just killing you, isn’t it?
Let us return to the Kevin Bacon question.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.47/64 It’s just killing you, isn’t it?
Let us return to the Kevin Bacon question. We want to find actors with an # infinite and
# with =8 .
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.47/64 Infinite Kevin Bacon number
# is someone with infinite . Thomas Alva Edison only appeared in one movie (a brief documentary) and was the only actor.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.48/64 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:
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.48/64 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).
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.48/64 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)
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.49/64 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.
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.49/64 Kevin Bacon number 8
William Rufus Shafter also appeared in two films:
Surrender of General Toral (1898) with Joseph Wheeler.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.50/64 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.
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.50/64 Conclusion
# Russell Alexander Alger has =6,
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.51/64 Conclusion
# Russell Alexander Alger has =6,
# Joseph Wheeler has =7, and
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.51/64 Conclusion
# Russell Alexander Alger has =6,
# Joseph Wheeler has =7, and
# William Rufus Shafter has =8. The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.51/64 The chain
8 William Rufus Shafter was in Surrender of General Toral (1898) with Joseph Wheeler
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.52/64 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
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.52/64 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 6 Russell Alexander Alger was in President McKinley's Inspection of Camp Wikoff (1898) with President William McKinley
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.52/64 The chain
7 Joseph Wheeler was in General Wheeler and Secretary of War Alger at Camp Wikoff (1898) with Russell Alexander Alger 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
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.52/64 The chain
7 Joseph Wheeler was in General Wheeler and Secretary of War Alger at Camp Wikoff (1898) with Russell Alexander Alger 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)
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.52/64 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)
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.52/64 The chain
5 President William McKinley was in President McKinley Taking the Oath (1901) with U. S. Senator Marcus Hanna (R-OH) 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 (then) Vice President Theodore Roosevelt
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.52/64 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 (then) Vice President Theodore Roosevelt 3 Vice President Theodore Roosevelt was in Womanhood , the Glory of the Nation (1917) with Walter McGrail
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.52/64 The chain
3 Vice President Theodore Roosevelt was in Womanhood , the Glory of the Nation (1917) with Walter McGrail 2 Walter McGrail was in Dick Tracy vs. Crime Inc. (1941) with Wally Rose
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.52/64 The chain
3 Vice President Theodore Roosevelt was in Womanhood , the Glory of the Nation (1917) with Walter McGrail 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
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.52/64 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
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.52/64 The chain
1 Wally Rose was in Murder in the First (1995) with
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.52/64 The chain
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.52/64 Dead presidents
William McKinley has a unique distinction.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.53/64 Dead presidents
William McKinley has a unique distinction.
He was one of four presidents to be assassinated:
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.53/64 Dead presidents
William McKinley has a unique distinction.
He was one of four presidents to be assassinated:
McKinley Sep. 14, 1901
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.53/64 Dead presidents
William McKinley has a unique distinction.
He was one of four presidents to be assassinated:
McKinley Kennedy Sep. 14, Nov. 22, 1901 1963
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.53/64 Dead presidents
William McKinley has a unique distinction.
He was one of four presidents to be assassinated:
Lincoln McKinley Kennedy Apr. 15, Sep. 14, Nov. 22, 1865 1901 1963
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.53/64 Dead presidents
William McKinley has a unique distinction.
He was one of four presidents to be assassinated:
Lincoln Garfield McKinley Kennedy Apr. 15, Sep. 19, Sep. 14, Nov. 22, 1865 1881 1901 1963
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.53/64 Dead presidents
William McKinley has a unique distinction.
He was one of four presidents to be assassinated:
Lincoln Garfield McKinley Kennedy Apr. 15, Sep. 19, Sep. 14, Nov. 22, 1865 1881 1901 1963 Dude, what a downer.
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.53/64 Garfield (not the cat)
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.54/64 Garfield (not the cat)
Born in a log cabin in 1831.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.54/64 Garfield (not the cat)
Born in a log cabin in 1831 near Cleveland.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.54/64 Garfield (not the cat)
Born in a log cabin in 1831 near Cleveland.
18 years in the House.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.54/64 Garfield (not the cat)
Born in a log cabin in 1831 near Cleveland.
18 years in the House.
Elected in 1880.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.54/64 Garfield (not the cat)
Born in a log cabin in 1831 near Cleveland.
18 years in the House.
Elected in 1880.
Shot on July 2.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.54/64 Garfield (not the cat)
Born in a log cabin in 1831 near Cleveland.
18 years in the House.
Elected in 1880.
Shot on July 2, died on September 19.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.54/64 Garfield (not the cat)
Born in a log cabin in 1831 near Cleveland.
18 years in the House.
Elected in 1880.
Shot on July 2, died on September 19.
Amateur mathematician.
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.54/64 Published mathematician
As a Congressman, Garfield got a publication credit:
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.55/64 Published mathematician
As a Congressman, Garfield got a publication credit: J.A. Garfield, The New England Journal of Education, 3, Boston, 1876, p. 161.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.55/64 Published mathematician
As a Congressman, Garfield got a publication credit: J.A. Garfield, The New England Journal of Education, 3, Boston, 1876, p. 161. Garfield found a proof of the Pythagorean theorem:
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.55/64 Published mathematician
As a Congressman, Garfield got a publication credit: J.A. Garfield, The New England Journal of Education, 3, Boston, 1876, p. 161. Garfield found a proof of the Pythagorean theorem:
b c c a
a b
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.55/64 Garfield’s proof
1 b c c 2 3 a a b
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.56/64 Garfield’s proof
1 b c c 2 3 a a b
area (trapezoid)
= area ( 1) + area ( 2) + area ( 3) △ △ △
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.56/64 Garfield’s proof
1 b c c 2 3 a a b
area (trapezoid)
= area ( 1) + area ( 2) + area ( 3) △ △ △
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.56/64 Garfield’s proof
1 b c c 2 3 a a b
1 (a + b)(a + b) 2 = area ( 1) + area ( 2) + area ( 3) △ △ △
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.56/64 Garfield’s proof
1 b c c 2 3 a a b
1 (a + b)(a + b) 2 = area ( 1) + area ( 2) + area ( 3) △ △ △
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.56/64 Garfield’s proof
1 b c c 2 3 a a b
1 (a + b)(a + b) 2 1 2 = c + area ( 2) + area ( 3) 2 △ △
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.56/64 Garfield’s proof
1 b c c 2 3 a a b
1 (a + b)(a + b) 2 1 2 = c + area ( 2) + area ( 3) 2 △ △
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.56/64 Garfield’s proof
1 b c c 2 3 a a b
1 (a + b)(a + b) 2 1 2 1 = c + ab + area ( 3) 2 2 △
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.56/64 Garfield’s proof
1 b c c 2 3 a a b
1 (a + b)(a + b) 2 1 2 1 = c + ab + area ( 3) 2 2 △
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.56/64 Garfield’s proof
1 b c c 2 3 a a b
1 (a + b)(a + b) 2 1 2 1 1 = c + ab + ab 2 2 2
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.56/64 Garfield’s proof
1 b c c 2 3 a a b
1 1 2 1 1 (a + b)(a + b) = c + ab + ab 2 2 2 2
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.56/64 Garfield’s proof
1 b c c 2 3 a a b
2 (a + b)(a + b) = c + ab + ab
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.56/64 Garfield’s proof
1 b c c 2 3 a a b
2 2 2 a +2ab + b = c + ab + ab
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.56/64 Garfield’s proof
1 b c c 2 3 a a b
2 2 2 a + b = c
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.56/64 Bacon and Erdos˝
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.57/64 Bacon and Erdos˝
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.57/64 Bacon and Erdos˝
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.57/64 Bacon and Erdos˝
How are THESE guys re- lated?
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.57/64 Celebrity nerds
There are some mathematicians and physicists who have Bacon numbers and low Erdos˝ numbers:
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.58/64 Celebrity nerds
There are some mathematicians and physicists who have Bacon numbers and low Erdos˝ numbers:
Brian Greene ( =3, =2) was in Frequency (2000).
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.58/64 Celebrity nerds
There are some mathematicians and physicists who have Bacon numbers and low Erdos˝ numbers:
Brian Greene ( =3, =2) was in Frequency (2000).
Dave Bayer ( =3, =2) was in A Beautiful Mind (2001).
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.58/64 Nerd celebrities
Danica McKellar
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.59/64 Nerd celebrities
Danica McKellar, math nerd.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.59/64 Nerd celebrities
Danica McKellar, math nerd. Best known for: The Wonder Years (1988-1993) and
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.59/64 Nerd celebrities
Danica McKellar, math nerd. Best known for: The Wonder Years (1988-1993) and The West Wing (2002-present).
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.59/64 Nerd celebrities
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.59/64 Nerd celebrities
Danica McKellar was in Intermission (2004) with Susan Leslie.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.59/64 Nerd celebrities
Susan Leslie was in Beauty Shop (2005) with
Danica McKellar was in Intermission (2004) with Susan Leslie.
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.59/64 Danica’s Math Career
4 Danica McKellar wrote Percolation and Gibbs State Multiplicity for Ferromagnetic Ashkin-Teller Models in Two Dimensions, which appeared in Journal of Physics A: Mathematics and General, with Winn and Chayes.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.60/64 Danica’s Math Career
3 Lincoln Chayes wrote No directed fractal percolation in zero area, which appeared in The Journal of Statistical Physics, with Peres and Pemantle.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.60/64 Danica’s Math Career
2 Robin Pemantle wrote Metrics on compositions and coincidences among renewal sequences, which appeared in The IMA Volumes in Mathematics and its Applications, with Diaconis, Holmes, Lalley and Janson.
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.60/64 Danica’s Math Career
1 Svante Janson wrote A note on triangle-free graphs, which appeared in The IMA Volumes in Mathematics and its Applications, with Łuczak, Spencer and Paul Erdos˝
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.60/64 Danica’s Math Career
1 Svante Janson wrote A note on triangle-free graphs, which appeared in The IMA Volumes in Mathematics and its Applications, with Łuczak, Spencer and Paul Erdos˝
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.60/64 Erdos,˝ Hollywood Leading Man
4 Paul Erdos˝ was in N is a Number (1993) with Gene Patterson
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.61/64 Erdos,˝ Hollywood Leading Man
4 Paul Erdos˝ was in N is a Number (1993) with Gene Patterson 3 Gene Patterson was in Box of Moon Light (1996) with Lisa Blount
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.61/64 Erdos,˝ Hollywood Leading Man
4 Paul Erdos˝ was in N is a Number (1993) with Gene Patterson 3 Gene Patterson was in Box of Moon Light (1996) with Lisa Blount 2 Lisa Blount was in Femme Fatale (1991) with Colin Firth
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.61/64 Erdos,˝ Hollywood Leading Man
3 Gene Patterson was in Box of Moon Light (1996) with Lisa Blount 2 Lisa Blount was in Femme Fatale (1991) with Colin Firth 1 Colin Firth was in Where the Truth Lies (2005) with Kevin Bacon
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.61/64 Erdos,˝ Hollywood Leading Man
2 Lisa Blount was in Femme Fatale (1991) with Colin Firth 1 Colin Firth was in Where the Truth Lies (2005) with Kevin Bacon
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.61/64 Revised title
The small world problem:
Kevin Bacon, Paul Erdos,˝ William McKinley and Ryan Martin
– 4 degrees of graph theory
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.62/64 Revised title
The small world problem:
Kevin Bacon, Paul Erdos,˝ William McKinley , Danica McKellar and Ryan Martin
– 5 degrees of graph theory
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.62/64 Revised title
The small world problem:
, , , and
– 5 degrees of graph theory
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.62/64 Tying it all together
And, just when you thought this whole talk was just a disjointed mess that didn’t fit at all together . . .
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.63/64 Tying it all together
And, just when you thought this whole talk was just a disjointed mess that didn’t fit at all together . . .
▽The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.63/64 Tying it all together
And, just when you thought this whole talk was just a disjointed mess that didn’t fit at all together . . .
It does!
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.63/64 Thanks
Thank you for letting me talk today.
Ryan Martin Iowa State University [email protected]
The file for this talk is available online at my website: http://www.math.iastate.edu/rymartin These slides were created by the Prosper document preparation system.
The small world problem: Kevin Bacon, Paul Erdos,˝ William McKinley, and me – 5 degrees of graph theory – p.64/64