The Small World. problem and so .. 6 much more! 666 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 and so much more! – p.1/56 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 André Kézdy, University of Louisville Maria Axenovich
▽The Small World problem and so much more! – p.1/56 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 André Kézdy, University of Louisville Maria Axenovich, 412 Carver
The Small World problem and so much more! – p.1/56 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 and so much more! – p.2/56 Six? degrees of separation
In the Kevin Bacon Game, it is postulated that the center of the Hollywood universe is
▽The Small World problem and so much more! – p.2/56 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 and so much more! – p.2/56 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 and so much more! – p.2/56 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 and so much more! – p.2/56 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 and so much more! – p.2/56 Bacon number
The actor’s Bacon number is the fewest number of steps it takes to connect that actor to
▽The Small World problem and so much more! – p.3/56 Bacon number
# The actor’s is the fewest number of steps it takes to connect that actor to
Kevin Bacon.
▽The Small World problem and so much more! – p.3/56 Bacon number
The actor’s Bacon number is the fewest number of steps it takes to connect that actor to
▽The Small World problem and so much more! – p.3/56 Bacon number
An actor can have infinite Kevin Bacon number.
▽The Small World problem and so much more! – p.3/56 Bacon number
# An actor can have infinite .
(For example, a TV actor who appears in no movie credits.)
The Small World problem and so much more! – p.3/56 Example: Kevin Costner
Kevin Costner is linked to Kevin Bacon because both appeared in
JFK (1991).
▽The Small World problem and so much more! – p.4/56 Example: Kevin Costner
is linked to because both appeared in
▽The Small World problem and so much more! – p.4/56 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 and so much more! – p.4/56 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 and so much more! – p.4/56 Example: Kevin Costner
Kevin Costner is linked to Kevin Bacon because both appeared in
JFK (1991).
# So, ’s is 1
The Small World problem and so much more! – p.4/56 Illustration: Kevin Costner
x x
▽The Small World problem and so much more! – p.5/56 Illustration: Kevin Costner
x x
▽The Small World problem and so much more! – p.5/56 Illustration: Kevin Costner
x x
▽The Small World problem and so much more! – p.5/56 Illustration: Kevin Costner
x x
The Small World problem and so much more! – p.5/56 E.g.: Keanu “Woah!” Reeves
We know that
appeared with in
▽The Small World problem and so much more! – p.6/56 E.g.: Keanu “Woah!” Reeves
We know that Keanu Reeves appeared with Al Pacino in The Devil's Advocate (1997)
appeared with in
▽The Small World problem and so much more! – p.6/56 E.g.: Keanu “Woah!” Reeves
We know that Keanu Reeves appeared with Al Pacino in The Devil's Advocate (1997)
appeared with in
▽The Small World problem and so much more! – p.6/56 E.g.: Keanu “Woah!” Reeves
We know that Keanu Reeves appeared with Al Pacino in The Devil's Advocate (1997) Al Pacino appeared with Christopher Walken in Gigli (2003)
appeared with in
▽The Small World problem and so much more! – p.6/56 E.g.: Keanu “Woah!” Reeves
We know that Keanu Reeves appeared with Al Pacino in The Devil's Advocate (1997) Al Pacino appeared with Christopher Walken in Gigli (2003) Christopher Walken appeared with Courtney Love in Basquiat (1996)
appeared with in
▽The Small World problem and so much more! – p.6/56 E.g.: Keanu “Woah!” Reeves
We know that Keanu Reeves appeared with Al Pacino in The Devil's Advocate (1997) Al Pacino appeared with Christopher Walken in Gigli (2003) Christopher Walken appeared with Courtney Love in Basquiat (1996) Courtney Love appeared with Kevin Bacon in Trapped (2002)
The Small World problem and so much more! – p.6/56 A chain: Keanu “Neo” Reeves
x x x x x
▽The Small World problem and so much more! – p.7/56 A chain: Keanu “Neo” Reeves
x x x x x
▽The Small World problem and so much more! – p.7/56 A chain: Keanu “Neo” Reeves
x x x x x
▽The Small World problem and so much more! – p.7/56 A chain: Keanu “Neo” Reeves
x x x x x
▽The Small World problem and so much more! – p.7/56 A chain: Keanu “Neo” Reeves
x x x x x
▽The Small World problem and so much more! – p.7/56 A chain: Keanu “Neo” Reeves
x x x x x
▽The Small World problem and so much more! – p.7/56 A chain: Keanu “Neo” Reeves
x x x x x
▽The Small World problem and so much more! – p.7/56 A chain: Keanu “Neo” Reeves
x x x x x
▽The Small World problem and so much more! – p.7/56 A chain: Keanu “Neo” Reeves
x x x x x
▽The Small World problem and so much more! – p.7/56 A chain: Keanu “Neo” Reeves
x x x x x
The Small World problem and so much more! – p.7/56 More: Keanu “Constantine” Reeves
But it is also true that
appeared with ??? in .
▽The Small World problem and so much more! – p.8/56 More: Keanu “Constantine” Reeves
But it is also true that
appeared with in .
▽The Small World problem and so much more! – p.8/56 More: Keanu “Constantine” Reeves
But it is also true that
Keanu Reeves appeared with Clint Howard in Parenthood (1989).
appeared with in
▽The Small World problem and so much more! – p.8/56 More: Keanu “Constantine” Reeves
But it is also true that
Keanu Reeves appeared with Clint Howard in Parenthood (1989).
Clint Howard appeared with Kevin Bacon in My Dog Skip (2000).
The Small World problem and so much more! – p.8/56 Better: Keanu “Excellent!” Reeves
x x x
▽The Small World problem and so much more! – p.9/56 Better: Keanu “Excellent!” Reeves
x x x
▽The Small World problem and so much more! – p.9/56 Better: Keanu “Excellent!” Reeves
x x x
▽The Small World problem and so much more! – p.9/56 Better: Keanu “Excellent!” Reeves
x x x
▽The Small World problem and so much more! – p.9/56 Better: Keanu “Excellent!” Reeves
x x x
▽The Small World problem and so much more! – p.9/56 Better: Keanu “Excellent!” Reeves
x x x
The Small World problem and so much more! – p.9/56 Can we do even better?
has never appeared in a film with .
▽The Small World problem and so much more! – p.10/56 Can we do even better?
Kevin Bacon has never appeared in a film with Keanu Reeves. # So, ’ is
▽The Small World problem and so much more! – p.10/56 Can we do even better?
Kevin Bacon has never appeared in a film with Keanu Reeves.
So, Keanu Reeves’ Kevin Bacon number is ???
▽The Small World problem and so much more! – p.10/56 Can we do even better?
Kevin Bacon has never appeared in a film with Keanu Reeves.
So, Keanu Reeves’ Kevin Bacon number is 2
The Small World problem and so much more! – p.10/56 In sum: Keanu “Bogus” Reeves
((hh ((( hhh ((( hhh ((( hhh PP x PP x PPhh (( x hhhh(((( x x x
▽The Small World problem and so much more! – p.11/56 In sum: Keanu “Bogus” Reeves
((hh ((( hhh ((( hhh ((( hhh PP x PP x PPhh (( x hhhh(((( x x x
▽The Small World problem and so much more! – p.11/56 In sum: Keanu “Bogus” Reeves
((hh ((( hhh ((( hhh ((( hhh PP x PP x PPhh (( x hhhh(((( x x x
The Small World problem and so much more! – p.11/56 Experimental data
# of actors 0 1 1 1806 2 145024 3 395126 4 95497 5 7451
▽The Small World problem and so much more! – p.12/56 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 and so much more! – p.12/56 What about the high numbers?
As we said before, there are actors with infinite # .
▽The Small World problem and so much more! – p.13/56 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 and so much more! – p.13/56 Kevin Bacon not so special
Most successful actors follow the same pattern:
▽The Small World problem and so much more! – p.14/56 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 and so much more! – p.14/56 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 and so much more! – p.14/56 Ask Christopher Walken
What do you think so far, Mr. Walken?
▽The Small World problem and so much more! – p.15/56 Ask Christopher Walken
What do you think so far, Mr. Walken?
▽The Small World problem and so much more! – p.15/56 Ask Christopher Walken
What do you think so far, Mr. Walken?
Pardon me?
▽The Small World problem and so much more! – p.15/56 Ask Christopher Walken
What do you think so far, Mr. Walken?
Pardon me?
The Small World problem and so much more! – p.15/56 Our model
We will represent actors by vertices
▽The Small World problem and so much more! – p.16/56 Our model
We will represent actors by vertices
x
▽The Small World problem and so much more! – p.16/56 Our model
We will represent actors by vertices
x and connect them with edges
▽The Small World problem and so much more! – p.16/56 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 and so much more! – p.16/56 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 and so much more! – p.16/56 Model parameters
There are n actors.
▽The Small World problem and so much more! – p.17/56 Model parameters
There are n actors. Fix a constant d.
▽The Small World problem and so much more! – p.17/56 Model parameters
There are n actors. Fix a constant d. We will begin with an arbitrary graph H such that . . .
▽The Small World problem and so much more! – p.17/56 Model parameters
There are n actors. Fix a constant d. We will begin with an arbitrary graph H such that ...... in H, each actor is connected to at least dn other actors.
▽The Small World problem and so much more! – p.17/56 Model parameters
There are n actors. Fix a constant d. We will begin with an arbitrary graph H such that ...... in H, each actor is connected to at least dn other actors. The constant d can be extremely tiny:
▽The Small World problem and so much more! – p.17/56 Model parameters
There are n actors. Fix a constant d. We will begin with an arbitrary graph H such that ...... 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 and so much more! – p.17/56 Model parameters
There are n actors. Fix a constant d. We will begin with an arbitrary graph H such that ...... 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 and so much more! – p.17/56 Model parameters
There are n actors. Fix a constant d. We will begin with an arbitrary graph H such that ...... in H, each actor is connected to at least dn other actors. The constant d can be extremely tiny:
100 0.1, 0.01, 10−10
It just needs to be independent of n.
The Small World problem and so much more! – p.17/56 Random casting
We add f(n) random casting connections.
▽The Small World problem and so much more! – p.18/56 Random casting
We add f(n) random casting connections.
What does random mean?
The Small World problem and so much more! – p.18/56 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 and so much more! – p.19/56 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 and so much more! – p.19/56 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 and so much more! – p.19/56 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 and so much more! – p.19/56 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 and so much more! – p.19/56 The models
For our purposes, these produce the same results.
The question:
▽The Small World problem and so much more! – p.20/56 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 and so much more! – p.20/56 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 and so much more! – p.20/56 Proposition
Proposition. If f(n) as n , then → ∞ → ∞
▽The Small World problem and so much more! – p.21/56 Proposition
Proposition. If f(n) as n , then → ∞ → ∞ Pr (diam ) 1 ≤ →
▽The Small World problem and so much more! – p.21/56 Proposition
Proposition. If f(n) as n , then → ∞ → ∞ Pr (diam 7) 1 ≤ →
▽The Small World problem and so much more! – p.21/56 Proposition
Proposition. If f(n) as n , then → ∞ → ∞ Pr (diam 7) 1 ≤ →
Important points:
▽The Small World problem and so much more! – p.21/56 Proposition
Proposition. If f(n) as n , then → ∞ → ∞ Pr (diam 7) 1 ≤ →
Important points: Recall f(n) is the number of random connections.
▽The Small World problem and so much more! – p.21/56 Proposition
Proposition. If f(n) as n , then → ∞ → ∞ Pr (diam 7) 1 ≤ →
Important points: Recall f(n) is the number of random connections. “7” doesn’t depend on d at all.
▽The Small World problem and so much more! – p.21/56 Proposition
Proposition. If f(n) as n , then → ∞ → ∞ Pr (diam 7) 1 ≤ →
Important points: Recall f(n) is the number of random connections. “7” doesn’t depend on d at all. f(n) can be very small:
▽The Small World problem and so much more! – p.21/56 Proposition
Proposition. If f(n) as n , then → ∞ → ∞ Pr (diam 7) 1 ≤ →
Important points: Recall f(n) is the number of random connections. “7” doesn’t depend on d at all. f(n) can be very small: √n
▽The Small World problem and so much more! – p.21/56 Proposition
Proposition. If f(n) as n , then → ∞ → ∞ Pr (diam 7) 1 ≤ →
Important points: Recall f(n) is the number of random connections. “7” doesn’t depend on d at all. f(n) can be very small: √n, log n
▽The Small World problem and so much more! – p.21/56 Proposition
Proposition. If f(n) as n , then → ∞ → ∞ Pr (diam 7) 1 ≤ →
Important points: Recall f(n) is the number of random connections. “7” doesn’t depend on d at all. f(n) can be very small: √n, log n, √log log log n
The Small World problem and so much more! – p.21/56 Proof of diameter 7
This is not hard to prove.
▽The Small World problem and so much more! – p.22/56 Proof of diameter 7
This is not hard to prove. Take G and construct v1,
v 1
▽The Small World problem and so much more! – p.22/56 Proof of diameter 7
This is not hard to prove. Take G and construct v1, v2,
v 1 v 2
▽The Small World problem and so much more! – p.22/56 Proof of diameter 7
This is not hard to prove. Take G and construct v1, v2, v3,
v 1v 2 v 3
▽The Small World problem and so much more! – p.22/56 Proof of diameter 7
This is not hard to prove. Take G and construct v1, v2, v3,...
. . .
v 1v 2v 3
▽The Small World problem and so much more! – p.22/56 Proof of diameter 7
This is not hard to prove. Take G and construct v1, v2, v3,... such that dist(vi, vj) 3 for i = j. ≥ 6
. . .
The Small World problem and so much more! – p.22/56 The decomposition
So, since they are of distance 3,
▽The Small World problem and so much more! – p.23/56 The decomposition
So, since they are of distance 3, their neighborhoods do not intersect.
▽The Small World problem and so much more! – p.23/56 The decomposition
So, since they are of distance 3, their neighborhoods do not intersect. The process will end after n/(dn + 1) steps. ≤ ⌊ ⌋
▽The Small World problem and so much more! – p.23/56 The decomposition
So, since they are of distance 3, their neighborhoods do not intersect. The process will end after n/(dn + 1) steps. ≤ ⌊ ⌋ The random edges guarantee at least one edge between each pair of the neighborhoods.
The Small World problem and so much more! – p.23/56 The path of length 7
So, every vertex is of distance at most 2 from some vi.
u w
vi vj
▽The Small World problem and so much more! – p.24/56 The path of length 7
So, every vertex is of distance at most 2 from some vi.
u w
vi vj
▽The Small World problem and so much more! – p.24/56 The path of length 7
So, every vertex is of distance at most 2 from some vi. One edge is between each pair of neighborhoods.
u w
vi vj
▽The Small World problem and so much more! – p.24/56 The path of length 7
So, every vertex is of distance at most 2 from some vi. One edge is between each pair of neighborhoods.
u w
vi vj
▽The Small World problem and so much more! – p.24/56 The path of length 7
So, every vertex is of distance at most 2 from some vi. One edge is between each pair of neighborhoods. And this gives the path of length 7:
u w
vi vj
▽The Small World problem and so much more! – p.24/56 The path of length 7
So, every vertex is of distance at most 2 from some vi. One edge is between each pair of neighborhoods. And this gives the path of length 7:
u w
vi vj
▽The Small World problem and so much more! – p.24/56 The path of length 7
So, every vertex is of distance at most 2 from some vi. One edge is between each pair of neighborhoods. And this gives the path of length 7:
u w
vi vj
▽The Small World problem and so much more! – p.24/56 The path of length 7
So, every vertex is of distance at most 2 from some vi. One edge is between each pair of neighborhoods. And this gives the path of length 7:
u w
vi vj
▽The Small World problem and so much more! – p.24/56 The path of length 7
So, every vertex is of distance at most 2 from some vi. One edge is between each pair of neighborhoods. And this gives the path of length 7:
u w
vi vj
▽The Small World problem and so much more! – p.24/56 The path of length 7
So, every vertex is of distance at most 2 from some vi. One edge is between each pair of neighborhoods. And this gives the path of length 7:
u w
vi vj
▽The Small World problem and so much more! – p.24/56 The path of length 7
So, every vertex is of distance at most 2 from some vi. One edge is between each pair of neighborhoods. And this gives the path of length 7:
u w
vi vj
The Small World problem and so much more! – p.24/56 Even better
There’s a better result, which is harder to prove:
▽The Small World problem and so much more! – p.25/56 Even better
There’s a better result, which is harder to prove:
Theorem. [BFKM] If f(n) as n , then → ∞ → ∞ Pr (diam 5) 1 ≤ →
▽The Small World problem and so much more! – p.25/56 Even better
There’s a better result, which is harder to prove:
Theorem. [BFKM] If f(n) as n , then → ∞ → ∞ Pr (diam 5) 1 ≤ →
To prove the theorem, you need the Regularity lemma.
▽The Small World problem and so much more! – p.25/56 Even better
There’s a better result, which is harder to prove:
Theorem. [BFKM] If f(n) as n , then → ∞ → ∞ Pr (diam 5) 1 ≤ →
To prove the theorem, you need the Regularity lemma.
The Regularity lemma is ’s powerful and complicated graph theoretic tool.
▽The Small World problem and so much more! – p.25/56 Even better
There’s a better result, which is harder to prove:
Theorem. [BFKM] If f(n) as n , then → ∞ → ∞ Pr (diam 5) 1 ≤ →
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 and so much more! – p.25/56 Best possible?
The theorem is “tight”:
▽The Small World problem and so much more! – p.26/56 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 and so much more! – p.26/56 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 and so much more! – p.27/56 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 and so much more! – p.27/56 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 and so much more! – p.27/56 5 degrees!
So, if you have a system, with a linear minimum degree and a little bit of randomness, the diameter will go to 5!
▽The Small World problem and so much more! – p.28/56 5 degrees!
So, if you have a system, with a linear minimum degree and a little bit of randomness, the diameter will go to 5!
What do you think of that?
▽The Small World problem and so much more! – p.28/56 5 degrees!
So, if you have a system, with a linear minimum degree and a little bit of randomness, the diameter will go to 5!
What do you think of that?
▽The Small World problem and so much more! – p.28/56 5 degrees!
So, if you have a system, with a linear minimum degree and a little bit of randomness, the diameter will go to 5!
What do you think of that?
▽The Small World problem and so much more! – p.28/56 5 degrees!
So, if you have a system, with a linear minimum degree and a little bit of randomness, the diameter will go to 5!
What do you think of that?
Whatever!
Let’s move to a new problem.
The Small World problem and so much more! – p.28/56 Graph editing problem
We have the idea of editing graphs. It is similar to the editing of DNA strings in biology.
▽The Small World problem and so much more! – p.29/56 Graph editing problem
Begin with a graph on a fixed vertex set. Edit it by deleting edges,
A graph.
▽The Small World problem and so much more! – p.29/56 Graph editing problem
Begin with a graph on a fixed vertex set. Edit it by deleting edges,
Editing of the graph.
▽The Small World problem and so much more! – p.29/56 Graph editing problem
Begin with a graph on a fixed vertex set. Edit it by deleting edges, and adding edges.
Further editing of the graph.
The Small World problem and so much more! – p.29/56 Original question
We were asked to consider a bipartite graph G = (A, B; E),
▽The Small World problem and so much more! – p.30/56 Original question
We were asked to consider a bipartite graph G = (A, B; E), A = B = n, | | | |
A
B
▽The Small World problem and so much more! – p.30/56 Original question
We were asked to consider a bipartite graph G = (A, B; E), A = B = n, and the number of edit | | | | operations to forbid a “W ” from being induced.
A
B
An induced “W ”
▽The Small World problem and so much more! – p.30/56 Original question
We were asked to consider a bipartite graph G = (A, B; E), A = B = n, and the number of edit | | | | operations to forbid a “W ” from being induced.
A
B
An induced “W ”(P5)
▽The Small World problem and so much more! – p.30/56 Original question
We were asked to consider a bipartite graph G = (A, B; E), A = B = n, and the number of edit | | | | operations to forbid a “W ” from being induced.
An induced “W ”(P5)
▽The Small World problem and so much more! – p.30/56 Original question
We were asked to consider a bipartite graph G = (A, B; E), A = B = n, and the number of edit | | | | operations to forbid a “W ” from being induced.
A
B
An induced “W ”(P5)
▽The Small World problem and so much more! – p.30/56 Original question
We were asked to consider a bipartite graph G = (A, B; E), A = B = n, and the number of edit | | | | operations to forbid a “W ” from being induced.
A
B
An induced “W ”(P5) and an induced “W ” upside-down.
▽The Small World problem and so much more! – p.30/56 Original question
We were asked to consider a bipartite graph G = (A, B; E), A = B = n, and the number of edit | | | | operations to forbid a “W ” from being induced.
A
B
Edit operations are between A and B.
The Small World problem and so much more! – p.30/56 Answer
Theorem. [AM] The maximum number of edits to remove all induced copies of W from an n n bipartite graph is at least ×
▽The Small World problem and so much more! – p.31/56 Answer
Theorem. [AM] The maximum number of edits to remove all induced copies of W from an n n bipartite graph is at least ×
n2 (1 o(1)) − 2
▽The Small World problem and so much more! – p.31/56 Answer
Theorem. [AM] The maximum number of edits to remove all induced copies of W from an n n bipartite graph is at least ×
n2 (1 o(1)) . − 2
This is best possible: We could either eliminate the edges or put in all of them, whichever is less than n2/2.
▽The Small World problem and so much more! – p.31/56 Answer
Theorem. [AM] The maximum number of edits to remove all induced copies of W from an n n bipartite graph is at least ×
n2 (1 o(1)) . − 2
This is (asymptotically) best possible: We could either eliminate the edges or put in all of them, whichever is less than n2/2.
▽The Small World problem and so much more! – p.31/56 Answer
Theorem. [AM] The maximum number of edits to remove all induced copies of W from an n n bipartite graph is at least ×
n2 (1 o(1)) . − 2
This is (asymptotically) best possible: We could either eliminate the edges or put in all of them, whichever is less than n2/2. The same theorem holds when W is any nontrivial bipartite graph.
▽The Small World problem and so much more! – p.31/56 Answer
Theorem. [AM] The maximum number of edits to remove all induced copies of W from an n n bipartite graph is at least ×
n2 (1 o(1)) . − 2
The same theorem holds when W is any nontrivial bipartite graph. The proof comes via . . .
▽The Small World problem and so much more! – p.31/56 Answer
Theorem. [AM] The maximum number of edits to remove all induced copies of W from an n n bipartite graph is at least ×
n2 (1 o(1)) . − 2
The same theorem holds when W is any nontrivial bipartite graph. The proof comes via the Regularity lemma.
The Small World problem and so much more! – p.31/56 The real question
Let us generalize.
▽The Small World problem and so much more! – p.32/56 The real question
Let us generalize.
Take any fixed graph H and an n-vertex graph G.
▽The Small World problem and so much more! – p.32/56 The real question
Let us generalize.
Take any fixed graph H and an n-vertex graph G. (Not just bipartite.)
▽The Small World problem and so much more! – p.32/56 The real question
Let us generalize.
Take any fixed graph H and an n-vertex graph G. (Not just bipartite.)
How many edit operations does it take to make sure no induced copy of H is left in G?
The Small World problem and so much more! – p.32/56 Self-complementary
Let H be the cycle on 5 vertices, C5.
▽The Small World problem and so much more! – p.33/56 Self-complementary
Let H be the cycle on 5 vertices, C5.
This is a self-complementary graph.
▽The Small World problem and so much more! – p.33/56 Self-complementary
Let H be the cycle on 5 vertices, C5. Theorem. [AKM] The number of edit operations needed to ensure no induced copy of C5 in an n-vertex graph is at least
▽The Small World problem and so much more! – p.33/56 Self-complementary
Let H be the cycle on 5 vertices, C5. Theorem. [AKM] The number of edit operations needed to ensure no induced copy of C5 in an n-vertex graph is at least
1 n (1 o(1)) , − 42
▽The Small World problem and so much more! – p.33/56 Self-complementary
Let H be any self-complementary graph. Theorem. [AKM] The number of edit operations needed to ensure no induced copy of H in an n-vertex graph is at least
1 n (1 o(1)) , − 42 and this is best possible.
▽The Small World problem and so much more! – p.33/56 Self-complementary
Let H be any self-complementary graph. Theorem. [AKM] The number of edit operations needed to ensure no induced copy of H in an n-vertex graph is at least
1 n (1 o(1)) , − 2(k 1)2 −
▽The Small World problem and so much more! – p.33/56 Self-complementary
Let H be any self-complementary graph. Theorem. [AKM] The number of edit operations needed to ensure no induced copy of H in an n-vertex graph is at least
1 n (1 o(1)) , − 2(k 1)2 − where k is a value that depends on H only.
▽The Small World problem and so much more! – p.33/56 Self-complementary
Let H be any self-complementary graph. Theorem. [AKM] The number of edit operations needed to ensure no induced copy of H in an n-vertex graph is at least
1 n (1 o(1)) , − 2(k 1)2 − where k is a value that depends on H only. This is best possible.
▽The Small World problem and so much more! – p.33/56 Self-complementary
Let H be any self-complementary graph. Theorem. [AKM] The number of edit operations needed to ensure no induced copy of H in an n-vertex graph is at least
1 n (1 o(1)) , − 2(k 1)2 − where k is a value that depends on H only. This is best possible.
The proof comes via . . .
▽The Small World problem and so much more! – p.33/56 Self-complementary
Let H be any self-complementary graph. Theorem. [AKM] The number of edit operations needed to ensure no induced copy of H in an n-vertex graph is at least
1 n (1 o(1)) , − 2(k 1)2 − where k is a value that depends on H only. This is best possible.
The proof comes via the Regularity lemma.
The Small World problem and so much more! – p.33/56 Editing distance
Although the problem is open for many graphs H, there would be no hope of solving it without the Regularity lemma.
▽The Small World problem and so much more! – p.34/56 Editing distance
Although the problem is open for many graphs H, there would be no hope of solving it without the Regularity lemma.
Well?
▽The Small World problem and so much more! – p.34/56 Editing distance
Although the problem is open for many graphs H, there would be no hope of solving it without the Regularity lemma.
Well?
▽The Small World problem and so much more! – p.34/56 Editing distance
Although the problem is open for many graphs H, there would be no hope of solving it without the Regularity lemma.
Well?
The Small World problem and so much more! – p.34/56 Regularity Lemma
So, what is this Regularity lemma?
’s Regularity lemma was used to prove a theorem in number theory.
▽The Small World problem and so much more! – p.35/56 Regularity Lemma
So, what is this Regularity lemma?
Endre Szemerédi’s Regularity lemma was used to prove a theorem in number theory.
▽The Small World problem and so much more! – p.35/56 Regularity Lemma
So, what is this Regularity lemma?
Endre Szemerédi’s Regularity lemma was used to prove a theorem in number theory. It’s a powerful tool for proving results that are asymptotic or apply to very large graphs.
▽The Small World problem and so much more! – p.35/56 Regularity Lemma
Endre Szemerédi’s Regularity lemma was used to prove a theorem in number theory. It’s a powerful tool for proving results that are asymptotic or apply to very large graphs.
It can prove an amazing array of results.
▽The Small World problem and so much more! – p.35/56 Regularity Lemma
Endre Szemerédi’s Regularity lemma was used to prove a theorem in number theory. It’s a powerful tool for proving results that are asymptotic or apply to very large graphs.
It can prove an amazing array of results.
The Small World problem and so much more! – p.35/56 Large graphs? How large?
Q: How big a graph are we talkin’?
▽The Small World problem and so much more! – p.36/56 Large graphs? How large?
Q: How big a graph are we talkin’?
A:
▽The Small World problem and so much more! – p.36/56 Large graphs? How large?
Q: How big a graph are we talkin’?
A: Huge.
The Small World problem and so much more! – p.36/56 OK, smart guy, how huge?
▽The Small World problem and so much more! – p.37/56 OK, smart guy, how huge?
Applying the Regularity Lemma to a graph requires it to have many vertices.
▽The Small World problem and so much more! – p.37/56 OK, smart guy, how huge?
Applying the Regularity Lemma to a graph requires it to have many vertices. The proof of the Regularity Lemma gives a tower function for M(ǫ), the number of vertices required to get the desired ǫ-regular partition .
▽The Small World problem and so much more! – p.37/56 OK, smart guy, how huge?
Applying the Regularity Lemma to a graph requires it to have many vertices. The proof of the Regularity Lemma gives a tower function for M(ǫ), the number of vertices required to get the desired ǫ-regular partition (whatever that is).
▽The Small World problem and so much more! – p.37/56 OK, smart guy, how huge?
Applying the Regularity Lemma to a graph requires it to have many vertices. The proof of the Regularity Lemma gives a tower function for M(ǫ), the number of vertices required to get the desired ǫ-regular partition (whatever that is). What the proof gives is:
4(1/ǫ) 5(1/ǫ) ·4 ·5 ·· ·· 44 55 4 M(ǫ) 5 ≤ ≤
▽The Small World problem and so much more! – p.37/56 OK, smart guy, how huge?
Applying the Regularity Lemma to a graph requires it to have many vertices. The proof of the Regularity Lemma gives a tower function for M(ǫ), the number of vertices required to get the desired ǫ-regular partition (whatever that is). What the proof gives is:
4(1/ǫ) 5(1/ǫ) ·4 ·5 ·· ·· 10 44 55 10 5 4 M(ǫ) 5 5 ǫ ≤ ≤ ǫ
The Small World problem and so much more! – p.37/56 Does it need to be so big?
Sorry, some kind of tower function is necessary.
▽The Small World problem and so much more! – p.38/56 Does it need to be so big?
Sorry, some kind of tower function is necessary.
Theorem [Gowers, 1997] For any ǫ > 0, there is a graph so that any application of the Regularity Lemma requires that the number of clusters is at least a number which is a tower of twos of height proportional to log(1/ǫ).
▽The Small World problem and so much more! – p.38/56 Does it need to be so big?
Sorry, some kind of tower function is necessary.
Theorem [Gowers, 1997] For any ǫ > 0, there is a graph so that any application of the Regularity Lemma requires that the number of clusters is at least a number which is a tower of twos of height proportional to log(1/ǫ).
So, there’s a minor drawback.
The Small World problem and so much more! – p.38/56 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 and so much more! – p.39/56 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 and so much more! – p.39/56 Erdos˝ number project
The Erdos˝ number project is concerned with the distance of mathematicians from Paul Erdos.˝
▽The Small World problem and so much more! – p.40/56 Erdos˝ number project
# The project is concerned with the distance of mathematicians from Paul Erdos.˝
▽The Small World problem and so much more! – p.40/56 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 and so much more! – p.40/56 Most prolific authors
: 1401 papers (Erdos˝ number )
▽The Small World problem and so much more! – p.41/56 Most prolific authors
: 1401 papers (Erdos˝ number )
Drumi Bainov: 782 (Erdos˝ number )
▽The Small World problem and so much more! – p.41/56 Most prolific authors
: 1401 papers (Erdos˝ number )
Drumi Bainov: 782 (Erdos˝ number )
Leonard Carlitz: 730 (Erdos˝ number )
▽The Small World problem and so much more! – p.41/56 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 and so much more! – p.41/56 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 and so much more! – p.41/56 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 and so much more! – p.41/56 Erdos˝ number statistics
wrote 1401 papers in Math Reviews.
▽The Small World problem and so much more! – p.42/56 Erdos˝ number statistics
wrote 1401 papers in Math Reviews.
There are 337, 000 vertices (authors) in the graph.
▽The Small World problem and so much more! – p.42/56 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 and so much more! – p.42/56 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 and so much more! – p.42/56 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 and so much more! – p.42/56 Experimental data
# 0 1 1 502 2 5713 3 26422 4 62136 5 66157
▽The Small World problem and so much more! – p.43/56 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 and so much more! – p.43/56 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 and so much more! – p.43/56 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 and so much more! – p.43/56 The unknown mathematician
\ x\ \ \ \ \ \ \ \ H ¨ H ¨ H ¨ x H x ¨ x H ¨ H ¨ HH¨¨ x
▽The Small World problem and so much more! – p.44/56 The unknown mathematician
\ x\ \ \ \ \ \ \ \ H ¨ H ¨ H ¨ x H x ¨ x H ¨ H ¨ HH¨¨ x
▽The Small World problem and so much more! – p.44/56 The unknown mathematician
\ x\ \ \ \ \ \ \ \ H ¨ H ¨ H ¨ x H x ¨ x H ¨ H ¨ HH¨¨ x
▽The Small World problem and so much more! – p.44/56 The unknown mathematician
\ x\ \ \ \ \ \ \ \ H ¨ H ¨ H ¨ x H x ¨ x H ¨ H ¨ HH¨¨ x
▽The Small World problem and so much more! – p.44/56 The unknown mathematician
\ x\ \ \ \ \ \ \ \ H ¨ H ¨ H ¨ x H x ¨ x H ¨ H ¨ HH¨¨ x
▽The Small World problem and so much more! – p.44/56 The unknown mathematician
\ x\ \ \ \ \ \ \ \ H ¨ H ¨ H ¨ x H x ¨ x H ¨ H ¨ HH¨¨ x
The Small World problem and so much more! – p.44/56 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 and so much more! – p.45/56 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 and so much more! – p.45/56 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 and so much more! – p.46/56 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 and so much more! – p.46/56 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 and so much more! – p.47/56 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 and so much more! – p.47/56 Computer networks
Graphs model much more serious stuff.
I.e.,
computer networks,
▽The Small World problem and so much more! – p.48/56 Computer networks
Graphs model much more serious stuff.
I.e.,
computer networks,
shipping routes,
▽The Small World problem and so much more! – p.48/56 Computer networks
Graphs model much more serious stuff.
I.e.,
computer networks,
shipping routes,
distribution networks.
The Small World problem and so much more! – p.48/56 Network question
In networks we are concerned with one particular quantity:
▽The Small World problem and so much more! – p.49/56 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 and so much more! – p.49/56 Same model
n computers
▽The Small World problem and so much more! – p.50/56 Same model
n computers
in H, each computer is connected to dn others ≥
▽The Small World problem and so much more! – p.50/56 Same model
n computers
in H, each computer is connected to dn others ≥
add f(n) random connections
▽The Small World problem and so much more! – p.50/56 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 and so much more! – p.50/56 Connectivity theorem
Theorem. [BFKM] Let k be a function of n that is n. ≪
▽The Small World problem and so much more! – p.51/56 Connectivity theorem
Theorem. [BFKM] Let k be a function of n that is n. (That is, k grows more slowly than n as n .) ≪ → ∞
▽The Small World problem and so much more! – p.51/56 Connectivity theorem
Theorem. [BFKM] 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 and so much more! – p.51/56 Connectivity theorem
Theorem. [BFKM] 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 and so much more! – p.51/56 Connectivity theorem
Theorem. [BFKM] 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 and so much more! – p.51/56 Bottom line
A way to interpret this theorem is:
▽The Small World problem and so much more! – p.52/56 Bottom line
A way to interpret this theorem is:
If you need k-connectivity,
▽The Small World problem and so much more! – p.52/56 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 and so much more! – p.52/56 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 and so much more! – p.52/56 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 and so much more! – p.53/56 Other properties
We’ve used this model to investigate other properties:
Hamilton cycle
▽The Small World problem and so much more! – p.54/56 Other properties
We’ve used this model to investigate other properties:
Hamilton cycle
Small cliques as subgraphs
▽The Small World problem and so much more! – p.54/56 Other properties
We’ve used this model to investigate other properties:
Hamilton cycle
Small cliques as subgraphs
Chromatic number
▽The Small World problem and so much more! – p.54/56 Other properties
We’ve used this model to investigate other properties:
Hamilton cycle
Small cliques as subgraphs
Chromatic number
Some use the Regularity lemma, some use other techniques.
The Small World problem and so much more! – p.54/56 Coming to a classroom near you!
Extremal Graph Theory
12:40-2:00 TR
Sign up today!
▽The Small World problem and so much more! – p.55/56 Coming to a classroom near you!
Extremal Graph Theory
12:40-2:00 TR
Sign up today (OK, tomorrow)!
▽The Small World problem and so much more! – p.55/56 Coming to a classroom near you!
Extremal Graph Theory
12:40-2:00 TR
Sign up today (OK, tomorrow)! http://orion.math.iastate.edu/rymartin/ISU690I/ISU690Isyl.shtml
The Small World problem and so much more! – p.55/56 More Cowbell!
The Small World problem and so much more! – p.56/56