Six degrees of graph theory:Kevin Bacon, Paul Erdos, William

McKinley and meRyan Martin

rymartin@iastate.edu

Assistant Professor

Mathematics Department

Iowa State University

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 1/61

Joint WorkThis talk is based on joint work with

• Tom Bohman,Carnegie Mellon University

• Alan Frieze,Carnegie Mellon University

• Michael Krivelevich,Tel Aviv University

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 2/61

Six? degrees of separationIn the Kevin Bacon Game, it is postulated that thecenter of the Hollywood universe is

Kevin Bacon.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 3/61

Six? degrees of separationIn the Kevin Bacon Game, it is postulated that thecenter of the Hollywood universe is

.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 3/61

Six? degrees of separationIn the Kevin Bacon Game, it is postulated that thecenter of the Hollywood universe is

Kevin Bacon.

This is false. It is Rod Steiger.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 3/61

Six? degrees of separationIn the Kevin Bacon Game, it is postulated that thecenter of the Hollywood universe is

Kevin Bacon.

This is false. It is Rod Steiger.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 3/61

Six? degrees of separationIn the Kevin Bacon Game, it is postulated that thecenter of the Hollywood universe is

Kevin Bacon.

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

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 3/61

Six? degrees of separationIn the Kevin Bacon Game, it is postulated that thecenter of the Hollywood universe is

Kevin Bacon.

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

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

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 3/61

Bacon numberThe actor’s

Bacon number

is the fewest number of steps it takes to connect thatactor to

Kevin Bacon.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 4/61

Bacon numberThe actor’s

#is the fewest number of steps it takes to connect thatactor to

.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 4/61

Bacon numberThe actor’s

Bacon number

is the fewest number of steps it takes to connect thatactor to

Kevin Bacon.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 4/61

Bacon numberAn actor can have infinite

#.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 4/61

Bacon numberAn actor can have infinite

#.

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

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 4/61

Example: Kevin Costner

Kevin Costner is linked to Kevin Bacon because bothappeared in

JFK (1991).

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 5/61

Example: Kevin Costner

is linked to because bothappeared in

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 5/61

Example: Kevin Costneris linked to because both appeared in

JFK (1991).

So, Kevin Costner’s Kevin Bacon number is

???

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 5/61

Example: Kevin Costneris linked to because both appeared in

JFK (1991).

So, Kevin Costner’s Kevin Bacon number is

1

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 5/61

Example: Kevin Costneris linked to because both appeared in

JFK (1991).

So, ’s#

is

1

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 5/61

Illustration: Kevin Costner

x x

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 6/61

Illustration: Kevin Costner

x x

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 6/61

Illustration: Kevin Costner

x x

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 6/61

Illustration: Kevin Costner

x x

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 6/61

E.g.: Henry “Fonz” WinklerWe know that

• appeared with in

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 7/61

E.g.: Henry “Fonz” WinklerWe know that

• Henry Winkler appeared with Michael Keaton inNight Shift (1982)

• appeared with in

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 7/61

E.g.: Henry “Fonz” WinklerWe know that

• Henry Winkler appeared with Michael Keaton inNight Shift (1982)

• Michael Keaton appeared with Kim Basinger inBatman (1989)

• appeared with in

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 7/61

E.g.: Henry “Fonz” WinklerWe know that

• Henry Winkler appeared with Michael Keaton inNight Shift (1982)

• Michael Keaton appeared with Kim Basinger inBatman (1989)

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

• appeared with in

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 7/61

E.g.: Henry “Fonz” WinklerWe know that

• Henry Winkler appeared with Michael Keaton inNight Shift (1982)

• Michael Keaton appeared with Kim Basinger inBatman (1989)

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

• Mickey Rourke appeared with Kevin Bacon inDiner (1982)

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 7/61

A chain: Henry “Fonz” Winkler

x x x x x

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 8/61

A chain: Henry “Fonz” Winkler

x x x x x

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 8/61

A chain: Henry “Fonz” Winkler

x x x x x

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 8/61

A chain: Henry “Fonz” Winkler

x x x x x

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 8/61

A chain: Henry “Fonz” Winkler

x x x x x

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 8/61

A chain: Henry “Fonz” Winkler

x x x x x

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 8/61

A chain: Henry “Fonz” Winkler

x x x x x

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 8/61

A chain: Henry “Fonz” Winkler

x x x x x

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 8/61

A chain: Henry “Fonz” Winkler

x x x x x

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 8/61

A chain: Henry “Fonz” Winkler

x x x x x

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 8/61

More: Henry “Fonz” WinklerBut it is also true that

• appeared with??? in(2000).

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 9/61

More: Henry “Fonz” WinklerBut it is also true that

• appeared with in(2000).

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 9/61

More: Henry “Fonz” WinklerBut it is also true that

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

• appeared with in(2000)

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 9/61

More: Henry “Fonz” WinklerBut it is also true that

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

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

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 9/61

Better: Henry “Fonz” Winkler

x x x

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 10/61

Better: Henry “Fonz” Winkler

x x x

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 10/61

Better: Henry “Fonz” Winkler

x x x

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 10/61

Better: Henry “Fonz” Winkler

x x x

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 10/61

Better: Henry “Fonz” Winkler

x x x

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 10/61

Better: Henry “Fonz” Winkler

x x x

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 10/61

Can we do even better?

has never appeared in a film with

.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 11/61

Can we do even better?

Kevin Bacon has never appeared in a film withHenry Winkler.

So, ’s#

is

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 11/61

Can we do even better?

Kevin Bacon has never appeared in a film withHenry Winkler.

So, Henry Winkler’s Kevin Bacon number is

???

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 11/61

Can we do even better?

Kevin Bacon has never appeared in a film withHenry Winkler.

So, Henry Winkler’s Kevin Bacon number is

2

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 11/61

In sum: Henry “Fonz” Winkler

PPPPPPhhhhhh((((((

������

((((((((

(((

hhhhhhhh

hhhx xx xxx

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 12/61

In sum: Henry “Fonz” Winkler

PPPPPPhhhhhh((((((

������

((((((((

(((

hhhhhhhh

hhhx xx xxx

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 12/61

In sum: Henry “Fonz” Winkler

PPPPPPhhhhhh((((((

������

((((((((

(((

hhhhhhhh

hhhx xx xxx

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 12/61

Experimental data

# of actors

0 11 18062 1450243 3951264 954975 7451

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 13/61

Experimental data

# of actors

0 11 18062 1450243 3951264 954975 7451

# of actors

6 9337 1068 13

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 13/61

What about the high numbers?As we said before, there are actors with infinite

#.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 14/61

What about the high numbers?As we said before, there are actors with infinite

#.

The actors with large#

are obscure and thereason why is fairly obvious.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 14/61

Kevin Bacon not so specialMost successful actors follow the same pattern:

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 15/61

Kevin Bacon not so specialMost successful actors follow the same pattern:

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

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 15/61

Kevin Bacon not so specialMost successful actors follow the same pattern:

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

Why?

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 15/61

Our modelWe will represent actors byvertices

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 16/61

Our modelWe will represent actors byvertices

x

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 16/61

Our modelWe will represent actors byvertices

xand connect them withedges

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 16/61

Our modelWe will represent actors byvertices

xand connect them withedges

x xif they appeared in the same film.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 16/61

Our modelWe will represent actors byvertices

xand connect them withedges

x xif they appeared in the same film.

This is agraph.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 16/61

Model parameters• There aren actors.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 17/61

Model parameters• There aren actors.• Fix a constantd.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 17/61

Model parameters• There aren actors.• Fix a constantd.• We will begin with anarbitrary graphH.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 17/61

Model parameters• There aren actors.• Fix a constantd.• We will begin with anarbitrary graphH.• In H, each actor is connected to at leastdn other

actors.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 17/61

Model parameters• There aren actors.• Fix a constantd.• We will begin with anarbitrary graphH.• In H, each actor is connected to at leastdn other

actors.• The constantd can be extremely tiny:

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 17/61

Model parameters• There aren actors.• Fix a constantd.• We will begin with anarbitrary graphH.• In H, each actor is connected to at leastdn other

actors.• The constantd can be extremely tiny:

0.1

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 17/61

Model parameters• There aren actors.• Fix a constantd.• We will begin with anarbitrary graphH.• In H, each actor is connected to at leastdn other

actors.• The constantd can be extremely tiny:

0.1, 0.01

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 17/61

Model parameters• There aren actors.• Fix a constantd.• We will begin with anarbitrary graphH.• In H, each actor is connected to at leastdn other

actors.• The constantd can be extremely tiny:

0.1, 0.01, 0.000001

It just needs to be independent ofn.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 17/61

Random castingWe addf(n) random casting connections.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 18/61

Random castingWe addf(n) random casting connections.

What doesrandommean?

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 18/61

Random edgesLet N be the number of pairs with no connectionbetween them (non-edges). We can createm newrandom edges in two ways:

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 19/61

Random edgesLet N be the number of pairs with no connectionbetween them (non-edges). We can createm newrandom edges in two ways:

• For every set ofm unconnected pairs, choose oneset uniformly at random.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 19/61

Random edgesLet N be the number of pairs with no connectionbetween them (non-edges). We can createm newrandom edges in two ways:

• For every set ofm unconnected pairs, choose oneset uniformly at random.

• Connect a previously unconnected pair,independently, with probabilitym/N .

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 19/61

Random edgesLet N be the number of pairs with no connectionbetween them (non-edges). We can createm newrandom edges in two ways:

• For every set ofm unconnected pairs, choose oneset uniformly at random.

• Connect a previously unconnected pair,independently, with probabilitym/N (coin flips).

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 19/61

Random edgesLet N be the number of pairs with no connectionbetween them (non-edges). We can createm newrandom edges in two ways:

• For every set ofm unconnected pairs, choose oneset uniformly at random.

• Connect a previously unconnected pair,independently, with probabilitym/N (coin flips).The average number of new connections ism.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 19/61

The modelsFor our purposes, these produce the same results.

The question:

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 20/61

The modelsFor our purposes, these produce the same results.

The question:

What is the longest distance between anypair of actors ?

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 20/61

The modelsFor our purposes, these produce the same results.

The question:

What is the longest distance between anypair of actors(diameter)?

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 20/61

The theoremTheorem. If f(n) → ∞ asn → ∞, then

Pr (diam ≤ 5) → 1

Important points:• Recallf(n) is the number of random

connections.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 21/61

The theoremTheorem. If f(n) → ∞ asn → ∞, then

Pr (diam ≤ 5) → 1

Important points:• Recallf(n) is the number of random

connections.• “5” doesn’t depend ond at all.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 21/61

The theoremTheorem. If f(n) → ∞ asn → ∞, then

Pr (diam ≤ 5) → 1

Important points:• Recallf(n) is the number of random

connections.• “5” doesn’t depend ond at all.• f(n) can be very small:

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 21/61

The theoremTheorem. If f(n) → ∞ asn → ∞, then

Pr (diam ≤ 5) → 1

Important points:• Recallf(n) is the number of random

connections.• “5” doesn’t depend ond at all.• f(n) can be very small:

√n

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 21/61

The theoremTheorem. If f(n) → ∞ asn → ∞, then

Pr (diam ≤ 5) → 1

Important points:• Recallf(n) is the number of random

connections.• “5” doesn’t depend ond at all.• f(n) can be very small:

√n, log n

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 21/61

The theoremTheorem. If f(n) → ∞ asn → ∞, then

Pr (diam ≤ 5) → 1

Important points:• Recallf(n) is the number of random

connections.• “5” doesn’t depend ond at all.• f(n) can be very small:

√n, log n,

√log log log n

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 21/61

Qualifying the modelThis is a nice result on a pretty good model.

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

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 22/61

The proofTo prove the theorem, you need theRegularityLemma.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 23/61

The proofTo prove the theorem, you need theRegularityLemma.

The Regularity Lemma is ’spowerful and complicated graph theoretic tool.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 23/61

The proofTo prove the theorem, you need theRegularityLemma.

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

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 23/61

Best possible?

The theorem is “tight”:

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 24/61

Best possible?

The theorem is “tight”:

If there aren’t an infinite number of edgesadded, then someH ’s will be disconnected.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 24/61

What about closer connections?

• To getdiam ≤ 4, you need randomconnections.

• To getdiam ≤ 3, you need randomconnections.

• To getdiam ≤ 2, you need randomconnections.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 25/61

What about closer connections?

• To getdiam ≤ 4, you needc1 log n randomconnections.

• To getdiam ≤ 3, you needc1 log n randomconnections.

• To getdiam ≤ 2, you need randomconnections.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 25/61

What about closer connections?

• To getdiam ≤ 4, you needc1 log n randomconnections.

• To getdiam ≤ 3, you needc1 log n randomconnections.

• To getdiam ≤ 2, you needc2n log n randomconnections.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 25/61

Applying this knowledge

To have it be very likely that everyone isconnected by a path of no more than 5acquaintances, just arrange a few randommeetings.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 26/61

Applying this knowledge

To have it be very likely that everyone isconnected by a path of no more than 5acquaintances, just arrange a few randommeetings.

Think about people at Central College.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 26/61

Central College cliques'

&

$

%xxx

xxx xx

x xxx

x xxx

x

x

EEEE����������

��������PP

PPP��LLLL�������QQb

bb CCBB���DDD�

��EEE�����hhh

������

����

������A

AAAAA

((((!!!

!!ll

ll

ll

AAAA

JJJJJJJJ

``````

bbbAAA

ZZ

���TTTT

PP

Mathematicians

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 27/61

Central College cliques'

&

$

%xxx

xxx xx

x xxx

x xxx

x

x

EEEE����������

��������PP

PPP��LLLL�������QQb

bb CCBB���DDD�

��EEE�����hhh

������

����

������A

AAAAA

((((!!!

!!ll

ll

ll

AAAA

JJJJJJJJ

``````

bbbAAA

ZZ

���TTTT

PP

'

&

$

%xx

x

xx

xxx

x

x

x xx

xx

@@

eeeee

xxLLLLL

���

��

��

���

eee

���

((((llllJJJ

ZZ

aaaa

hhhhhx!!!!AAAA

���

hhhhbb

bb

``"

"""

((((

���

bbbbCC

��` QQ

(((HHHHH���

��

Mathematicians Social Scientists

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 27/61

More Central College cliques

xx

x

'

&

$

%

xx

x

xx x

x

x

x

xxx

xx``PP

hhhh

""""

aaaa

But for us to getdiameter≤ 5, wedo need each per-son to know atleastdn others be-fore we add fewrandom edges.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 28/61

More Central College cliques

xx

x

'

&

$

%

xx

x

xx x

x

x

x

xxx

xx``PP

hhhh

""""

aaaa

But for us to getdiameter≤ 5, wedo need each per-son to know atleastdn others be-fore we add fewrandom edges.

Computer Scientists

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 28/61

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

six degrees of separation.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 29/61

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

six degrees of separation.

In fact, it isFIVE degrees of separation

and there’s an actual proof!

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 29/61

Erdos numberOne of the most prolific mathematicians of the 20thcentury was

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

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 30/61

Erdos numberOne of the most prolific mathematicians of the 20thcentury was

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

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 30/61

Erdos number project

TheErdos number project is concerned with thedistance of mathematicians from Paul Erdos.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 31/61

Erdos number project

The#

project is concerned with thedistance of mathematicians from Paul Erdos.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 31/61

Erdos number project

The#

project is concerned with thedistance of mathematicians from Paul Erdos.

Two mathematicians are connected if theyco-authored a paper together and that paper appears inMathematical Reviews, accessible by MathSciNet.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 31/61

Most prolific authors

• : 1401 papers (Erdos number )

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 32/61

Most prolific authors

• : 1401 papers (Erdos number )

• Drumi Bainov:782 (Erdos number )

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 32/61

Most prolific authors

• : 1401 papers (Erdos number )

• Drumi Bainov:782 (Erdos number )

• Leonard Carlitz:730 (Erdos number )

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 32/61

Most prolific authors

• : 1401 papers (Erdos number )

• Drumi Bainov:782 (Erdos number )

• Leonard Carlitz:730 (Erdos number )

• Lucien Godeaux:644 (Erdos number )

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 32/61

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 )

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 32/61

Most prolific authors

• : 1401 papers (Erdos number0)

• Drumi Bainov:782 (Erdos number4)

• Leonard Carlitz:730 (Erdos number2)

• Lucien Godeaux:644 (Erdos number∞)

• Saharon Shelah:600 (Erdos number1)

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 32/61

Erdos number statistics

• wrote1401 papers in Math Reviews.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 33/61

Erdos number statistics

• wrote1401 papers in Math Reviews.

• There are337, 000 vertices (authors) in the graph.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 33/61

Erdos number statistics

• wrote1401 papers in Math Reviews.

• There are337, 000 vertices (authors) in the graph.

• There are about496, 000 edges.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 33/61

Erdos number statistics

• wrote1401 papers in Math Reviews.

• There are337, 000 vertices (authors) in the graph.

• There are about496, 000 edges.

• Average number of authors per paper:1.45

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 33/61

Erdos number statistics

• wrote1401 papers in Math Reviews.

• There are337, 000 vertices (authors) in the graph.

• There are about496, 000 edges.

• Average number of authors per paper:1.45

• Average number of papers per author:6.87

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 33/61

Experimental data

#0 11 5022 57133 264224 621365 661576 322807 10431

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 34/61

Experimental data

#0 11 5092 69843 264224 621365 661576 322807 10431

#8 32149 95310 26211 9412 2313 414 715 1

(Most recent data)

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 34/61

Experimental data

#0 11 5092 69843 264224 621365 661576 322807 10431

#8 32149 95310 26211 9412 2313 414 715 1 (R. G. Kamalov)

(Most recent data)

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 34/61

The unknown mathematician

xxx

x

xPPPPPPPPPPP

,,,,,,,,,,,

�����������

ll

ll

ll

ll

lll

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 35/61

The unknown mathematician

xxx

x

xPPPPPPPPPPP

,,,,,,,,,,,

�����������

ll

ll

ll

ll

lll

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 35/61

The unknown mathematician

xxx

x

xPPPPPPPPPPP

,,,,,,,,,,,

�����������

ll

ll

ll

ll

lll

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 35/61

The unknown mathematician

xxx

x

xPPPPPPPPPPP

,,,,,,,,,,,

�����������

ll

ll

ll

ll

lll

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 35/61

The unknown mathematician

xxx

x

xPPPPPPPPPPP

,,,,,,,,,,,

�����������

ll

ll

ll

ll

lll

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 35/61

The unknown mathematician

xxx

x

xPPPPPPPPPPP

,,,,,,,,,,,

�����������

ll

ll

ll

ll

lll

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 35/61

Computer networksGraphs model much more serious stuff.

I.e.,

• computer networks,

• shipping routes,

• distribution networks.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 36/61

Network questionIn networks we are concerned with one particularquantity:

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 37/61

Network questionIn networks we are concerned with one particularquantity:

connectivity: A connected graph isk-connected ifremovingany set ofk − 1 vertices (and allrelevant edges) leaves the graph connected.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 37/61

Same model• n computers

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 38/61

Same model• n computers

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

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 38/61

Same model• n computers

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

• addf(n) random connections

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 38/61

Same model• n computers

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

• addf(n) random connections

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

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 38/61

Connectivity theoremTheorem. Let k be a function ofn that is≪ n. Let Hhave the property that each vertex is connected to atleastdn other vertices.

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

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 39/61

Connectivity theoremTheorem. Let k be a function ofn that is≪ n. Let Hhave the property that each vertex is connected to atleastdn other vertices.

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

• If d < 1/2, there is anH0 such that for everyk ≪ n, f(n) = k − 1 ensures that the graph failsto bek-connected, with high probability.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 39/61

Bottom lineA way to interpret this theorem is:

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 40/61

Bottom lineA way to interpret this theorem is:

If you needk-connectivity,

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 40/61

Bottom lineA way to interpret this theorem is:

If you needk-connectivity,

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

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 40/61

Bottom lineA way to interpret this theorem is:

If you needk-connectivity,

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

If fewer thank random edges are added,k-connectivity does not necessarily occur.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 40/61

Worst caseWhat is thatH0?

H0 =

�� S

S��S

S""""b

bbbx x

xxx x����AAAAb

bbb""""

�� S

S��S

S""""b

bbbx x

xxx x����AAAAb

bbb""""

�� S

S��S

S""""b

bbbx x

xxx x����AAAAb

bbb""""

�� S

S��S

S""""b

bbbx x

xxx x����AAAAb

bbb""""

�� S

S��S

S""""bb

bbx x

xxx x����AAAAb

bbb""""

�� S

S��S

S""""bb

bbx x

xxx x����AAAAb

bbb""""

�� S

S��S

S""""b

bbbx x

xxx x����AAAAb

bbb""""

�� S

S��S

S""""b

bbbx x

xxx x����AAAAb

bbb""""

Disjoint cliques give the worst case.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 41/61

Other propertiesWe’ve used this model to investigate other properties:

• Hamilton cycle

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 42/61

Other propertiesWe’ve used this model to investigate other properties:

• Hamilton cycle

• Small cliques as subgraphs

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 42/61

Other propertiesWe’ve used this model to investigate other properties:

• Hamilton cycle

• Small cliques as subgraphs

• Chromatic number

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 42/61

What next?Shall we conclude with

• more mathematics,

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 43/61

What next?Shall we conclude with

• more mathematics,

• people with high Bacon numbers,

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 43/61

What next?Shall we conclude with

• more mathematics,

• people with high Bacon numbers,

• mathematicians and dead presidents, or

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 43/61

What next?Shall we conclude with

• more mathematics,

• people with high Bacon numbers,

• mathematicians and dead presidents, or

• connections between Bacon numbers and Erdosnumbers?

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 43/61

Intersecting hypergraphsI work on the question of random intersectinghypergraphs.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 44/61

Intersecting hypergraphsI work on the question of random intersectinghypergraphs.

We take subsets at random so that, with eachselection, every pair of subsets has a nonemptyintersection.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 44/61

Intersecting hypergraphsI work on the question of random intersectinghypergraphs.

We take subsets at random so that, with eachselection, every pair of subsets has a nonemptyintersection.

Eventually, we run out of eligible subsets.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 44/61

Intersecting hypergraphsI work on the question of random intersectinghypergraphs.

We take subsets at random so that, with eachselection, every pair of subsets has a nonemptyintersection.

Eventually, we run out of eligible subsets.

What do we end up with?

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 44/61

Intersecting hypergraphsI work on the question of random intersectinghypergraphs.

We take subsets, of sizer, from [n] at random.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 44/61

Intersecting hypergraphsWe take subsets, of sizer, from [n] at random.

• If r ≪ n1/3, then all subsets contain the samevertex.

Size=

(

n − 1

r − 1

)

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 44/61

Intersecting hypergraphsWe take subsets, of sizer, from [n] at random.

• If r ≪ n1/3, then all subsets contain the samevertex.

Size=

(

n − 1

r − 1

)

• If n1/3 ≪ r ≪ n5/12, it’s determined by a randomvariablet.

Size∼(

r2

n

)t (n − 1

r − 1

)

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 44/61

It’s just killing you, isn’t it?Let us return to the Kevin Bacon question.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 45/61

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 .

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 45/61

Infinite Kevin Bacon number

is someone with infinite#

.

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

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 46/61

Infinite Kevin Bacon number

is someone with infinite#

.

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

Not soon coming to DVD:

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 46/61

Infinite Kevin Bacon number

is someone with infinite#

.

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

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

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 46/61

Kevin Bacon number 7Joseph Wheeler appeared in two films:

• General Wheeler and Secretary of WarAlger at Camp Wikoff (1898), a shortdocumentary, in which he appeared with RussellAlexander Alger (Kevin Bacon number 6)

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 47/61

Kevin Bacon number 7Joseph Wheeler appeared in two films:

• General Wheeler and Secretary of WarAlger at Camp Wikoff (1898), a shortdocumentary, in which he appeared with RussellAlexander Alger (Kevin Bacon number 6)

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

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 47/61

Kevin Bacon number 8William Rufus Shafter also appeared in two films:

• Surrender of General Toral (1898) withJoseph Wheeler.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 48/61

Kevin Bacon number 8William Rufus Shafter also appeared in two films:

• Surrender of General Toral (1898) withJoseph Wheeler.

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

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 48/61

Conclusion

• Russell Alexander Alger has#

=6,

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 49/61

Conclusion

• Russell Alexander Alger has#

=6,

• Joseph Wheeler has#

=7, and

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 49/61

Conclusion

• Russell Alexander Alger has#

=6,

• Joseph Wheeler has#

=7, and

• William Rufus Shafter has#

=8.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 49/61

The chain

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

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 50/61

The chain

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

7 Joseph Wheeler was inGeneral Wheeler andSecretary of War Alger at Camp Wikoff(1898) with Russell Alexander Alger

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 50/61

The chain

6 Russell Alexander Alger was inPresidentMcKinley’s Inspection of Camp Wikoff(1898) with President William McKinley

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 50/61

The chain

6 Russell Alexander Alger was inPresidentMcKinley’s Inspection of Camp Wikoff(1898) with President William McKinley

5 President William McKinley was inPresidentMcKinley Taking the Oath (1901) withU. S. Senator Marcus Hanna

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 50/61

The chain

6 Russell Alexander Alger was inPresidentMcKinley’s Inspection of Camp Wikoff(1898) with President William McKinley

5 President William McKinley was inPresidentMcKinley Taking the Oath (1901) withU. S. Senator Marcus Hanna (R-OH)

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 50/61

The chain

4 U. S. Senator Marcus Hanna (R-OH) was inOpening of the Pan-American ExpositionShowing Vice President RooseveltLeading the Procession (1901) withPresident Theodore Roosevelt

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 50/61

The chain

4 U. S. Senator Marcus Hanna (R-OH) was inOpening of the Pan-American ExpositionShowing Vice President RooseveltLeading the Procession (1901) withPresident Theodore Roosevelt

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

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 50/61

The chain

2 Walter McGrail was inDick Tracy vs. CrimeInc. (1941) with Wally Rose

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 50/61

The chain

2 Walter McGrail was inDick Tracy vs. CrimeInc. (1941) with Wally Rose

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

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 50/61

The chain

2 Walter McGrail was inDick Tracy vs. CrimeInc. (1941) with Wally Rose

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

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 50/61

Dead presidentsWilliam McKinley has a unique distinction.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 51/61

Dead presidentsWilliam McKinley has a unique distinction.

He was one of four presidents to be assassinated:

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 51/61

Dead presidentsWilliam McKinley has a unique distinction.

He was one of four presidents to be assassinated:

McKinleySep. 14,

1901

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 51/61

Dead presidentsWilliam McKinley has a unique distinction.

He was one of four presidents to be assassinated:

McKinleySep. 14,

1901

KennedyNov. 22,

1963

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 51/61

Dead presidentsWilliam McKinley has a unique distinction.

He was one of four presidents to be assassinated:

LincolnApr. 15,

1865

McKinleySep. 14,

1901

KennedyNov. 22,

1963

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 51/61

Dead presidentsWilliam McKinley has a unique distinction.

He was one of four presidents to be assassinated:

LincolnApr. 15,

1865

GarfieldSep. 19,

1881

McKinleySep. 14,

1901

KennedyNov. 22,

1963

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 51/61

Garfield (not the cat)

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 52/61

Garfield (not the cat)

• Born in a log cabin in1831.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 52/61

Garfield (not the cat)

• Born in a log cabin in1831 near Cleveland.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 52/61

Garfield (not the cat)

• Born in a log cabin in1831 near Cleveland.

• 18 years in the House.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 52/61

Garfield (not the cat)

• Born in a log cabin in1831 near Cleveland.

• 18 years in the House.

• Elected in 1880.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 52/61

Garfield (not the cat)

• Born in a log cabin in1831 near Cleveland.

• 18 years in the House.

• Elected in 1880.

• Shot on July 2.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 52/61

Garfield (not the cat)

• Born in a log cabin in1831 near Cleveland.

• 18 years in the House.

• Elected in 1880.

• Shot on July 2, died onSeptember 19.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 52/61

Garfield (not the cat)

• Born in a log cabin in1831 near Cleveland.

• 18 years in the House.

• Elected in 1880.

• Shot on July 2, died onSeptember 19.

• Amateurmathematician.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 52/61

Published mathematicianAs a Congressman, Garfield got a publication credit:

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 53/61

Published mathematicianAs a Congressman, Garfield got a publication credit:

J.A. Garfield,The New England Journal of Education,3, Boston, 1876, p. 161.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 53/61

Published mathematicianAs 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:

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 53/61

Published mathematicianAs 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 cca

a b

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 53/61

Garfield’s proof

2 cca

a b

b 13

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 54/61

Garfield’s proof

2 cca

a b

b 13

area of trapezoid= area of triangle 1+ area of triangle 2+ area of triangle 3

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 54/61

Garfield’s proof

2 cca

a b

b 13

area of trapezoid= area of triangle 1+ area of triangle 2+ area of triangle 3

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 54/61

Garfield’s proof

2 cca

a b

b 13

1

2(a + b)(a + b) = area of triangle 1

+ area of triangle 2+ area of triangle 3

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 54/61

Garfield’s proof

a

a b

b

c c 32

1

2(a + b)(a + b) = area of triangle 1

+ area of triangle 2+ area of triangle 3

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 54/61

Garfield’s proof

a

a b

b

c c 32

1

2(a + b)(a + b) =

1

2c2

+ area of triangle 2+ area of triangle 3

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 54/61

Garfield’s proof

cca

a b

b

3

1

2(a + b)(a + b) =

1

2c2

+ area of triangle 2+ area of triangle 3

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 54/61

Garfield’s proof

cca

a b

b

3

1

2(a + b)(a + b) =

1

2c2

+1

2ab

+ area of triangle 3

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 54/61

Garfield’s proof

3b

b cca

a

1

2(a + b)(a + b) =

1

2c2

+1

2ab

+ area of triangle 3

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 54/61

Garfield’s proof

3b

b cca

a

1

2(a + b)(a + b) =

1

2c2

+1

2ab

+1

2ab

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 54/61

Garfield’s proof

cca

a b

b

1

2(a + b)(a + b) =

1

2c2

+1

2ab

+1

2ab

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 54/61

Garfield’s proof

cca

a b

b

1

2(a + b)(a + b) =

1

2c2 +

1

2ab +

1

2ab

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 54/61

Garfield’s proof

cca

a b

b

(a + b)(a + b) = c2 + ab + ab

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 54/61

Garfield’s proof

cca

a b

b

a2 + 2ab + b2 = c2 + ab + ab

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 54/61

Garfield’s proof

cca

a b

b

a2 + b2 = c2

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 54/61

Bacon and Erdos

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 55/61

Bacon and Erdos

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 55/61

Bacon and Erdos

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 55/61

Bacon and Erdos

How are THESE guysrelated?

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 55/61

Celebrity nerdsThere are some mathematicians and physicists whohave Bacon numbers and low Erdos numbers:

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 56/61

Celebrity nerdsThere are some mathematicians and physicists whohave Bacon numbers and low Erdos numbers:

•Brian Greene ( =3, =2) was in

Frequency(2000).

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 56/61

Celebrity nerdsThere are some mathematicians and physicists whohave Bacon numbers and low Erdos numbers:

•Brian Greene ( =3, =2) was in

Frequency(2000).

•Dave Bayer ( =3, =2) was inA

Beautiful Mind(2001).

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 56/61

Nerd celebrities

Danica McKellar

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 57/61

Nerd celebrities

Danica McKellar, math nerd.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 57/61

Nerd celebrities

Danica McKellar, math nerd. Best known for: TheWonder Years (1988-1993) and

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 57/61

Nerd celebrities

Danica McKellar, math nerd. Best known for: TheWonder Years (1988-1993) and The West Wing(2002-present).

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 57/61

Nerd celebrities

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 57/61

Nerd celebrities

•Danica McKellar wasin Intermission (2004)with Susan Leslie.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 57/61

Nerd celebrities

•Danica McKellar wasin Intermission (2004)with Susan Leslie.

•Susan Leslie was inBeauty Shop (2005)with

.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 57/61

Danica’s Math Career

4 Danica McKellar wrote

Percolation and Gibbs State Multiplicityfor Ferromagnetic Ashkin-Teller Modelsin Two Dimensions,

which appeared in

Journal of Physics A: Mathematics and General,

with Winn and Chayes.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 58/61

Danica’s Math Career

3 Lincoln Chayes wrote

No directed fractal percolation in zeroarea,

which appeared in

The Journal of Statistical Physics,

with Peres and Pemantle.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 58/61

Danica’s Math Career

2 Robin Pemantle wrote

Metrics on compositions andcoincidences among renewal sequences,

which appeared in

The IMA Volumes in Mathematics and itsApplications,

with Diaconis, Holmes, Lalley and Janson.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 58/61

Danica’s Math Career

1 Svante Janson wrote

A note on triangle-free graphs,

which appeared in

The IMA Volumes in Mathematics and itsApplications,

with Łuczak, Spencer and Paul Erdos

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 58/61

Danica’s Math Career

1 Svante Janson wroteA note on triangle-freegraphs, which appeared inThe IMA Volumes inMathematics and its Applications, with Łuczak,Spencer and Paul Erdos

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 58/61

Erdos, Hollywood Leading Man

4 Paul Erdos was inN is a Number (1993) withGene Patterson

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 59/61

Erdos, Hollywood Leading Man

4 Paul Erdos was inN is a Number (1993) withGene Patterson

3 Gene Patterson was inBox of Moon Light(1996) with Lisa Blount

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 59/61

Erdos, Hollywood Leading Man

4 Paul Erdos was inN is a Number (1993) withGene Patterson

3 Gene Patterson was inBox of Moon Light(1996) with Lisa Blount

2 Lisa Blount was inFemme Fatale (1991) withColin Firth

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 59/61

Erdos, Hollywood Leading Man

3 Gene Patterson was inBox of Moon Light(1996) with Lisa Blount

2 Lisa Blount was inFemme Fatale (1991) withColin Firth

1 Colin Firth was inWhere the Truth Lies(2005) with Kevin Bacon

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 59/61

Erdos, Hollywood Leading Man

2 Lisa Blount was inFemme Fatale (1991) withColin Firth

1 Colin Firth was inWhere the Truth Lies(2005) with Kevin Bacon

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 59/61

Tying it all togetherAnd, just when you thought this whole talk was just adisjointed mess that didn’t fit at all together . . .

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 60/61

Tying it all togetherAnd, just when you thought this whole talk was just adisjointed mess that didn’t fit at all together . . .

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 60/61

Tying it all togetherAnd, just when you thought this whole talk was just adisjointed mess that didn’t fit at all together . . .

It does!

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 60/61

ThanksThank you for letting me talk today.

Ryan MartinIowa State University

rymartin@iastate.edu

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.

Six degrees of graph theory:Kevin Bacon, Paul Erdos, William McKinley and me – p. 61/61