Party hard! - The maths of connections

Party hard!
The maths of connections
Colva Roney-Dougal
University of St Andrews
March 23rd, 2013
Colva Roney-Dougal
Party hard!
Connection 1: Friendship
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The party problem
Question
How many people need to come to a party, to guarantee that at
least three of them all know each other, or at least three of them
are mutual strangers?
Let’s do an experiment!
With this group of 6 people, we succeeded in finding a trio. What
can we say, in general?
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Introducing graphs
We’ll represent the people as dots, called vertices.
If two people know each other, draw a red line (a red edge)
between them. If two people don’t know each other, draw a blue
edge between them.
Set of vertices and edges is a graph.
We want to find the smallest number of vertices, such that
however we colour the edges, we’ll always find either a red or a
blue triangle.
If only 5 people are at our
party, we can fail to find three
friends or three strangers.
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Six people suffice!
The Pigeonhole Principle
Suppose we have p pigeons and h holes, and let s be p/h (the
“squash factor”).
If we put all of the pigeons in holes, then at least one hole contains
at least s pigeons.
Pigeons ↔ other people
Consider two holes:
“Knows Colva” and
“Doesn’t know Colva”.
Colva
At least one hole
contains at least three
pigeons.
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I want more people to be friends!
For 3 mutual friends or strangers we need 6
people. We write this R(3) = 6.
What about four mutual friends or strangers?
It’s not too hard to show that R(4) = 18.
The numbers R(n) are the Ramsey numbers:
Ramsey proved that R(n) is finite for all n.
Frank Ramsey
What about R(5)?
The best we can say is 43 ≤ R(5) ≤ 49.
We know that 102 ≤ R(6) ≤ 165.
Paul Erdős
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An infinite party?
Infinite Ramsey Theorem
Given infinitely many vertices, we can always find infinitely many
connected all with red edges or all with blue edges.
A prime number is a number that is divisible only by itself and 1.
Every positive number factorises uniquely into primes: 12 = 2 · 2 · 3.
A Number-Theoretic Consequence
There exists an infinite set S of positive whole numbers, such that
for all pairs m, n of numbers in S, the sum m + n has an even
number of prime factors, including multiplicity.
Infinitely many red edges: all fine. Infinitely many blues: double!
Challenge: Find such a set!
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Connection 2: Marriage
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A theorem on marriage
Hall’s Marriage Theorem (Phillip Hall, 1935)
Consider n women, W1 , . . . , Wn . Suppose each woman Wi has a
list Mi of the men she would happily marry.
(Each man will happily marry any woman who wants him.)
Every woman can be happily married if and only if
for each set W of women, the union of their lists Mi of men
contains at least |W| men.
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An application: Sudoku
The Independent: There’s no mathematics involved.
Use logic and reasoning to solve the puzzle.
2
2
2
1
1
2
3
8
2
8
Consider the bottom left
sub-square.
9
9
3
5
3
4
1
2
8
1
3
4
9
6
7
5
5
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The women are the unfilled
cells: (7, 1), (7, 2), (7, 3),
(8, 1) and (9, 3).
The men are the unused
numbers: 4, 6, 7, 8, 9.
Party hard!
An application: Sudoku
The Independent: There’s no mathematics involved.
Use logic and reasoning to solve the puzzle.
2
2
2
1
1
2
3
8
1
3
5
7
8
9
9
3
5
3
4
1
2
8
8
2
4
9
(7, 1)
4
(7, 2)
6
(7, 3)
7
(8, 1)
8
(9, 3)
9
6
7
5
A critical set W of women, is a set that are willing to marry
exactly |W| men. We find critical sets of women, and use them.
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Longer distance connections
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The Kevin Bacon Game
Given an actor, find the shortest path from them to Kevin Bacon,
using only films in the Internet Movie Database (IMDB).
The minimum number of films is the actor’s Bacon number.
was in
“The big picture”
with
Kevin Bacon
John Cleese
Let’s make a graph!
Vertices: every actor in IMDB.
Edges: between actors who’ve appeared in a film together.
Bacon number is the number of edges in a path from actor to
Bacon. (Infinity if no path exists.)
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Bacon and more
“Candy”
Brando
1968
“Skum Rocks”
Ringo
2013
Bacon
The Erdős graph has as vertices everyone who’s published an
academic paper. Two people have an edge between them if they
have published together.
Nina
Colva
Max
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Akos Seress
Erdős
Erdős–Bacon numbers?
Erdős number + Bacon number = Erdős–Bacon number.
Most people’s Erdős–Bacon number is infinity.
Someone with an Erdős–Bacon number of 6 is:
Natalie Portman
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Connection 3: Disease
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The spread of disease
Imagine you’re a farmer, planting an apple orchard.
Blight can arrive in your orchard, via birds and insects.
Once there, it can spread on the wind from a tree to its neighbours.
More trees means more apples, but the closer together the trees
are planted, the more likely it is that blight will spread.
Question
How closely should you plant the trees?
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Some experiments
Let’s run some experiments on a big square grid.
p = 1/4
p = 1/2
p = 3/4
With p = 1/4, one infection spreads to at most 4 other trees.
With p = 1/2, the biggest cluster has size 36, and more than half
the trees are in three big clusters.
With p = 3/4, one infection covers almost the whole orchard.
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Percolation theory
The study of problems like this is percolation theory.
Instead of edges, we can percolate through the vertices: make
them “open” with probability p and “closed” otherwise.
Applications of percolation theory include:
The spread of wild fires.
The spread of human diseases: flu and SARS.
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Infinity and probability
For many applications we consider infinite sets of vertices.
The system percolates if after putting in the edges with fixed
probability p, infinitely many vertices are all joined together.
Infinite orchard: You might think that the
probability of an infinite set of infected
trees changes smoothly from 0 to 1, as we
plant the trees closer together.
In fact, the probability of an infinite
set of infected trees jumps like this.
The point where it jumps is the
critical probability, pc .
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The critical probability
In 1960, Ted Harris proved that pc ≤ 1/2 for our infinite grid.
In 1980, Harry Kesten proved that pc is exactly 1/2.
We’d love to know more about what happens when p is exactly pC .
Wendelin Werner and Stanislav Smirnov both won Fields medals
for work relating to percolation theory.
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Why do the galaxies look like they do?
One answer (Schulman & Seiden):
Each bit of a galaxy contains stars,
and regions of gas, that could
collapse to form stars.
Something needs to happen to
trigger that collapse.
A supernova!
This sends a
shockwave through space,
triggering star formation.
Many years later, some of these
new stars will go nova in their turn:
galactic percolation.
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