Mathematical Ideas that
Shaped the World
Graphs and Networks
Plan for this class
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What was the famous Königsberg bridge
problem?
What is a graph?
Why was the 4-colour theorem controversial?
How are soap bubbles and slime mould good
at town planning?
Why is it so hard for salesmen to be efficient?
How does Google work?
The Seven Bridges of
Königsberg
The Seven Bridges of Königsberg
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Once upon a time there
was a city called
Königsberg in Prussia.
It was founded in 1255 by
the Teutonic Knights, and
was the capital of East
Prussia until 1945.
It was a centre of learning
for centuries, being home
to Goldbach, Hilbert, Kant
and Wagner.
The Seven Bridges of Königsberg
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Running through the city was the River
Pregel.
It separated the city into two mainland areas
and two large islands.
There were 7 bridges
connecting the various
areas of land.
The Seven Bridges of Königsberg
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The residents of Königsberg wondered
whether they could wander around the city,
crossing each of the seven bridges once and
only once.
Can you find a way?
Leonhard Euler (1707 – 1783)
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Born in Basel, Switzerland,
and was expected to become
a pastor like his father.
Studied Hebrew and
theology at university, but
got private maths lessons
from Johann Bernoulli.
In 1727 got a job in the
medical section at the Uni of
St Petersburg…
Leonhard Euler (1707 – 1783)
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…but in the chaos
surrounding the death of
Empress Catherine I, he
managed to sneak into the
maths department.
Got married in 1733 and had
13 children, of whom 5
survived to adulthood.
In 1741 moved to Berlin,
where he spent 25 years.
Leonhard Euler (1707 – 1783)
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Published over 500 books and
papers in his lifetime, with
another 400 posthumously.
Invented the notation i, π, e,
sin, cos, f(x) and more!
Lost sight in both eyes but
became more productive,
saying
“now I have fewer distractions”
Back to Königsberg…
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In 1736 Euler turned his mind to the problem
of the bridges of Königsberg.
He realised that it didn’t matter how you
walked around the land, or where exactly the
bridges were.
It only mattered how many bridges there
were between each bit of land, and in what
order you crossed them.
Reformulating the problem
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With this observation, we can re-draw the
bridges of Königsberg as follows:
A
B
D
C
Conditions for a solution
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Euler’s Eureka! moment was realising that
whenever you cross into a bit of land, you
also have to cross back out of it.
Therefore, for a bridge tour to be possible,
there must be an even number of bridges
coming out of every bit of land.
(Except for the starting and finishing points.)
An impossible problem!
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If we look again at the map of Königsberg,
we see that there are an odd number of
bridges coming out of every bit of land, so
such a walk around the city is impossible.
A
B
D
C
Königsberg extra
Look at your
handout to
learn about
these
characters.
Can you make
them all
happy?
Postscript on Königsberg
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Königsberg was heavily bombed during
World War II.
The city was taken over by Russia and renamed Kaliningrad.
Two of the 7 bridges were destroyed:
Question:
Is the bridge problem
possible now?
The beginning of graph theory
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By solving the problem the way he did, Euler
invented the subject of graph theory.
A graph is a collection of nodes and edges.
It doesn’t matter how
long the edges are or
where the nodes are;
it only matters which
edges are connected
to which nodes.
node
edge
Examples of graphs
Train maps
Examples of graphs
Social networks
Examples of graphs
Chemical models
The Four Colour Theorem
The Four-Colour Problem
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Proposed by Francis Guthrie in 1852 and
remained unsolved for more than a century.
Can any map be coloured with 4 colours so
that no two adjacent regions have the
same colour?
Example of a 4-colouring
Why not 3 colours?
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A simple example shows that it impossible to
always colour a map with only 3 colours.
Why not 5 colours?
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It was proved by 1890 that every map can be
coloured with at most 5 colours.
The difficult part of the problem was to show
that there was no map sufficiently
complicated as to need 5 colours.
Martin Gardner set the following graph as a
problem to his readers. Can you colour it
using only 4 colours?
Martin Gardner’s map
In terms of graphs
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The 4-colour problem can be phrased in
terms of graphs.
Each region of the map becomes a node,
with two nodes being connected by an edge
if and only if the regions are adjacent on the
map.
The problem becomes: can you colour the
nodes with 4 colours so that an edge never
connects two nodes of the same colour?
Maps to graphs: example
A proof?
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In 1976, two men called Kenneth Appel and
Wolfgang Haken announced that they had a
proof of the conjecture.
A controversial result
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They had made a computer program to
check the 4-colouring of all possible
examples (1,936 of them!).
It was the first mathematical theorem to be
proved with computer help, and aroused
much controversy.
An inelegant result
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One critic said
“A good mathematical proof is like a
poem. This is a telephone directory!”
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However, the proof is now widely
accepted and computers are used in
many areas of pure mathematics.
Building efficient graphs
Building the shortest graphs
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Very often we have a set of points and want
to find the shortest collection of edges that
connect them up. For example,
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Roads/railways connecting towns
Telephone/internet cables
Gas pipes
Connections in electronic circuits
Neurons connecting bits of your brain
The shortest graph?
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Suppose we have 4 towns that we wish to
connect up. Which of these do you think is
shortest?
An unexpected solution
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If we’re restricted to roads between towns,
then the first graph is the shortest.
But there is a better solution, which we can
find using a bit of perspex and some soap
bubbles…
Soap bubbles know best
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So the best solution is to create two ghost
towns!
Steiner graph
How do they do it?
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We currently have no (fast) algorithm for
finding the shortest Steiner graph between a
given number of points.
Nature, on the other hand, is quite good at it.
http://www.youtube.com/watch?v=0lpsLCgCp2Q
Slime mould is better than politicians
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Scientists studied slime mould growing in a
region shaped like Tokyo, placing food
sources where the major regional cities would
be.
The resulting slime mould network was
remarkably similar to the Tokyo train
network.
In some respects it was actually better!
Slime mould networks
Tokyo train network
Slime mould
Finding the shortest route
The Chinese postman problem
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Now suppose that the towns and roads are
fixed, and we know the distances between
them.
The Chinese postman problem asks: what is
the shortest route that travels over every road
at least once and returns to the start?
5
3
8
4
5
9
8
6
9
The Chinese postman problem
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Here’s how to solve the problem:
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If the graph is Eulerian (i.e. an even number of
edges out of every node) then each edge can be
walked exactly once, so we are done.
If not, find the shortest distances between the
nodes with odd numbers of edges, and add extra
edges to turn it into an Eulerian graph.
Chinese postman: example
B
5
C
5
3
A
9
8
D
8
4
9
F
6
E
The travelling salesman problem
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If, instead, you are a travelling salesman, you
wish to find the route that allows you to visit
each town exactly once (and then return to
the start).
This problem was posed as long
ago as 1800 by the Irish
mathematician Hamilton, and
rose drastically in popularity
in the 1950s and 60s.
The Icosian Game
(Or the travel version!)
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This is the
poster for a
contest run by
Proctor &
Gamble in 1962.
There were 33
cities in this
problem.
A tantalising problem
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Unlike the Chinese postman problem,
nobody has ever found a fast algorithm for
solving the Travelling Salesman Problem
(TSP).
Deciding whether there is a route shorter
than a given length is an “NP-complete”
problem.
Finding a good algorithm is currently worth
$1 million!
Methods of solving the TSP
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Brute force – try all possible routes and pick
the fastest one.
Caveat: using today’s fastest supercomputer,
solving the 33-city problem using this
method would take about 100 trillion years!
Methods of solving the TSP
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Branch and bound algorithms – divide the
problem into smaller graphs and try to
eliminate edges that can’t be part of the
solution.
The record set with this kind of exact method is
85,900 cities, which took over 126 CPU years
to compute in 2006.
Methods of solving the TSP
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Heuristics: find ‘good’ solutions which are
highly likely to be close to the perfect
solution. For example,
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The nearest neighbour algorithm lets the
salesman pick the nearest unvisited city every
time.
Find any route, then rearrange edges to find a
shorter one.
Methods of solving the TSP
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Heuristic algorithms can find solutions to
TSP with millions of cities in a fairly short
amount of time.
Caveat: these solutions may not always be
the best possible.
Have a go!
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Humans are surprisingly good at finding
solutions to TSP quite quickly.
Play the following game online to see how
good you are!
http://www.tsp.gatech.edu/games/tspOnePlayer.html
Applications
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The Travelling Salesman Problem has lots of
applications in our lives:
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Logistics of delivering goods
Drilling holes in circuit boards
Genome sequencing
Programming space telescopes like Hubble
Collecting post from postboxes every day
Making travel itineraries
X-ray crystallography
How does Google work?
The internet to Google
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Google sees the internet is a giant graph.
Each webpage is a node, and two pages are
joined by an edge if there is a link from one
page to the other.
Note: the edges in the internet graph have a
direction.
The algorithm that Google uses to rank its
searches is called PageRank.
How does PageRank work?
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Idea: the more links a page has pointing to
it, the more important it is.
Second idea: if an important page links to
your page, this is worth more than if an
unimportant page links to you.
For example, Wikipedia referencing you is
worth more than Haggis The Sheep
referencing you.
Example
Using PageRank to make money!
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People who understand PageRank can make
it very lucrative for them.
For example, businesses or individuals with a
high page rank can sell links to those wishing
to boost their page rank.
Businesses have also used similar algorithms
to rank universities in the job market.
Social networking
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Graphs are also important to social
networking sites like Facebook.
By analysing the preferences of your ‘friends’
and pages that you ‘like’, Facebook can target
its advertising very effectively.
Making recommendations
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Similarly, shops like Amazon use graphs to
make suggestions for future shopping.
In 2009 the US company Netflix awarded $1
million to the people who best improved
their recommendation algorithm.
(One of the problems was that they could
never predict whether someone would like
the film Napoleon Dynamite!)
Lessons to take home
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That Graph Theory is an incredibly important
part of modern-day life.
That a solution to a single graph theory
problem can have many different real-world
applications.
That slime mould is often cleverer than
humans.
That problems in graph theory can be worth
a lot of money!
References
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Here are some good places to read more
about the subjects in today’s lecture.
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Four Colour Theorem: http://nrich.maths.org/6291
“Four Colours Suffice” by Robin Wilson, Penguin
Books, 2003
A comprehensive website about the Travelling
Salesman problem: http://www.tsp.gatech.edu/
A New York Times article about the NetFlix prize:
http://www.nytimes.com/2008/11/23/magazine/2
3Netflix-t.html