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Mathematical Ideas that Shaped the World Graphs and Networks

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Plan for this class 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?

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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 It was a centre of learning for centuries, being home to Goldbach, Hilbert, Kant and Wagner.

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The Seven Bridges of Königsberg 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.

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The Seven Bridges of Königsberg 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?

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Leonhard Euler (1707 – 1783) 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…

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Leonhard Euler (1707 – 1783) …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.

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Leonhard Euler (1707 – 1783) 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”

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Back to Königsberg… 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.

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Reformulating the problem With this observation, we can re-draw the bridges of Königsberg as follows: A B C D

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Conditions for a solution 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.)

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An impossible problem! 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 C D

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Königsberg extra Look at your handout to learn about these characters. Can you make them all happy?

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Postscript on Königsberg Königsberg was heavily bombed during World War II. The city was taken over by Russia and re- named Kaliningrad. Two of the 7 bridges were destroyed: Question: Is the bridge problem possible now?

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The beginning of graph theory 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

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Examples of graphs Train maps

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Examples of graphs Social networks

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Examples of graphs Chemical models

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The Four Colour Theorem

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The Four-Colour Problem 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?

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Example of a 4-colouring

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Why not 3 colours? A simple example shows that it impossible to always colour a map with only 3 colours.

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Why not 5 colours? 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?

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Martin Gardner’s map

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In terms of graphs 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?

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Maps to graphs: example

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A proof? In 1976, two men called Kenneth Appel and Wolfgang Haken announced that they had a proof of the conjecture.

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A controversial result 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.

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An inelegant result One critic said “A good mathematical proof is like a poem. This is a telephone directory!” However, the proof is now widely accepted and computers are used in many areas of pure mathematics.

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Building efficient graphs

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Building the shortest graphs Very often we have a set of points and want to find the shortest collection of edges that connect them up. For example, Roads/railways connecting towns Telephone/internet cables Gas pipes Connections in electronic circuits Neurons connecting bits of your brain

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The shortest graph? Suppose we have 4 towns that we wish to connect up. Which of these do you think is shortest?

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An unexpected solution 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…

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Soap bubbles know best So the best solution is to create two ghost towns! Steiner graph

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How do they do it? 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.

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Slime mould is better than politicians 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!

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Slime mould networks Tokyo train network Slime mould

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Finding the shortest route

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The Chinese postman problem 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?

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The Chinese postman problem Here’s how to solve the problem: 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.

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Chinese postman: example A BC D E F

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The travelling salesman problem 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.

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The Icosian Game

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(Or the travel version!)

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This is the poster for a contest run by Proctor & Gamble in There were 33 cities in this problem.

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A tantalising problem 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!

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Methods of solving the TSP 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!

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Methods of solving the TSP 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.

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Methods of solving the TSP Heuristics: find ‘good’ solutions which are highly likely to be close to the perfect solution. For example, 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.

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Methods of solving the TSP 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.

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Have a go! Humans are surprisingly good at finding solutions to TSP quite quickly. Play the following game online to see how good you are!

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Applications The Travelling Salesman Problem has lots of applications in our lives: 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

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How does Google work?

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The internet to Google 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.

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How does PageRank work? 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.

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Example

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Using PageRank to make money! 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.

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Social networking 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.

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Making recommendations 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!)

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Lessons to take home 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!

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References Here are some good places to read more about the subjects in today’s lecture. Four Colour Theorem: “Four Colours Suffice” by Robin Wilson, Penguin Books, 2003 A comprehensive website about the Travelling Salesman problem: A New York Times article about the NetFlix prize: 3Netflix-t.html 3Netflix-t.html

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