How to get from A to E: using networks to unravel the past, - PowerPoint PPT Presentation

How to get from A to E: using networks to unravel the past, present, and future Rebecca Cotton-Barratt Admissions Coordinator and Schools Liaison Officer Mathematical Institute University of Oxford 24 June 2014 1 How to get from A to E 24 June 2014

What is a graph? 2 How to get from A to E 24 June 2014

What is a graph? • Set of vertices v and edges e • Can be complete, directed, weighted, simple and/or connected How to get from A to E 24 June 2014 3

Bridges of Königsberg How to get from A to E 24 June 2014 4

Handshaking Lemma Every finite undirected graph has an even number of vertices of odd degree How to get from A to E 24 June 2014 5

Eulerian Circuits and Paths • Circuit: every vertex must have an even degree • Path: at most two vertices can have an odd degree How to get from A to E 24 June 2014 6

Utilities Problem Is it possible to connect three houses to three utility supplies without the supply pipes crossing? How to get from A to E 24 June 2014 7

Euler’s Formula F + V – E = 2 2E = F 1 + 2F 2 + 3F 3 + 4F 4 + … How to get from A to E 24 June 2014 8

Bipartite Graphs F + V – E = 2 2E = 4F 4 + 6F 6 + 8F 8 + … How to get from A to E 24 June 2014 9

Utilities Problem F = 2 + E – V F = 2 + 9 – 6 F = 5 How to get from A to E 24 June 2014 10

Utilities Problem F = 2 + E – V F = 2 + 9 – 6 F = 5 2E = 4F 4 + 6F 6 + 8F 8 + … 18 = 4F 4 + ... > 20 How to get from A to E 24 June 2014 11

Theorem on Friends and Strangers Can you find a group of 5 people of whom no 3 are mutual acquaintances or mutual strangers? How to get from A to E 24 June 2014 12

Theorem on Friends and Strangers How to get from A to E 24 June 2014 13

World War I “The enemy of my enemy is my friend.” Unbalanced triangles. How to get from A to E 24 June 2014 14

World War I How to get from A to E 24 June 2014 15

Rock, Paper, Scissors • Chinese Han dynasty game • Exported to Japan as Frog, Snake, Snail. How to get from A to E 24 June 2014 21

Rock, Paper, Scissors • Chinese Han dynasty game • Exported to Japan as Frog, Snake, Snail. How to get from A to E 24 June 2014 22

Variations: Changing Arrows • 3 move game • Good old rock, nothing beats rock. ~ Bart Simpson How to get from A to E 24 June 2014 23

Variations: Adding a Move How to get from A to E 24 June 2014 24

Variations: Adding More Moves General Properties • There is one winner between every pair of moves • Ties are not allowed How to get from A to E 24 June 2014 25

General Games Possible Question • Is there always a move that beats everything else? How to get from A to E 24 June 2014 26

General Games Possible Question • Is there always a move that beats everything else? • No! How to get from A to E 24 June 2014 27

Threatening Behaviour Definition • A move a threatens another move b if a beats another move c which beats b . How to get from A to E 24 June 2014 28

Threatening Behaviour New Question • Is there always a move which beats or threatens every other move? How to get from A to E 24 June 2014 29

Proof: Is There Always a Move Which Beats or Threatens Every Other Move? 1. Take the move which beats the most other moves. (Note: We don’t need to know what beats what exactly.) How to get from A to E 24 June 2014 30

Proof: Is There Always a Move Which Beats or Threatens Every Other Move? 2. Does T. rex threaten every red? (Hint: What does it mean for T. rex to threaten every red?) How to get from A to E 24 June 2014 31

Proof: Is There Always a Move Which Beats or Threatens Every Other Move? 3. What happens if some red beats every blue? How to get from A to E 24 June 2014 32

Proof: Is There Always a Move Which Beats or Threatens Every Other Move? 4. Then that move beats more moves than T. rex How to get from A to E 24 June 2014 33

Proof: Is There Always a Move Which Beats or Threatens Every Other Move? 1. Take the move which beats the most other moves. (Note: We don’t need to know what beats what exactly.) How to get from A to E 24 June 2014 34

King Chicken Theorem • The King Chicken theorem states that in a flock of chickens there is always a chicken which pecks or threatens every other chicken. How to get from A to E 24 June 2014 35

Further Questions • If a chicken is pecked by another chicken, is one of the chickens that pecks that chicken a king? • Can there be precisely two king chickens in a flock? How to get from A to E 24 June 2014 36

Big Brother Surveillance How to get from A to E 24 June 2014 37

Networks in the News How to get from A to E 24 June 2014 38

Travelling Salesman Problem How to get from A to E 24 June 2014 39

Travelling Salesman Problem How to get from A to E 24 June 2014 40

Google Page Rank How to get from A to E 24 June 2014 41

University Mathematics Applied Mathematics Statistics Pure Mathematics Differential Equations Probability Logic Numerical Analysis Optimization Set Theory Classical Mechanics Mathematical Genetics Algebra (e.g. groups) Fluid Dynamics Analysis (e.g. calculus) Mathematical Physics Geometry Information Theory Topology Cryptography Number Theory Mathematical Biology How to get from A to E 24 June 2014 42

Joint Degrees • Mathematics and • Mathematics and Statistics (3/4 years) Philosophy (3/4 years) • Same first year as Maths • Logic/set theory from natural bridge • Then greater range of • Philosophy options (e.g. Stats options epistemology or • Mathematics and language) relate well with CompSci (3/4 years) maths • Pure Maths options and CompSci from a flexible mathematical viewpoint How to get from A to E 24 June 2014 43

Thanks for Listening! Any Questions? Visit us at: www.maths.ox.ac.uk How to get from A to E 24 June 2014 44