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
World War I How to get from A to E 24 June 2014 16
World War I How to get from A to E 24 June 2014 17
World War I How to get from A to E 24 June 2014 18
World War I How to get from A to E 24 June 2014 19
World War I How to get from A to E 24 June 2014 20
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
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