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

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

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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

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How to get from A to E 24 June 2014

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

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What is a graph?

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What is a graph?

  • Set of vertices v and

edges e

  • Can be complete,

directed, weighted, simple and/or connected

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Bridges of Königsberg

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Handshaking Lemma

Every finite undirected graph has an even number

  • f vertices of odd degree

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Eulerian Circuits and Paths

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  • Circuit: every vertex must

have an even degree

  • Path: at most two

vertices can have an odd degree

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Utilities Problem

Is it possible to connect three houses to three utility supplies without the supply pipes crossing?

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Euler’s Formula

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F + V – E = 2 2E = F1 + 2F2 + 3F3 + 4F4 + …

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Bipartite Graphs

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F + V – E = 2 2E = 4F4 + 6F6 + 8F8 + …

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Utilities Problem

F = 2 + E – V F = 2 + 9 – 6 F = 5

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Utilities Problem

F = 2 + E – V F = 2 + 9 – 6 F = 5 2E = 4F4 + 6F6 + 8F8 + … 18 = 4F4 + ... > 20

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Theorem on Friends and Strangers

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Can you find a group of 5 people of whom no 3 are mutual acquaintances or mutual strangers?

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Theorem on Friends and Strangers

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World War I

“The enemy of my enemy is my friend.”

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Unbalanced triangles.

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World War I

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World War I

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World War I

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World War I

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World War I

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World War I

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Rock, Paper, Scissors

  • Chinese Han dynasty

game

  • Exported to Japan as

Frog, Snake, Snail.

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Rock, Paper, Scissors

  • Chinese Han dynasty

game

  • Exported to Japan as

Frog, Snake, Snail.

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Variations: Changing Arrows

  • 3 move game
  • Good old rock, nothing

beats rock. ~ Bart Simpson

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Variations: Adding a Move

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Variations: Adding More Moves

General Properties

  • There is one winner

between every pair of moves

  • Ties are not allowed

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General Games

Possible Question

  • Is there always a move

that beats everything else?

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General Games

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Possible Question

  • Is there always a move

that beats everything else?

  • No!
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Threatening Behaviour

Definition

  • A move a threatens

another move b if a beats another move c which beats b.

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Threatening Behaviour

New Question

  • Is there always a move

which beats or threatens every other move?

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Proof: Is There Always a Move Which Beats

  • r 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.)

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Proof: Is There Always a Move Which Beats

  • r Threatens Every Other Move?
  • 2. Does T. rex threaten

every red?

(Hint: What does it mean for T. rex to threaten every red?)

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Proof: Is There Always a Move Which Beats

  • r Threatens Every Other Move?
  • 3. What happens if some

red beats every blue?

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Proof: Is There Always a Move Which Beats

  • r Threatens Every Other Move?
  • 4. Then that move beats more moves than T. rex

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Proof: Is There Always a Move Which Beats

  • r 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.)

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King Chicken Theorem

  • The King Chicken

theorem states that in a flock of chickens there is always a chicken which pecks or threatens every

  • ther chicken.

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Further Questions

  • If a chicken is pecked by

another chicken, is one

  • f the chickens that

pecks that chicken a king?

  • Can there be precisely

two king chickens in a flock?

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Big Brother Surveillance

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Networks in the News

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Travelling Salesman Problem

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Travelling Salesman Problem

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Google Page Rank

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University Mathematics

Statistics

Probability Optimization Mathematical Genetics

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Pure Mathematics

Logic Set Theory Algebra (e.g. groups) Analysis (e.g. calculus) Geometry Topology Number Theory

Applied Mathematics

Differential Equations Numerical Analysis Classical Mechanics Fluid Dynamics Mathematical Physics Information Theory Cryptography Mathematical Biology

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Joint Degrees

  • Mathematics and

Philosophy (3/4 years)

  • Logic/set theory from

natural bridge

  • Philosophy options (e.g.

epistemology or language) relate well with maths

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  • Mathematics and

Statistics (3/4 years)

  • Same first year as Maths
  • Then greater range of

Stats options

  • Mathematics and

CompSci (3/4 years)

  • Pure Maths options and

CompSci from a flexible mathematical viewpoint

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Thanks for Listening!

Any Questions? Visit us at: www.maths.ox.ac.uk

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