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How to amaze your friends the link between mazes, murder and organised crime Chris Budd Chris Budd

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Most maths is taught as a subject which is Precise Careful Safe Logical Today we will look at combining maths and Murder Suicide Love Conquest Crime How an ancient idea has 21st Century applications!

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The Minotaur Who does this remind you of?

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Cretan Labyrinth

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Seed for the Cretan Labyrinth: an example of a space filling curve. Alumni Investigation: which seeds lead to which types of labyrinth?

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Ancient stone labyrinth Alumni Project: Build one of these in your garden

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Mappa Mundi Maiden Castle Alumni Field Trip: Visit one of these

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The rise of the maze Maze at Hatfield House

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Hampton Court Maze Alternative Alumni Field Trip: Visit one of these and lose your family and friends

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How to solve a maze and amaze your friends: I Method one.. The hand on the hedge method X Always keep turning left (or right)

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X X Add more hedges

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If a bird can walk on a hedge from the entrance to the centre then you can solve the maze using the hand on the hedge method Alumni Challenge: Prove this using an inductive argument

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To solve a more general maze we need to learn about Networks Bridges of Konigsberg A B D C Euler and friend

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Network Even Node Odd Node Edge

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Euler Circuit Can we find a route through the network which passes through each edge once and only once? Theorem: 1.Always if the network only has even nodes 2. Yes, if the network has two odd nodes, starting and finishing at an odd node 3. Otherwise never Proof: Tricky, but leads to a way to crack a maze

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A B C A B C D D E E F F G G H I H I J J K K L L M M N N Centre Maze Network

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A B C D E F G H I J K L M N Centre More generally.. Double up each of the paths Apply the Euler circuit theory

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Using a network and some peanuts you can solve any maze Start at the entrance and take any path If you come to a new node then leave a peanut and take any new path If you come to an old node (or the end of a blind alley) and you are on a new path then turn back along this path If you come to an old node and you are on an old path then take a new path (or another old path if no new path exists) Never go down a path more than twice Alumni Challenge 2: Try this and Good Luck!

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Networks have many other applications Social: Friendship, Sexual partners, FACEBOOK Organisational: Management, crime Technological: World-wide-web, Internet, power grid Information: DNA, Protein-Protein, Citations, word-of- mouth, gossip, retail Ecological: Food chains, disease & infection Moreno 1934 Systematic study of networks is a key application of maths to the 21st Century!!!

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Friends (social network) Bob MaryDave Alice Richard Basil Chloe Six degrees of separation??

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School social network in USA

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The Internet/world-wide-web Google exploits features of this to work, especially using the page-rank algorithm and eigenvalues of the adjacency matrices

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Food web Scientific collaborations Sexual contacts

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Different types of network developed to model various situations : this is what mathematicians do with their time …. Random: Erdos- Renyi Small-world: Watts-Strogatz (social) Barabasi-Albert: Internet, World-wide-web, Facebook

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Mathematical features of a network : Nodal degree: Number of edges k attached to a node Proportion: P(k) probability a node has degree k Scaling laws and scale free networks: Random network but more often see Network resilience: How does the network connectivity eg. The average path length/node degree change as nodes/edges are removed? Understanding this is vital to the study of disease, power generation and to ….

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Organised crime! How can we break up a crime network?

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Method 1: Delete members at random at each time step

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Method 2. Attack the member towards whom the previous victim had the highest degree of attachment. Assume that all victims are arrested, not murdered, and they inform the police of the person they most respect/trust/know.

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Finally …Facebook Friends.. An amazing result!

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Conclusions: Networks are a fascinating area of maths With a rich history which we can explore And great relevance to many aspects of the modern world including crime Which have the potential for almost limitless investigations, field trips and construction projects Truly they are a way to amaze your friends!

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Networks lead naturally to interesting mathematical investigations of great practical relevance 1.Shortest distance l between two nodes: lowest number of nodes between them 6 degrees max of separation is claimed ?? Is this really true ?? 2.Degree of a node = number of edges that meet the node ?? What proportion p of nodes have degree k ??

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p k

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Professor Yashar Ganjali Department of Computer Science University of Toronto

Professor Yashar Ganjali Department of Computer Science University of Toronto

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