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MIDDLE COLLEGE 2013 Why are you less popular than your friends?

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Today’s plan: to answer the question as to why you are less popular than average Generate 3 networks; 2 random and 1 preferential attachment Calculate the measures of degree distribution, clustering coefficient and path length With 10 people how many connections can there be in total?

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Random Graph n = 5 p = ½ 12345 1X 2 X 3 X 4 X 5 X

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Preferential Attachment Graph The rich get richer Red Die White123456 1 123456 2 789101112 3 131415161718 4 192021222324 5 252627282930 Start with dyad, each end labeled 1,2 Add node with 2 edges, one edge at a time, labeling ends sequentially Kite graph with 10 ends labeled Add 6 new nodes labeling the new ends as you add them Complete the Adjacency Matrix below and draw the network

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2 types of networks Random Formed when links occur with probability p Hump degree distribution centred at np Preferential attachment Formed when ‘rich get richer’ Power law degree distribution You have two networks

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Clustering Coefficient The probability that two randomly selected neighbors of a node are connected to each other. The proportion of the number of triangular subgraphs among neighbors to the possible number of triangular subgraphs.

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The Formula

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Example

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Degree is popularity Pick a random node from your preferential attachment graph (1-10) Find the average degree of its friends Compare to its degree Is anyone more popular than average? Why?

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Triangle numbers Where did the 45 possible edges come from? What is the sum of the first n numbers?

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MIDDLE COLLEGE 2013 Pascal’s Triangle

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Blaise Pascal French Mathematician 1623-1662 (died at the age of 39) Invented the Mechanical Calculator (Pascaline) while still a teenager.

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Pascal’s Triangle Each entry is equal to the sum of the two values directly above it. A formula can be obtained from the pattern in order to find an appropriate set of values for any given row.

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Patterns Diagonals Powers Odds and Evens Powers of 11 Prime Numbers Hockey Stick Fibonacci’s Sequence

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The Triangle Entries

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Creating the triangle using the combinatorial

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Binomial Coefficients

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Binomial Expansion (a+b) 1 (a+b) 2 (a+b) 3

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Binomial Coin Flipping Each person flip a coin 10 times, listing the heads and tails HHTHTTHTHH How many different lists are there? How many H do I expect? How many lists have 0 H, 1H, 2H’s? How many of the 2 10 lists have 5 H’s?

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Binomial Coin Flipping Say you did 4 coin flips. How many H do I expect? How many lists have 0 H, 1H, 2H, 3H, 4H’s? See connection with a random graph? If you flip 5 coins how many have 2H’s? Use your lists from 4 flips.

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Combinations Say you have 3 books, Harry Potter, Lord of the Rings and Differential Equations, an Introduction How many ways can I choose 2 books? (1 book?) How many ways can I choose 2 of 4 things? How many ways can I choose 8 of 10 things?

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Combinations How many ways can I order 3 things in a row? How many ways can I choose 3 of 10 things? How many ways can I choose r of n things? n! factorial

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Proof by Induction We want to prove that Pascal’s triangle gives you the number of ways you can choose r from n items Steps: Show it’s true for the small numbers Assume it’s true for a row in the triangle Show it must be true for the next row. Proof that every number is interesting?

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Plinko How many paths are there to each tube? Notice the Gaussian curve forming How much would you pay for the right to get $10 if the ball ended up in a tube greater than 8? What is the probability it ends up in tube greater than 8?

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Stocks and Options What’s a stock? Stocks can go up or down See graph of real stock What’s a call option? Strike price Graph the value of a call at expiry How much should a call cost?

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Pricing a call Selling something short. Eliminating risk Consider a portfolio with 1 call option and Δ units of shorted stock V = C – ΔS Today the stock is worth $100, tomorrow $104 or $92 with 50/50 chance

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Binomial method of pricing Build the formula to price a call given u = 1+ a, d = 1- a The stock is at $100 now, the call expires in 3 days with an exercise price of $100. a = 0.05 Sketch the payout at the time of expiry. Price the call and then sketch the profit diagram

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