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

2. 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?

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

5. 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

6. 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

7. 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.

8. The Formula

9. Example

10. 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?

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

12. MIDDLE COLLEGE 2013 Pascal’s Triangle

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

14. 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.

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

16. The Triangle Entries

17. Creating the triangle using the combinatorial

18. Binomial Coefficients

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

20. 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?

21. 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.

22. 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?

23. 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

24. 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?

25. 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?

26. 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?

27. 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

28. 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