Middle College 2013 - PowerPoint Presentation

Middle College2013 Why are you less popular than your friends?

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 lengthWith 10 people how many connections can there be in total?

Random Graphn = 5 p = ½   1 2 3 4 5 1 X         2   X       3     X     4       X   5         X

Preferential Attachment Graph The rich get richer Red Die White 1 2 3 4 5611234562789101112313141516171841920212223245252627282930 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

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

Clustering Coefficient T he 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.

The Formula the number of edges between the neighbors of node the degree (number of neighbors) of node  

Example 4  

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

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

Middle College2013 Pascal’s Triangle

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

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.

Patterns Diagonals PowersOdds and Evens Powers of 11Prime NumbersHockey StickFibonacci’s Sequence

The Triangle Entries The entries are found by the combinatorial: A factorial is the product of a natural number with all of its successive natural number values. Example:  

Creating the triangle using the combinatorial And so on…  

Binomial Coefficients The triangle allows us to find the coefficients needed in any binomial expansion: Think about it: It is easy to multiply the perfect cubed binomial above. But what if we have a much larger power? Do we really want to multiply a binomial out 10 times? 15 times? 100 times?    

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

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?

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.

Combinations Say you have 3 books, Harry Potter , Lord of the Rings and Differential Equations, an IntroductionHow 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?

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

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 numbersAssume it’s true for a row in the triangleShow it must be true for the next row. Proof that every number is interesting?

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?

Stocks and Options What’s a stock? Stocks can go up or down See graph of real stockWhat’s a call option? Strike priceGraph the value of a call at expiryHow much should a call cost?

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

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