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 length

With 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 GraphThe rich get richer

Red Die

White 1 2 3 4 5 6

1 1 2 3 4 5 6

2 7 8 9 10 11 12

3 13 14 15 16 17 18

4 19 20 21 22 23 24

5 25 26 27 28 29 30

•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 np

Preferential attachment Formed when ‘rich get richer’ Power law degree distribution

You have two networks

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.

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 friendsCompare 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 Mathematician1623-1662 (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

DiagonalsPowersOdds and EvensPowers 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?

( 𝑥+𝑦 )𝑛=∑𝑘=0

𝑛

(𝑛𝑘)𝑥𝑛 𝑦𝑛−1

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 210 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 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?

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 – ΔS

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- 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