Why are you less popular than your friends?
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MIDDLE COLLEGE2013
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 Graphn = 5 p = ½
1 2 3 4 5
1 X
2 X
3 X
4 X
5 X
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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
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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
the number of edges between the neighbors of node
the degree (number of neighbors) of node
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Example
4
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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?
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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 COLLEGE2013
Pascal’s Triangle
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Blaise Pascal
French Mathematician1623-1662 (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
DiagonalsPowersOdds and EvensPowers of 11Prime NumbersHockey StickFibonacci’s Sequence
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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:
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Creating the triangle using the combinatorial
And so on…
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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
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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 210 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 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?
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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 stockWhat’s a call option?
Strike priceGraph the value of a call at expiryHow 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