Why are you less popular than
your friends?
MIDDLE COLLEGE
2013
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 Graph
n=5 p=½
1
1
2
3
4
5
2
3
4
5
X
X
X
X
X
Preferential Attachment Graph
The rich get richer
•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
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
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
𝐶𝑖 =
2 𝑒𝑗,𝑘
𝑘𝑖 𝑘𝑖 − 1
𝑒𝑗,𝑘 =the number of edges between the neighbors of node 𝑖
𝑘𝑖 =the degree (number of neighbors) of node 𝑖
Example
𝑒𝑗,𝑘 = 1
𝑘1 =4
𝐶1 =
2(1)
2
1
=
=
4(4 − 1) 12 6
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?
Triangle numbers
 Where did the 45 possible edges come from?
 What is the sum of the first n numbers?
Pascal’s Triangle
MIDDLE COLLEGE
2013
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
 Powers
 Odds and Evens
 Powers of 11
 Prime Numbers
 Hockey Stick
 Fibonacci’s Sequence
The Triangle Entries
 The entries are found by the combinatorial:
𝑛!
𝑛
=
𝑘
𝑘! 𝑛 − 𝑘 !
 A factorial is the product of a natural number with all
of its successive natural number values.
𝑛! = 𝑛 𝑛 − 1 𝑛 − 2 … 3 ∗ 2 ∗ 1
 Example:
5! = 5 ∗ 4 ∗ 3 ∗ 2 ∗ 1 = 120
Creating the triangle using the combinatorial
1
0
2
0
3
0
0
0
1
1
2
1
3
1
2
2
3
2
3
3
And so on…
Binomial Coefficients
 The triangle allows us to find the coefficients needed in
any binomial expansion:
𝑛
𝑥+𝑦
𝑛
=
𝑘=0
𝑛 𝑛 𝑛−1
𝑥 𝑦
𝑘
 Think about it:
𝑥 + 1 3 = 𝑥 3 + 3𝑥 2 + 3𝑥 + 1
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 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 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?
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 stock
 What’s a call option?
 Strike price
 Graph the value of a call at expiry
 How 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