Social Networks 101
PROF. JASON HARTLINE AND PROF. NICOLE IMMORLICA
Guessing game
Experiment:
There are three balls in this urn, either
or
Guessing game
Experiment:
This is
called a
blue urn.
This is
called a
yellow urn.
Guessing game
Experiment: When I call your name,
1. You (and only you) will see a random ball.
2. You must then guess if the urn is a blue urn
or a yellow (red) urn, and tell the class your
guess.
If you guess correctly, you will earn one point.
What should the 1st student do?
?
?
?
Guess that urn is same color as ball.
What should the 2nd student do?
1st guess
was blue.
?
?
?
What should the 2nd student do?
1st guess
was blue.
?
?
?
Guess that urn is same color as ball.
What should the 3rd student do?
1st and 2nd
guesses
were blue.
?
?
?
Guess that urn is blue
no matter what she sees!
What should the nth student do?
First (n-1)
guesses
were blue.
?
?
?
First 2
students told
the truth.
If the first two guesses are blue,
everyone should guess blue.
Staring up at the sky.
Choosing a restaurant in a strange town.
Self-reinforcing success of best-selling books.
Voting for popular candidates.
Information cascades
When:
1. People make decisions sequentially,
2. and observe actions of earlier people.
Information cascade: People abandon own info.
in favor of inferences based on others’ actions.
Rational
Imitation
Cascade
(not simply peer pressure)
Observations
1. Cascades are easy to start.
(every student makes same guess
so long as first two students make same guess)
Observations
2. Cascades can lead to bad outcomes.
(given blue urn, chance of seeing red ball is 1/3,
so first two students guess red with prob. (1/3)2 = 1/9)
Observations
3. Cascades are fragile.
(if a few students show the color of the ball they picked
to the entire class, the cascade can be reversed)
Conditional probability
How likely is it that the urn is yellow
given what you’ve seen and heard?
Strategy for urns
A player should guess yellow if,
Pr [ yellow urn | what she saw and heard ] > ½
and blue otherwise.
Analysis
From setup of experiment,
Pr [
] = Pr [
] = 1/2
Analysis
From composition of urns,
Pr [
|
] = Pr [
|
] = 2/3
Analysis
Suppose first student draws
Pr [
| ]=
:
Pr [ |
] x Pr [
Pr [ ]
]
Analysis
Suppose first student draws
Pr [
| ]=
:
Pr (2/3)
[ | x] x Pr
(1/2)
[ ]
Pr [ ]
Pr [ ] = Pr [(2/3)
| x] x (1/2)
Pr [ ]
+ Pr [(1/3)
| x] x (1/2)
Pr [ =] (1/2)
Analysis
Suppose first student draws
Pr [
:
| ] = (2/3) > (1/2)
then he should guess yellow.
Analysis
Suppose second student draws
Pr [
too:
| , ]
2nd student’s
draw
1st student’s
guess
Analysis
Suppose second student draws
Pr [
too:
| , ] > (1/2)
by a similar analysis, so she also guesses yellow.
Analysis
Suppose third student draws
Pr [
:
| , , ]
3rd student’s
draw
2nd student’s
guess
1st student’s
guess
Analysis
Suppose third student draws
Pr [
:
| , , ]=
(1/3) x (2/3) x (2/3)
Pr [ , , |
(1/2)
] x Pr [
Pr [ , , ]
]
Analysis
Pr [ , , ] =
(1/3) x (2/3) x (2/3)
Pr [ , , |
(2/3) x (1/3) x (1/3)
+ Pr [ , , |
(1/2)
] x Pr [
]
(1/2)
] x Pr [
]
Analysis
Suppose third student draws
Pr [
:
| , , ]=
(1/3) x (2/3) x (2/3) x (1/2)
(1/3) x (2/3) x (2/3) x (1/2) + (2/3) x (1/3) x (1/3) x (1/2)
= (2/3) > (1/2)
Analysis
The best strategy for the third
student is to guess yellow no
matter what he draws.
Other sequences
# yellow guesses
- # blue guesses
2
Yellow cascade starts.
1
0
-1
-2
1
2
3
4
Blue cascade starts.
5
6
7
student
Q. What is the largest city in the US?
A. New York, population 8,175,133
Q. What is the 2nd largest city in the US?
A. Los Angeles, population 3,729,621
Q. What is the 3rd largest city in the US?
A. Chicago, population 3,695,598
City populations
1. New York
2. Los Angeles
3. Chicago
4. Houston
5. Philadelphia
6. Phoenix
7. San Antonio
8. San Diego
9. Dallas
10. San Jose
8,175,133
3,792,621
2,695,598
2,099,451
1,526,006
1,445,632
1,327,407
1,307,402
1,197,816
945,942
City populations
1. New York
2. Los Angeles
3. Chicago
8,175,133
3,792,621
2,695,598
230. Berkeley, CA
112,580
240. Murfreesboro, TN
108,755
250. Ventura, CA
106,433
A few cities with
high population
Many cities with
low population
City populations
City populations
Power Law: The number of cities
with population at least k is
proportional to k-c for a constant c.
“fraction of items”
Power Law: The number of cities with
population > k is proportional to k-c.
“popularity = k”
Power Law: Fraction f(k) of items with
popularity k is proportional to k-c.
f(k)
k-c
log [f(k)]
log [k-c]
log [f(k)]
-c log [k]
A power law is a straight line
on a log-log plot.
City populations
Other examples
Why does data exhibit power laws?
Previously, …
Imitation
Cascade
Today
Imitation
Power law
Constructing the web
1. Pages are created in order, named 1, 2, …, N
2. When created, page j links to a page by
a) With probability p, picking a page i uniformly at
random from 1, …, j-1
b) With probability (1-p), pick page i uniformly at
random and link to the page that i links too
Imitation
The rich get richer
2 b) With prob. (1-p), pick page i uniformly at
random and link to the page that i links too
3/4
1/4
The rich get richer
2 b) With prob. (1-p), pick page i uniformly at
random and link to the page that i links too
Equivalently,
2 b) With prob. (1-p), pick a page
proportional to its in-degree and link to it
Rich get richer
Power law
Why is Harry Potter popular?
If we could re-play history, would we still read
Harry Potter, or would it be some other book?
Information cascades and the rich
Information cascade = so some people get a
little bit richer by chance
and then rich-get-richer dynamics = the random
rich people get a lot richer very fast
Music download site – 8 worlds
1. “Let’s go driving,”
Barzin
18
2. “Silence is sexy,”
Einstürzende
3
Neubauten
3. “Go it alone,”
Noonday
47
Underground
1. “Let’s go driving,”
Barzin
2. “Silence is sexy,”
Einstürzende
Neubauten
3. “Go it alone,”
Noonday
Underground
10.“Picadilly Lilly,”
Tiger Lillies
10.“Picadilly Lilly,”
Tiger Lillies
2
59
7
10
1