Contagion Complex Networks, SFI Summer School, June, 2010 Prof. Peter Dodds

Contagion
Contagion
Complex Networks, SFI Summer School, June, 2010
Introduction
Simple Disease
Spreading Models
Background
Prediction
Prof. Peter Dodds
Social Contagion
Models
Granovetter’s model
Department of Mathematics & Statistics
Center for Complex Systems
Vermont Advanced Computing Center
University of Vermont
Network version
Groups
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
References
Frame 1/80
Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License
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Outline
Introduction
Simple Disease Spreading Models
Background
Prediction
Contagion
Introduction
Simple Disease
Spreading Models
Background
Prediction
Social Contagion
Models
Granovetter’s model
Social Contagion Models
Granovetter’s model
Network version
Groups
Summary
Network version
Groups
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
References
Winning: it’s not for everyone
Superstars
Musiclab
References
Frame 2/80
Contagion
Contagion
Definition:
Simple Disease
Spreading Models
Introduction
I
I
(1) The spreading of a quality or quantity between
individuals in a population.
(2) A disease itself:
the plague, a blight, the dreaded lurgi, ...
Background
Prediction
Social Contagion
Models
Granovetter’s model
Network version
Groups
Summary
Winning: it’s not for
everyone
Two main classes of contagion:
1. Infectious diseases:
tuberculosis, HIV, ebola, SARS, influenza, ...
Superstars
Musiclab
References
2. Social contagion:
fashion, word usage, rumors, riots, religion, ...
Frame 3/80
Contagion models
Contagion
Introduction
Some large questions concerning network
contagion:
1. For a given spreading mechanism on a given
network, what’s the probability that there will be
global spreading?
2. If spreading does take off, how far will it go?
3. How do the details of the network affect the
outcome?
Simple Disease
Spreading Models
Background
Prediction
Social Contagion
Models
Granovetter’s model
Network version
Groups
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
References
4. How do the details of the spreading mechanism
affect the outcome?
5. What if the seed is one or many nodes?
Frame 4/80
Mathematical Epidemiology
The standard SIR model:
I
Three states:
I
I
I
S = Susceptible
I = Infected
R = Recovered
Contagion
Introduction
I
S(t) + I(t) + R(t) = 1
I
Presumes random
interactions
Simple Disease
Spreading Models
Background
Prediction
Social Contagion
Models
Granovetter’s model
Network version
Discrete time example:
Groups
Summary
1 − βI
Winning: it’s not for
everyone
S
Superstars
Transition Probabilities:
βI
Musiclab
References
ρ
I
r
R
1−ρ
1−r
β for being infected given
contact with infected
r for recovery
ρ for loss of immunity
Frame 6/80
Independent Interaction models
Reproduction Number R0 :
I
Introduction
R0 = expected number of infected individuals
resulting from a single initial infective.
I
Epidemic threshold: If R0 > 1, ‘epidemic’ occurs.
I
Example:
Simple Disease
Spreading Models
Background
Prediction
Social Contagion
Models
Granovetter’s model
Network version
Groups
1
Fraction infected
Contagion
Summary
Winning: it’s not for
everyone
0.8
Superstars
Musiclab
0.6
I
Continuous phase
transition.
I
Fine idea from a
simple model.
0.4
0.2
0
0
1
2
R0
3
References
4
Frame 7/80
Disease spreading models
Contagion
Introduction
Simple Disease
Spreading Models
Background
Prediction
For ‘novel’ diseases:
Social Contagion
Models
Granovetter’s model
Network version
1. Can we predict the size of an epidemic?
2. How important/useful is the reproduction number R0 ?
3. What is the population size N?
Groups
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
References
Frame 9/80
R0 and variation in epidemic sizes
Contagion
Introduction
Simple Disease
Spreading Models
Background
Prediction
R0 approximately the same for all of the following:
Social Contagion
Models
Granovetter’s model
I
1918-19 “Spanish Flu” ∼ 500,000 deaths in US
I
1957-58 “Asian Flu” ∼ 70,000 deaths in US
I
1968-69 “Hong Kong Flu” ∼ 34,000 deaths in US
Network version
Groups
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
I
2003 “SARS Epidemic” ∼ 800 deaths world-wide
References
Frame 10/80
Size distributions
Contagion
Introduction
Simple Disease
Spreading Models
Background
Elsewhere, event size distributions are important:
I
earthquakes (Gutenberg-Richter law)
I
city sizes, forest fires, war fatalities
I
wealth distributions
I
‘popularity’ (books, music, websites, ideas)
I
What about Epidemics?
Prediction
Social Contagion
Models
Granovetter’s model
Network version
Groups
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
References
Power laws distributions are common but not obligatory...
Frame 11/80
Feeling icky in Iceland
Contagion
Introduction
Simple Disease
Spreading Models
Caseload recorded monthly for range of diseases in
Iceland, 1888-1990
Background
Prediction
Social Contagion
Models
Granovetter’s model
Frequency
0.03
0.02
Network version
Iceland: measles
normalized count
0.01
Groups
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
0
1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990
Date
References
Treat outbreaks separated in time as ‘novel’ diseases.
Frame 12/80
Contagion
Measles
Introduction
75
Insert plots:
Complementary cumulative
frequency distributions:
2
N (ψ)
5
N>(ψ)
10
A
1
10
0
4
10
−5
10
10
−4
−3
10
3
10
−2
−1
10
ψ
N> (Ψ) ∝ Ψ−γ+1
2
1
Simple Disease
Spreading Models
Background
Prediction
Social Contagion
Models
Granovetter’s model
Network version
Groups
Summary
0
0
0.025
0.05
0.075
ψ
0.1
Ψ = fractional epidemic size
Winning: it’s not for
everyone
Superstars
Measured values of γ:
I
measles: 1.40 (low Ψ) and 1.13 (high Ψ)
I
Expect 2 ≤ γ < 3 (finite mean, infinite variance)
I
Distribution is rather flat...
Musiclab
References
Frame 13/80
Contagion
Resurgence—example of SARS
Introduction
Simple Disease
Spreading Models
160
Background
# New cases
D
Prediction
120
Social Contagion
Models
80
Granovetter’s model
Network version
40
Groups
Summary
0
Nov 16, ’02
Dec 16, ’02
Jan 15, ’03
Feb 14, ’03
Mar 16, ’03
Apr 15, ’03
May 15, ’03
Date of onset
Jun 14, ’03
Winning: it’s not for
everyone
Superstars
I
Epidemic discovers new ‘pools’ of susceptibles:
Resurgence.
I
Importance of rare, stochastic events.
Musiclab
References
Frame 14/80
A challenge
Contagion
Introduction
Simple Disease
Spreading Models
Background
Prediction
So... can a simple model produce
Social Contagion
Models
Granovetter’s model
Network version
Groups
1. broad epidemic distributions
and
2. resurgence ?
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
References
Frame 15/80
Contagion
Size distributions
Introduction
2000
A
R0=3
1500
N(ψ)
Simple Disease
Spreading Models
Background
Simple models
typically produce
bimodal or unimodal
size distributions.
1000
500
0
0
0.25
0.5
0.75
1
ψ
I
I
This includes network models:
random, small-world, scale-free, ...
Some exceptions:
Prediction
Social Contagion
Models
Granovetter’s model
Network version
Groups
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
References
1. Forest fire models
2. Sophisticated metapopulation models
Frame 16/80
Contagion
A toy agent-based model
Geography: allow people to move between contexts:
b=2
Introduction
Simple Disease
Spreading Models
Background
Prediction
Social Contagion
Models
l=3
x ij =2
Granovetter’s model
Network version
Groups
Summary
Winning: it’s not for
everyone
Superstars
i
n=8
I
P = probability of travel
I
Movement distance: Pr(d) ∝ exp(−d/ξ)
I
ξ = typical travel distance
j
Musiclab
References
Frame 17/80
Contagion
Example model output: size distributions
Introduction
683
1942
R0=3
R =12
400
N(ψ)
N(ψ)
400
300
200
Background
0
Prediction
300
Social Contagion
Models
200
Granovetter’s model
Network version
Groups
100
100
0
0
Simple Disease
Spreading Models
0.25
0.5
ψ
0.75
1
0
0
Summary
0.25
0.5
0.75
1
ψ
Winning: it’s not for
everyone
Superstars
Musiclab
References
I
Flat distributions are possible for certain ξ and P.
I
Different R0 ’s may produce similar distributions
I
Same epidemic sizes may arise from different R0 ’s
Frame 18/80
Contagion
# New cases
Standard model:
6000
D
R0=3
4000
Introduction
Simple Disease
Spreading Models
2000
Background
0
0
Prediction
500
1000
1500
t
Social Contagion
Models
Granovetter’s model
Network version
# New cases
Standard model with transport: Resurgence
400
G
Groups
Summary
Winning: it’s not for
everyone
R0=3
Superstars
200
Musiclab
References
0
0
500
1000
1500
t
I
Disease spread highly sensitive to population
structure
I
Rare events may matter enormously
Frame 19/80
Simple disease spreading models
Contagion
Introduction
Simple Disease
Spreading Models
Background
Prediction
Attempts to use beyond disease:
Social Contagion
Models
Granovetter’s model
I
Adoption of ideas/beliefs (Goffman & Newell, 1964)
I
Spread of rumors (Daley & Kendall, 1965)
I
I
Diffusion of innovations (Bass, 1969)
Spread of fanatical behavior (Castillo-Chávez &
Song, 2003)
Network version
Groups
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
References
Frame 20/80
Social Contagion
Contagion
Introduction
Simple Disease
Spreading Models
Background
Prediction
Social Contagion
Models
Granovetter’s model
Network version
Groups
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
References
Frame 21/80
Contagion
Social Contagion
Introduction
Simple Disease
Spreading Models
Examples abound:
Background
Prediction
being polite/rude
I
Harry Potter
strikes
I
voting
I
innovation
I
gossip
I
residential segregation
I
Rubik’s cube
I
ipods
I
religious beliefs
I
obesity
I
leaving lectures
I
I
Social Contagion
Models
Granovetter’s model
Network version
Groups
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
References
SIR and SIRS contagion possible
I
Classes of behavior versus specific behavior: dieting
Frame 22/80
Social Contagion
Contagion
Introduction
Simple Disease
Spreading Models
Background
Prediction
Social Contagion
Models
Two focuses for us:
I
Widespread media influence
I
Word-of-mouth influence
Granovetter’s model
Network version
Groups
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
References
Frame 23/80
The hypodermic model of influence:
Contagion
Introduction
Simple Disease
Spreading Models
Background
Prediction
Social Contagion
Models
Granovetter’s model
Network version
Groups
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
References
Frame 24/80
The two step model of influence:
Contagion
Introduction
Simple Disease
Spreading Models
Background
Prediction
Social Contagion
Models
Granovetter’s model
Network version
Groups
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
References
Frame 25/80
The general model of influence:
Contagion
Introduction
Simple Disease
Spreading Models
Background
Prediction
Social Contagion
Models
Granovetter’s model
Network version
Groups
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
References
Frame 26/80
Social Contagion
Contagion
Introduction
Why do things spread?
Simple Disease
Spreading Models
Background
Prediction
I
Because of system level properties?
I
Or properties of special individuals?
I
Is the match that lights the forest fire the key?
(Katz and Lazarsfeld; Gladwell)
I
Yes. But only because we are narrative-making
machines...
I
System/group properties harder to understand
I
Always good to examine what is said before and
after the fact...
Social Contagion
Models
Granovetter’s model
Network version
Groups
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
References
Frame 27/80
The Mona Lisa:
Contagion
Introduction
Simple Disease
Spreading Models
Background
Prediction
Social Contagion
Models
Granovetter’s model
Network version
Groups
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
References
I
“Becoming Mona Lisa: The Making of a Global
Icon”—David Sassoon
I
Not the world’s greatest painting from the start...
I
Escalation through theft, vandalism, parody, ...
Frame 28/80
The completely unpredicted fall of Eastern
Europe:
Contagion
Introduction
Simple Disease
Spreading Models
Background
Prediction
Social Contagion
Models
Granovetter’s model
Network version
Groups
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
References
Timur Kuran: “Now Out of Never: The Element of
Surprise in the East European Revolution of 1989”
Frame 29/80
Social Contagion
Contagion
Introduction
Simple Disease
Spreading Models
Background
Some important models:
I
Tipping models—Schelling (1971)
I
I
Simulation on checker boards
Idea of thresholds
I
Threshold models—Granovetter (1978)
I
Herding models—Bikhchandani, Hirschleifer, Welch
(1992)
Prediction
Social Contagion
Models
Granovetter’s model
Network version
Groups
Summary
Winning: it’s not for
everyone
Superstars
I
Musiclab
References
Social learning theory, Informational cascades,...
Frame 30/80
Social contagion models
Contagion
Introduction
Simple Disease
Spreading Models
Background
Prediction
Thresholds:
I
I
Basic idea: individuals adopt a behavior when a
certain fraction of others have adopted
‘Others’ may be everyone in a population, an
individual’s close friends, any reference group.
Social Contagion
Models
Granovetter’s model
Network version
Groups
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
I
Response can be probabilistic or deterministic.
I
Individual thresholds vary.
References
Frame 31/80
Social Contagion
Contagion
Introduction
Simple Disease
Spreading Models
Background
Prediction
Some possible origins of thresholds:
Social Contagion
Models
Granovetter’s model
I
Desire to coordinate, to conform.
I
Lack of information: impute the worth of a good or
behavior based on degree of adoption (social proof)
Economics: Network effects or network externalities
I
I
Telephones, Facebook, operating systems, ...
Network version
Groups
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
References
Frame 32/80
Contagion
Imitation
Introduction
Simple Disease
Spreading Models
Background
Prediction
“When people are free
to do as they please,
they usually imitate
each other.”
—Eric Hoffer
“The Passionate State
of Mind” [11]
Social Contagion
Models
Granovetter’s model
Network version
Groups
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
References
despair.com
Frame 33/80
Contagion
Granovetter’s threshold model:
Introduction
Action based on perceived behavior of others:
1
2.5
1
0.6
1.5
0.4
0.2
C
φt+1 = F (φt)
B
2
f (φ∗)
Pr(a
i,t+1
=1)
A
0.8
Simple Disease
Spreading Models
1
0.5
Background
0.8
Prediction
0.6
Social Contagion
Models
0.4
Granovetter’s model
0.2
Network version
Groups
0
0
φ∗i
φi,t
0
0
1
0.5
∗
1
0
0
φ
0.5
φt
1
Summary
Winning: it’s not for
everyone
Superstars
I
Two states: S and I.
I
φ = fraction of contacts ‘on’ (e.g., rioting)
I
Z
φt+1 =
0
I
φt
Musiclab
References
f (γ)dγ = F (γ)|φ0 t = F (φt )
This is a Critical Mass model
Frame 35/80
Contagion
Social Sciences: Threshold models
Introduction
3
Simple Disease
Spreading Models
1
Background
Prediction
2.5
0.8
Social Contagion
Models
2
Granovetter’s model
Network version
φt+1
f(γ)
0.6
1.5
Groups
Summary
0.4
1
Winning: it’s not for
everyone
0.2
0.5
Superstars
Musiclab
0
0
0.2
0.4
0.6
γ
I
0.8
1
0
0
0.2
0.4
φt
0.6
0.8
1
References
Example of single stable state model
Frame 36/80
Social Sciences—Threshold models
Contagion
Introduction
Simple Disease
Spreading Models
Background
Prediction
Social Contagion
Models
Implications for collective action theory:
Granovetter’s model
Network version
Groups
1. Collective uniformity 6⇒ individual uniformity
2. Small individual changes ⇒ large global changes
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
References
Frame 37/80
Threshold model on a network
Contagion
Introduction
Simple Disease
Spreading Models
Background
Prediction
Social Contagion
t=1
e
a
Granovetter’s model
Network version
a
d
b
t=1
Models
c
Groups
Summary
b
Winning: it’s not
c for
everyone
Superstars
Musiclab
References
I
All nodes have threshold φ = 0.2.
I
“A simple model of global cascades on random
networks”
D. J. Watts. Proc. Natl. Acad. Sci., 2002
Frame 39/80
d
Snowballing
Contagion
Introduction
Simple Disease
Spreading Models
Background
Prediction
The Cascade Condition:
Social Contagion
Models
Granovetter’s model
I
I
If one individual is initially activated, what is the
probability that an activation will spread over a
network?
What features of a network determine whether a
cascade will occur or not?
Network version
Groups
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
References
Frame 40/80
The most gullible
Contagion
Introduction
Simple Disease
Spreading Models
Background
Prediction
Vulnerables:
I
I
= Individuals who can be activated by just one
‘infected’ contact
For global cascades on random networks, must have
a global cluster of vulnerables
Social Contagion
Models
Granovetter’s model
Network version
Groups
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
I
Cluster of vulnerables = critical mass
I
Network story: 1 node → critical mass → everyone.
References
Frame 41/80
Contagion
Cascades on random networks
Introduction
Simple Disease
Spreading Models
1
Background
Final
cascade size
〈S〉
0.8
I
0.6
Fraction of
Vulnerables
0.4
0.2
No
Cascades
0
1
Cascades
Possible
2
3
Low influence
4
z
No
Cascades
5
6
7
High influence
I
I
Cascades occur
only if size of max
vulnerable cluster
> 0.
Prediction
Social Contagion
Models
Granovetter’s model
Network version
Groups
Summary
System may be
‘robust-yet-fragile’.
Winning: it’s not for
everyone
‘Ignorance’
facilitates
spreading.
References
Superstars
Musiclab
Example networks
Frame 42/80
Contagion
Cascade window for random networks
Introduction
Simple Disease
Spreading Models
30
Background
25
Prediction
1
0.8
〈S〉
no cascades
20
Social Contagion
Models
0.6
0.4
Granovetter’s model
influence z
0.2
15
0
Network version
1
2
3
4
5
6
7
z
Summary
Winning: it’s not for
everyone
10
5
Groups
Superstars
cascades
Musiclab
References
0
0.05
0.1
0.15
0.2
0.25
φ = uniform individual threshold
I
‘Cascade window’ widens as threshold φ decreases.
I
Lower thresholds enable spreading.
Frame 43/80
Cascade window for random networks
Contagion
Introduction
Simple Disease
Spreading Models
Background
Prediction
Social Contagion
Models
Granovetter’s model
Network version
Groups
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
References
Frame 44/80
Contagion
Analytic work
Introduction
I
Threshold model completely solved (by 2008):
I
Cascade condition: [22]
∞
X
k (k − 1)βk Pk /z ≥ 1.
Simple Disease
Spreading Models
Background
Prediction
Social Contagion
Models
Granovetter’s model
Network version
k =1
Groups
Summary
where βk = probability a degree k node is vulnerable.
I
Final size of spread figured out by Gleeson and
Calahane [9, 8] .
I
Solution involves finding fixed points of an iterative
map of the interval.
I
Spreading takes off: expansion
I
Spreading reaches a particular node: contraction
Winning: it’s not for
everyone
Superstars
Musiclab
References
Frame 45/80
Contagion
Expected size of spread
= active at t=0
t=4
= active at t=1
= active at t=2
= active at t=3
= active at t=4
Introduction
Simple Disease
Spreading Models
Background
Prediction
Social Contagion
Models
Granovetter’s model
Network version
Groups
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
i
ϕ = 1/3
References
Frame 46/80
Contagion
Early adopters—degree distributions
t =0
t =1
t =2
0.8
0.2
t=0
0.15
t =3
0.8
t=1
0.6
t=2
0.6
0.4
0.4
0.4
0.05
0.2
0.2
0.2
5
10
15
20
0
0
t =4
10
15
20
0.3
0.2
0.2
0.1
10
15
20
0
0
t = 12
0
0
5
10
15
20
0.2
0.1
0.1
0
0
5
10
15
20
t = 16
Granovetter’s model
Network version
0.1
0.05
5
10
15
20
0
0
Winning: it’s not for
everyone
5
10
15
20
5
10
15
20
0
0
Superstars
Musiclab
References
t = 18
0.15
0.1
0
0
Social Contagion
Models
Groups
0.2
t = 16
0.15
0.05
20
20
t = 18
0.2
t = 14
0.15
0
0
0.1
15
15
t = 10
0.3
0.05
10
10
t = 10
0.1
5
5
0.4
0.05
0
0
Background
Prediction
Summary
0.2
t = 12
20
0.2
t = 14
0.2
0.15
15
t=8
0.3
0.1
5
10
0.4
t=6
0.4
0.3
5
t =8
0.5
t=4
0.4
0
0
t =6
0.5
0
0
5
Simple Disease
Spreading Models
t=3
0.6
0.1
0
0
Introduction
0.8
5
10
15
20
Pk ,t versus k
Frame 47/80
Contagion
The power of groups...
Introduction
Simple Disease
Spreading Models
Background
Prediction
Social Contagion
Models
Granovetter’s model
Network version
“A few harmless flakes
working together can
unleash an avalanche
of destruction.”
Groups
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
References
Frame 49/80
despair.com
Group structure—Ramified random networks
Contagion
Introduction
Simple Disease
Spreading Models
Background
Prediction
Social Contagion
Models
Granovetter’s model
Network version
Groups
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
References
p = intergroup connection probability
q = intragroup connection probability.
Frame 50/80
Contagion
Generalized affiliation model
Introduction
Simple Disease
Spreading Models
Background
geography
occupation
age
0
Prediction
100
Social Contagion
Models
Granovetter’s model
Network version
Groups
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
a
b
c
d
e
References
(Blau & Schwartz, Simmel, Breiger)
Frame 51/80
Cascade windows for group-based networks
Contagion
Introduction
Single seed
Random set seed
Coherent group seed
Simple Disease
Spreading Models
Background
B
Prediction
C
Generalized Affiliation
Model networks
Random
Group networks
A
Social Contagion
Models
Granovetter’s model
Network version
Groups
Summary
D
E
F
Winning: it’s not for
everyone
Superstars
Musiclab
References
Frame 52/80
Assortativity in group-based networks
0.8
0.6
Contagion
Introduction
1
Average
Cascade size
Simple Disease
Spreading Models
Background
0.5
0
Prediction
0
4
8
Social Contagion
Models
12
Granovetter’s model
k
0.4
Network version
Groups
Summary
Degree distribution
for initially infected node
0.2
Winning: it’s not for
everyone
Superstars
Musiclab
References
0
0
5
Local influence
10
15
20
k
I
The most connected nodes aren’t always the most
‘influential.’
I
Degree assortativity is the reason.
Frame 53/80
Social contagion
Contagion
Introduction
Summary:
Simple Disease
Spreading Models
Background
I
‘Influential vulnerables’ are key to spread.
I
Early adopters are mostly vulnerables.
I
Vulnerable nodes important but not necessary.
I
Groups may greatly facilitate spread.
I
Extreme/unexpected cascades may occur in highly
connected networks
Prediction
Social Contagion
Models
Granovetter’s model
I
Many potential ‘influentials’ exist.
I
Average individuals may be more influential
system-wise than locally influential individuals.
I
‘Influentials’ are posterior constructs.
Network version
Groups
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
References
Frame 55/80
Social contagion
Contagion
Introduction
Implications:
Simple Disease
Spreading Models
Background
I
Focus on the influential vulnerables.
I
Create entities that many individuals ‘out in the wild’
will adopt and display rather than broadcast from a
few ‘influentials.’
Prediction
Social Contagion
Models
Granovetter’s model
Network version
Groups
Summary
Displaying can be passive = free (yo-yo’s, fashion),
or active = harder to achieve (political messages).
Winning: it’s not for
everyone
I
Accept that movement of entities will be out of
originator’s control.
References
I
Possibly only simple ideas can spread by
word-of-mouth.
(Idea of opinion leaders has spread well...)
I
Superstars
Musiclab
Frame 56/80
Social Contagion
Contagion
Introduction
Simple Disease
Spreading Models
Background
Prediction
Messing with social connections:
Social Contagion
Models
Granovetter’s model
I
Ads based on message content
(e.g., Google and email)
I
Buzz media
I
Facebook’s advertising (Beacon)
Network version
Groups
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
References
Arguably not always a good idea...
Frame 57/80
Contagion
The collective...
Introduction
Simple Disease
Spreading Models
Background
Prediction
Social Contagion
Models
Granovetter’s model
“Never Underestimate
the Power of Stupid
People in Large
Groups.”
Network version
Groups
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
References
despair.com
Frame 58/80
Where do superstars come from?
Contagion
Introduction
Simple Disease
Spreading Models
Background
Rosen (1981): “The Economics of Superstars”
Examples:
Prediction
Social Contagion
Models
Granovetter’s model
Network version
I
Full-time Comedians (≈ 200)
I
Soloists in Classical Music
I
Economic Textbooks (the usual myopic example)
Groups
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
References
I
Highly skewed distributions again...
Frame 60/80
Superstars
Contagion
Introduction
Rosen’s theory:
I
Individual quality q maps to reward R(q)
I
R(q) is ‘convex’ (d2 R/dq 2 > 0)
Two reasons:
I
1. Imperfect substitution:
A very good surgeon is worth many mediocre ones
2. Technology:
Media spreads & technology reduces cost of
reproduction of books, songs, etc.
I
Simple Disease
Spreading Models
Background
Prediction
Social Contagion
Models
Granovetter’s model
Network version
Groups
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
References
No social element—success follows ‘inherent quality’
Frame 61/80
Superstars
Contagion
Introduction
Simple Disease
Spreading Models
Background
Prediction
Adler (1985): “Stardom and Talent”
Social Contagion
Models
Granovetter’s model
I
Assumes extreme case of equal ‘inherent quality’
I
Argues desire for coordination in knowledge and
culture leads to differential success
I
Success is then purely a social construction
Network version
Groups
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
References
Frame 62/80
Dominance hierarchies
Chase et al. (2002): “Individual differences versus social
dynamics in the formation of animal dominance
hierarchies”
The aggressive female Metriaclima zebra ():
Contagion
Introduction
Simple Disease
Spreading Models
Background
Prediction
Social Contagion
Models
Granovetter’s model
Network version
Groups
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
References
Pecking orders for fish...
Frame 63/80
Dominance hierarchies
I
Fish forget—changing of dominance hierarchies:
Table 1. Percentage of groups with different numbers of fish
changing ranks between first and second hierarchies (n ! 22)
No. of fish changing ranks
0
2
3
4
I
Percentage of groups
27.3
36.4
18.2
18.2
rank on prior attributes of itself creates the linear structure of the
hierarchies. Although 50% of the fish changed ranks from one
hierarchy to the other, almost all the hierarchies were linear in
structure. Some factor other than differences in attributes seems
to have ensured high rates of linearity. In the next experiment,
we tested to determine whether that factor might be social
dynamics.
It might seem possible that ‘‘noise,’’ random fluctuations in
individuals’ attributes or behaviors, could account for the observed differences between the first and second hierarchies.
However, a careful consideration of the ways in which fluctuations might occur shows that this explanation is unlikely. For
example, what if the differences were assumed to have occurred
because some of the fish changed their ranks on attributes from
the 1.firstTransition
to the second
Fig.
patternshierarchies?
between ranksTo
of account
fish in thefor
firstour
andresults,
second
this assumption
would
require a mixture
of stability
instahierarchies.
Frequencies
of experimental
groups showing
each and
pattern
are
indicated
in parentheses.
arrowstimes
indicate
of right
rank.
bility in attribute
ranksOpen-headed
at just the right
andtransitions
in just the
Solid-headed
arrows
show
dominance
relationships
in
intransitive
triads;
all
proportion of groups. The rankings would have had to have been
the
fish for
in anall
intransitive
samefor
rank.
stable
the fish triad
in allshare
the the
groups
the day or two it took
them to form their first hierarchies (or we would not have seen
stable dominance
relationships
Then,
(one-sided
binomial
test: n ! by
22,our
P criterion).
" 0.001 and
P in
"three0.03,
quarters of the
(but 27%
not inofthe
one-quarter)
respectively).
Ingroups
this light,
theremaining
groups with
identical
various numbers
fish would have had to have swapped ranks
hierarchies
is veryofsmall.
on attributes in the 2-week period of separation so as to have
produced different second hierarchies. And finally, the rankings
Discussion. When we rewound the tape of the fish to form new
on attributeswe
for
all thedid
fish
inget
all the
the same
groups
would have
had
to
hierarchies,
usually
not
hierarchy
twice.
The
have become
oncepersisted
more forand
thethe
dayindividuals
or two it took
them
linearity
of thestable
structures
stayed
the
to form
hierarchies.
same,
buttheir
theirsecond
ranks did
not. Thus our results differ considerably
22 observations: about 3/4 of the time, hierarchy
changed
rank on prior attributes of itself creates th
hierarchies. Although 50% of the fish c
Contagion
hierarchy to the other,
almost all the hi
structure. Some factor other than differe
to have ensured high rates of linearity.
we tested to determine whether that
dynamics.
It might seem possible that ‘‘noise,’’
individuals’ attributes
or behaviors, cou
Introduction
served differences between the first a
However, a careful
consideration
Simple
Disease of the
tions might occur shows that this expla
Spreading
Models
example, what
if the differences
were as
because someBackground
of the fish changed their r
the first to thePrediction
second hierarchies? To
this assumption would require a mixtur
bility in attribute ranks at just the right ti
proportion ofSocial
groups. Contagion
The rankings woul
stable for all Models
the fish in all the groups fo
them to form Granovetter’s
their first hierarchies
(or w
model
stable dominance relationships by our cr
Network
version
quarters of the
groups
(but not in the
Groups
various numbers
of fish would have had
on attributes Summary
in the 2-week period of se
produced different second hierarchies. A
on attributes for all the fish in all the gr
Winning: it’s not for
have become stable once more for the d
to form theireveryone
second hierarchies.
Alternatively,
instead of attribute ra
Superstars
nance rank as in the prior attribute mo
Musiclab
of fish might be considered to have been
at one meeting one might dominate, b
References
there was some
chance that the other
problem with this model is that earlier
demonstrates that in situations in which
group has even a small chance of dom
probability of getting linear hierarchies
even in a more restrictive model in whic
are close in rank in the first hierarchies ha
of reversing their relationships, such
observed in this experiment, the probab
linear hierarchies as we observed is sti
available from the authors).
We know of only one other study (4
assembled groups to form initial hier
individuals for a period, and then reass
second hierarchy (but see Guhl, ref. 4
Frameencounters
64/80 between
groups had pairwise
bly). Unfortunately, their techniques of
sible to compare results, because they
between the frequency of aggressive acts
in pairs toward one another in the two
Contagion
Music Lab Experiment
Introduction
Simple Disease
Spreading Models
Background
Prediction
Social Contagion
Models
Granovetter’s model
Network version
Groups
Summary
48 songs
30,000 participants
multiple ‘worlds’
Inter-world variability
I
How probable is the world?
I
Can we estimate variability?
I
Superstars dominate but are unpredictable. Why?
Winning: it’s not for
everyone
Superstars
Musiclab
References
Frame 66/80
Music Lab Experiment
Contagion
Introduction
Simple Disease
Spreading Models
Background
Prediction
Social Contagion
Models
Granovetter’s model
Network version
Groups
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
References
Salganik et al. (2006) “An experimental study of inequality
and unpredictability in an artificial cultural market”
Frame 67/80
Contagion
Music Lab Experiment
Introduction
Simple Disease
Spreading Models
Background
Experiment 1
Experiments 2–4
Prediction
Social Contagion
Models
Granovetter’s model
Network version
Groups
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
References
Frame 68/80
Contagion
Music Lab Experiment
Introduction
Experiment 1
1
12
24
36
48
48
36
24
12
1
Rank market share in indep. world
Rank market share in influence worlds
Rank market share in influence worlds
Simple Disease
Spreading Models
Background
Experiment 2
Prediction
1
Social Contagion
Models
12
Granovetter’s model
Network version
24
Groups
Summary
Winning: it’s not for
everyone
36
Superstars
48
48
36
24
12
1
Musiclab
Rank market share in indep. world
References
I
Variability in final rank.
Frame 69/80
Contagion
Music Lab Experiment
Introduction
Simple Disease
Spreading Models
Experiment 1
Market share in influence worlds
Market share in influence worlds
Prediction
0.2
0.15
0.1
0.05
0
0
Background
Experiment 2
0.2
0.01
0.02
0.03
0.04
Market share in independent world
0.05
Social Contagion
Models
0.15
Granovetter’s model
Network version
0.1
Groups
Summary
Winning: it’s not for
everyone
0.05
Superstars
0
0
0.01
0.02
0.03
0.04
0.05
Musiclab
Market share in independent world
References
I
Variability in final number of downloads.
Frame 70/80
Contagion
Music Lab Experiment
Introduction
0.6
Experiment 1
Simple Disease
Spreading Models
Experiment 2
Gini coefficient G
Background
Prediction
0.4
Social Contagion
Models
Granovetter’s model
0.2
Network version
Groups
Summary
0
Social Influence
Indep.
Social Influence
Indep.
Winning: it’s not for
everyone
Superstars
I
Inequality as measured by Gini coefficient:
Musiclab
References
N
G=
N
s X
s
X
1
|mi − mj |
(2Ns − 1)
i=1 j=1
Frame 71/80
Contagion
Music Lab Experiment
Introduction
0.015
Experiment 1
Experiment 2
Simple Disease
Spreading Models
Unpredictability U
Background
Prediction
0.01
Social Contagion
Models
Granovetter’s model
0.005
Network version
Groups
Summary
0
Social
Influence
Independent
Social
Influence
Independent
Winning: it’s not for
everyone
Superstars
I
Musiclab
Unpredictability
References
U=
Ns X
Nw X
Nw
X
1
Ns
Nw
2
|mi,j − mi,k |
i=1 j=1 k =j+1
Frame 72/80
Music Lab Experiment
Sensible result:
I
Stronger social signal leads to greater following and
greater inequality.
Contagion
Introduction
Simple Disease
Spreading Models
Background
Prediction
Social Contagion
Models
Peculiar result:
I
Stronger social signal leads to greater
unpredictability.
Granovetter’s model
Network version
Groups
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
Very peculiar observation:
I
The most unequal distributions would suggest the
greatest variation in underlying ‘quality.’
I
But success may be due to social construction
through following...
References
Frame 73/80
Music Lab Experiment—Sneakiness
Contagion
Introduction
Exp. 3
Exp. 4
Unchanged world
Inverted worlds
500
Exp. 3
Song 1
Song 48
Song 48
200
Downloads
Downloads
Background
Song 2
200
300
Song 1
Song 48
400 752
Prediction
Social Contagion
Models
Granovetter’s model
Network version
100
Groups
Song 2
Song 2
Song 2
Song 47
50
Song 47
Song 48
1200 1600 2000 2400 2800
Song 47
Song 47
150
Song 1
Song 1
100
0
0
Exp. 4
Unchanged world
Inverted worlds
250
400
Simple Disease
Spreading Models
0
0
400
752
Subjects
1200 1600 2000 2400 2800
Subjects
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
I
Inversion of download count
I
The ‘pretend rich’ get richer ...
I
... but at a slower rate
References
Frame 74/80
References I
Contagion
Introduction
[1] M. Adler.
Stardom and talent.
American Economic Review, pages 208–212, 1985.
pdf ()
Simple Disease
Spreading Models
Background
Prediction
Social Contagion
Models
Granovetter’s model
Network version
[2] S. Bikhchandani, D. Hirshleifer, and I. Welch.
A theory of fads, fashion, custom, and cultural
change as informational cascades.
J. Polit. Econ., 100:992–1026, 1992.
Groups
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
References
[3] S. Bikhchandani, D. Hirshleifer, and I. Welch.
Learning from the behavior of others: Conformity,
fads, and informational cascades.
J. Econ. Perspect., 12(3):151–170, 1998. pdf ()
Frame 75/80
References II
[4] J. Carlson and J. Doyle.
Highly optimized tolerance: A mechanism for power
laws in design systems.
Phys. Rev. E, 60(2):1412–1427, 1999. pdf ()
[5] J. Carlson and J. Doyle.
Highly optimized tolerance: Robustness and design
in complex systems.
Phys. Rev. Lett., 84(11):2529–2532, 2000. pdf ()
[6] I. D. Chase, C. Tovey, D. Spangler-Martin, and
M. Manfredonia.
Individual differences versus social dynamics in the
formation of animal dominance hierarchies.
Proc. Natl. Acad. Sci., 99(8):5744–5749, 2002.
pdf ()
Contagion
Introduction
Simple Disease
Spreading Models
Background
Prediction
Social Contagion
Models
Granovetter’s model
Network version
Groups
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
References
Frame 76/80
References III
[7] M. Gladwell.
The Tipping Point.
Little, Brown and Company, New York, 2000.
[8] J. P. Gleeson.
Cascades on correlated and modular random
networks.
Phys. Rev. E, 77:046117, 2008. pdf ()
[9] J. P. Gleeson and D. J. Cahalane.
Seed size strongly affects cascades on random
networks.
Phys. Rev. E, 75:056103, 2007. pdf ()
[10] M. Granovetter.
Threshold models of collective behavior.
Am. J. Sociol., 83(6):1420–1443, 1978. pdf ()
Contagion
Introduction
Simple Disease
Spreading Models
Background
Prediction
Social Contagion
Models
Granovetter’s model
Network version
Groups
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
References
Frame 77/80
References IV
[11] E. Hoffer.
The Passionate State of Mind: And Other Aphorisms.
Buccaneer Books, 1954.
Contagion
Introduction
Simple Disease
Spreading Models
Background
Prediction
[12] E. Katz and P. F. Lazarsfeld.
Personal Influence.
The Free Press, New York, 1955.
Social Contagion
Models
Granovetter’s model
Network version
Groups
Summary
[13] T. Kuran.
Now out of never: The element of surprise in the east
european revolution of 1989.
World Politics, 44:7–48, 1991. pdf ()
[14] T. Kuran.
Private Truths, Public Lies: The Social
Consequences of Preference Falsification.
Harvard University Press, Cambridge, MA, Reprint
edition, 1997.
Winning: it’s not for
everyone
Superstars
Musiclab
References
Frame 78/80
References V
Contagion
Introduction
[15] J. D. Murray.
Mathematical Biology.
Springer, New York, Third edition, 2002.
[16] S. Rosen.
The economics of superstars.
Am. Econ. Rev., 71:845–858, 1981. pdf ()
[17] M. J. Salganik, P. S. Dodds, and D. J. Watts.
An experimental study of inequality and
unpredictability in an artificial cultural market.
Science, 311:854–856, 2006. pdf ()
Simple Disease
Spreading Models
Background
Prediction
Social Contagion
Models
Granovetter’s model
Network version
Groups
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
References
[18] T. Schelling.
Dynamic models of segregation.
J. Math. Sociol., 1:143–186, 1971.
Frame 79/80
References VI
[19] T. C. Schelling.
Hockey helmets, concealed weapons, and daylight
saving: A study of binary choices with externalities.
J. Conflict Resolut., 17:381–428, 1973. pdf ()
[20] T. C. Schelling.
Micromotives and Macrobehavior.
Norton, New York, 1978.
[21] D. Sornette.
Critical Phenomena in Natural Sciences.
Springer-Verlag, Berlin, 2nd edition, 2003.
[22] D. J. Watts.
A simple model of global cascades on random
networks.
Proc. Natl. Acad. Sci., 99(9):5766–5771, 2002.
pdf ()
Contagion
Introduction
Simple Disease
Spreading Models
Background
Prediction
Social Contagion
Models
Granovetter’s model
Network version
Groups
Summary
Winning: it’s not for
everyone
Superstars
Musiclab
References
Frame 80/80