Contagion - Complex Networks, SFI Summer School, June, 2010 · 2010-06-07 · Contagion...
Transcript of Contagion - Complex Networks, SFI Summer School, June, 2010 · 2010-06-07 · Contagion...
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 1/80
ContagionComplex Networks, SFI Summer School, June, 2010
Prof. Peter Dodds
Department of Mathematics & StatisticsCenter for Complex Systems
Vermont Advanced Computing CenterUniversity of Vermont
Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 2/80
Outline
Introduction
Simple Disease Spreading ModelsBackgroundPrediction
Social Contagion ModelsGranovetter’s modelNetwork versionGroupsSummary
Winning: it’s not for everyoneSuperstarsMusiclab
References
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 3/80
Contagion
Definition:I (1) The spreading of a quality or quantity between
individuals in a population.I (2) A disease itself:
the plague, a blight, the dreaded lurgi, ...
Two main classes of contagion:
1. Infectious diseases:tuberculosis, HIV, ebola, SARS, influenza, ...
2. Social contagion:fashion, word usage, rumors, riots, religion, ...
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 4/80
Contagion models
Some large questions concerning networkcontagion:
1. For a given spreading mechanism on a givennetwork, what’s the probability that there will beglobal spreading?
2. If spreading does take off, how far will it go?3. How do the details of the network affect the
outcome?4. How do the details of the spreading mechanism
affect the outcome?5. What if the seed is one or many nodes?
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 6/80
Mathematical Epidemiology
The standard SIR model:I Three states:
I S = SusceptibleI I = InfectedI R = Recovered
I S(t) + I(t) + R(t) = 1I Presumes random
interactions
Discrete time example:
I
R
SβI
1 − ρ
ρ
1 − βI
r1 − r
Transition Probabilities:
β for being infected givencontact with infectedr for recoveryρ for loss of immunity
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 7/80
Independent Interaction models
Reproduction Number R0:
I R0 = expected number of infected individualsresulting from a single initial infective.
I Epidemic threshold: If R0 > 1, ‘epidemic’ occurs.I Example:
0 1 2 3 40
0.2
0.4
0.6
0.8
1
R0
Frac
tion
infe
cted
I Continuous phasetransition.
I Fine idea from asimple model.
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 9/80
Disease spreading models
For ‘novel’ diseases:
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?
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 10/80
R0 and variation in epidemic sizes
R0 approximately the same for all of the following:
I 1918-19 “Spanish Flu” ∼ 500,000 deaths in USI 1957-58 “Asian Flu” ∼ 70,000 deaths in USI 1968-69 “Hong Kong Flu” ∼ 34,000 deaths in USI 2003 “SARS Epidemic” ∼ 800 deaths world-wide
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 11/80
Size distributions
Elsewhere, event size distributions are important:
I earthquakes (Gutenberg-Richter law)I city sizes, forest fires, war fatalitiesI wealth distributionsI ‘popularity’ (books, music, websites, ideas)I What about Epidemics?
Power laws distributions are common but not obligatory...
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 12/80
Feeling icky in Iceland
Caseload recorded monthly for range of diseases inIceland, 1888-1990
1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 19900
0.01
0.02
0.03
Date
Freq
uenc
y
Iceland: measlesnormalized count
Treat outbreaks separated in time as ‘novel’ diseases.
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 13/80
Measles
0 0.025 0.05 0.075 0.10
1
2
3
4
5
75
N (
ψ)
ψ
A
10−5
10−4
10−3
10−2
10−1
100
101
102
ψ
N>(ψ
)
Insert plots:Complementary cumulativefrequency distributions:
N>(Ψ) ∝ Ψ−γ+1
Ψ = fractional epidemic size
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...
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 14/80
Resurgence—example of SARS
D
Date of onset
# N
ew c
ases
Nov 16, ’02 Dec 16, ’02 Jan 15, ’03 Feb 14, ’03 Mar 16, ’03 Apr 15, ’03 May 15, ’03 Jun 14, ’03
160
120
80
40
0
I Epidemic discovers new ‘pools’ of susceptibles:Resurgence.
I Importance of rare, stochastic events.
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 15/80
A challenge
So... can a simple model produce
1. broad epidemic distributionsand
2. resurgence ?
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 16/80
Size distributions
0 0.25 0.5 0.75 10
500
1000
1500
2000A
ψ
N(ψ
)
R0=3
Simple modelstypically producebimodal or unimodalsize distributions.
I This includes network models:random, small-world, scale-free, ...
I Some exceptions:1. Forest fire models2. Sophisticated metapopulation models
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 17/80
A toy agent-based model
Geography: allow people to move between contexts:
b=2
i j
x ij =2l=3
n=8
I P = probability of travelI Movement distance: Pr(d) ∝ exp(−d/ξ)
I ξ = typical travel distance
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 18/80
Example model output: size distributions
0 0.25 0.5 0.75 10
100
200
300
400
1942
ψ
N(ψ
)
R0=3
0 0.25 0.5 0.75 10
100
200
300
400
683
ψ
N(ψ
)
R0=12
I Flat distributions are possible for certain ξ and P.I Different R0’s may produce similar distributionsI Same epidemic sizes may arise from different R0’s
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 19/80
Standard model:
0 500 1000 15000
2000
4000
6000
t
# N
ew c
ases D R
0=3
Standard model with transport: Resurgence
0 500 1000 15000
200
400
t
# N
ew c
ases G R
0=3
I Disease spread highly sensitive to populationstructure
I Rare events may matter enormously
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 20/80
Simple disease spreading models
Attempts to use beyond disease:
I Adoption of ideas/beliefs (Goffman & Newell, 1964)I Spread of rumors (Daley & Kendall, 1965)I Diffusion of innovations (Bass, 1969)I Spread of fanatical behavior (Castillo-Chávez &
Song, 2003)
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 21/80
Social Contagion Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 22/80
Social Contagion
Examples abound:
I being polite/rudeI strikesI innovationI residential segregationI ipodsI obesity
I Harry PotterI votingI gossip
I Rubik’s cubeI religious beliefsI leaving lectures
SIR and SIRS contagion possible
I Classes of behavior versus specific behavior: dieting
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 23/80
Social Contagion
Two focuses for us:I Widespread media influenceI Word-of-mouth influence
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 24/80
The hypodermic model of influence:
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 25/80
The two step model of influence: Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 26/80
The general model of influence:
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 27/80
Social Contagion
Why do things spread?
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 understandI Always good to examine what is said before and
after the fact...
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 28/80
The Mona Lisa:
I “Becoming Mona Lisa: The Making of a GlobalIcon”—David Sassoon
I Not the world’s greatest painting from the start...I Escalation through theft, vandalism, parody, ...
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 29/80
The completely unpredicted fall of EasternEurope:
Timur Kuran: “Now Out of Never: The Element ofSurprise in the East European Revolution of 1989”
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 30/80
Social Contagion
Some important models:
I Tipping models—Schelling (1971)I Simulation on checker boardsI Idea of thresholds
I Threshold models—Granovetter (1978)I Herding models—Bikhchandani, Hirschleifer, Welch
(1992)I Social learning theory, Informational cascades,...
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 31/80
Social contagion models
Thresholds:I Basic idea: individuals adopt a behavior when a
certain fraction of others have adoptedI ‘Others’ may be everyone in a population, an
individual’s close friends, any reference group.I Response can be probabilistic or deterministic.I Individual thresholds vary.
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 32/80
Social Contagion
Some possible origins of thresholds:
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)I Economics: Network effects or network externalities
I Telephones, Facebook, operating systems, ...
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 33/80
Imitation
despair.com
“When people are freeto do as they please,they usually imitateeach other.”
—Eric Hoffer“The Passionate Stateof Mind” [11]
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 35/80
Granovetter’s threshold model:
Action based on perceived behavior of others:
0 10
0.2
0.4
0.6
0.8
1
φi∗
A
φi,t
Pr(a
i,t+
1=1)
0 0.5 10
0.5
1
1.5
2
2.5B
φ∗
f (φ∗ )
0 0.5 10
0.2
0.4
0.6
0.8
1
φt
φ t+1 =
F (
φ t)
C
I Two states: S and I.I φ = fraction of contacts ‘on’ (e.g., rioting)I
φt+1 =
∫ φt
0f (γ)dγ = F (γ)|φt
0 = F (φt)
I This is a Critical Mass model
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 36/80
Social Sciences: Threshold models
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
2.5
3
γ
f(γ)
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
φt
φ t+1
I Example of single stable state model
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 37/80
Social Sciences—Threshold models
Implications for collective action theory:
1. Collective uniformity 6⇒ individual uniformity2. Small individual changes ⇒ large global changes
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 39/80
Threshold model on a network
t=1 t=2 t=3
c
a
bc
e
a
b
e
a
bc
e
d dd
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
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 40/80
Snowballing
The Cascade Condition:I If one individual is initially activated, what is the
probability that an activation will spread over anetwork?
I What features of a network determine whether acascade will occur or not?
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 41/80
The most gullible
Vulnerables:I = Individuals who can be activated by just one
‘infected’ contactI For global cascades on random networks, must have
a global cluster of vulnerablesI Cluster of vulnerables = critical massI Network story: 1 node → critical mass → everyone.
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 42/80
Cascades on random networks
1 2 3 4 5 6 70
0.2
0.4
0.6
0.8
1
z
⟨ S
⟩
Example networks
PossibleNo
Cascades
Low influence
Fraction ofVulnerables
cascade sizeFinal
CascadesNo Cascades
CascadesNo
High influence
I Cascades occuronly if size of maxvulnerable cluster> 0.
I System may be‘robust-yet-fragile’.
I ‘Ignorance’facilitatesspreading.
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 43/80
Cascade window for random networks
1 2 3 4 5 6 70
0.2
0.4
0.6
0.8
1
z
⟨ S
⟩
0.05 0.1 0.15 0.2 0.250
5
10
15
20
25
30
φ
z
cascades
no cascades
infl
uenc
e
= uniform individual threshold
I ‘Cascade window’ widens as threshold φ decreases.I Lower thresholds enable spreading.
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 44/80
Cascade window for random networks
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 45/80
Analytic work
I Threshold model completely solved (by 2008):I Cascade condition: [22]
∞∑k=1
k(k − 1)βkPk/z ≥ 1.
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: expansionI Spreading reaches a particular node: contraction
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 46/80
Expected size of spread
i
ϕ = 1/3
t=4= active at t=0
= active at t=1
= active at t=2
= active at t=3
= active at t=4
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 47/80
Early adopters—degree distributions
t = 0 t = 1 t = 2 t = 3
0 5 10 15 200
0.05
0.1
0.15
0.2
t = 0
0 5 10 15 200
0.2
0.4
0.6
0.8
t = 1
0 5 10 15 200
0.2
0.4
0.6
0.8
t = 2
0 5 10 15 200
0.2
0.4
0.6
0.8
t = 3
t = 4 t = 6 t = 8 t = 10
0 5 10 15 200
0.1
0.2
0.3
0.4
0.5
t = 4
0 5 10 15 200
0.1
0.2
0.3
0.4
0.5
t = 6
0 5 10 15 200
0.1
0.2
0.3
0.4
t = 8
0 5 10 15 200
0.1
0.2
0.3
0.4
t = 10
t = 12 t = 14 t = 16 t = 18
0 5 10 15 200
0.05
0.1
0.15
0.2
t = 12
0 5 10 15 200
0.05
0.1
0.15
0.2
t = 14
0 5 10 15 200
0.05
0.1
0.15
0.2
t = 16
0 5 10 15 200
0.05
0.1
0.15
0.2
t = 18
Pk ,t versus k
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 49/80
The power of groups...
despair.com
“A few harmless flakesworking together canunleash an avalancheof destruction.”
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 50/80
Group structure—Ramified random networks
p = intergroup connection probabilityq = intragroup connection probability.
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 51/80
Generalized affiliation model
100
eca b d
geography occupation age
0
(Blau & Schwartz, Simmel, Breiger)
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 52/80
Cascade windows for group-based networks
Gen
eral
ized
Affili
atio
n
A
Gro
up n
etwo
rks
Single seed Coherent group seed
Mod
el n
etwo
rks
Random set seed
Rand
om
F
C
D E
B
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 53/80
Assortativity in group-based networks
0 5 10 15 200
0.2
0.4
0.6
0.8
k
0 4 8 120
0.5
1
k
AverageCascade size
Local influence
Degree distributionfor initially infected node
I The most connected nodes aren’t always the most‘influential.’
I Degree assortativity is the reason.
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 55/80
Social contagion
Summary:
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 networksI Many potential ‘influentials’ exist.I Average individuals may be more influential
system-wise than locally influential individuals.I ‘Influentials’ are posterior constructs.
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 56/80
Social contagion
Implications:
I Focus on the influential vulnerables.I Create entities that many individuals ‘out in the wild’
will adopt and display rather than broadcast from afew ‘influentials.’
I Displaying can be passive = free (yo-yo’s, fashion),or active = harder to achieve (political messages).
I Accept that movement of entities will be out oforiginator’s control.
I Possibly only simple ideas can spread byword-of-mouth.
(Idea of opinion leaders has spread well...)
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 57/80
Social Contagion
Messing with social connections:
I Ads based on message content(e.g., Google and email)
I Buzz mediaI Facebook’s advertising (Beacon)
Arguably not always a good idea...
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 58/80
The collective...
despair.com
“Never Underestimatethe Power of StupidPeople in LargeGroups.”
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 60/80
Where do superstars come from?
Rosen (1981): “The Economics of Superstars”
Examples:
I Full-time Comedians (≈ 200)I Soloists in Classical MusicI Economic Textbooks (the usual myopic example)
I Highly skewed distributions again...
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 61/80
Superstars
Rosen’s theory:
I Individual quality q maps to reward R(q)
I R(q) is ‘convex’ (d2R/dq2 > 0)I Two reasons:
1. Imperfect substitution:A very good surgeon is worth many mediocre ones
2. Technology:Media spreads & technology reduces cost ofreproduction of books, songs, etc.
I No social element—success follows ‘inherent quality’
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 62/80
Superstars
Adler (1985): “Stardom and Talent”
I Assumes extreme case of equal ‘inherent quality’I Argues desire for coordination in knowledge and
culture leads to differential successI Success is then purely a social construction
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 63/80
Dominance hierarchies
Chase et al. (2002): “Individual differences versus socialdynamics in the formation of animal dominancehierarchies”The aggressive female Metriaclima zebra (�):
Pecking orders for fish...
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 64/80
Dominance hierarchies
I Fish forget—changing of dominance hierarchies:
(one-sided binomial test: n ! 22, P " 0.001 and P " 0.03,respectively). In this light, 27% of the groups with identicalhierarchies is very small.
Discussion. When we rewound the tape of the fish to form newhierarchies, we usually did not get the same hierarchy twice. Thelinearity of the structures persisted and the individuals stayed thesame, but their ranks did not. Thus our results differ considerablyfrom those predicted by the prior attributes hypothesis. The factthat more identical hierarchies occurred than expected by chancealone supports the hypothesis that rank on prior attributesinfluences rank within hierarchies but not the hypothesis that
rank on prior attributes of itself creates the linear structure of thehierarchies. Although 50% of the fish changed ranks from onehierarchy to the other, almost all the hierarchies were linear instructure. Some factor other than differences in attributes seemsto have ensured high rates of linearity. In the next experiment,we tested to determine whether that factor might be socialdynamics.
It might seem possible that ‘‘noise,’’ random fluctuations inindividuals’ attributes or behaviors, could account for the ob-served differences between the first and second hierarchies.However, a careful consideration of the ways in which fluctua-tions might occur shows that this explanation is unlikely. Forexample, what if the differences were assumed to have occurredbecause some of the fish changed their ranks on attributes fromthe first to the second hierarchies? To account for our results,this assumption would require a mixture of stability and insta-bility in attribute ranks at just the right times and in just the rightproportion of groups. The rankings would have had to have beenstable for all the fish in all the groups for the day or two it tookthem to form their first hierarchies (or we would not have seenstable dominance relationships by our criterion). Then, in three-quarters of the groups (but not in the remaining one-quarter)various numbers of fish would have had to have swapped rankson attributes in the 2-week period of separation so as to haveproduced different second hierarchies. And finally, the rankingson attributes for all the fish in all the groups would have had tohave become stable once more for the day or two it took themto form their second hierarchies.
Alternatively, instead of attribute rank determining domi-nance rank as in the prior attribute model, dominance in pairsof fish might be considered to have been probabilistic, such thatat one meeting one might dominate, but at a second meetingthere was some chance that the other might dominate. Theproblem with this model is that earlier mathematical analysisdemonstrates that in situations in which one of each pair in agroup has even a small chance of dominating the other, theprobability of getting linear hierarchies is quite low (34). Andeven in a more restrictive model in which only pairs of fish thatare close in rank in the first hierarchies have modest probabilitiesof reversing their relationships, such as the level (0.25) weobserved in this experiment, the probability of getting as manylinear hierarchies as we observed is still very low (details areavailable from the authors).
We know of only one other study (47) in which researchersassembled groups to form initial hierarchies, separated theindividuals for a period, and then reassembled them to form asecond hierarchy (but see Guhl, ref. 48, for results in whichgroups had pairwise encounters between assembly and reassem-bly). Unfortunately, their techniques of analysis make it impos-sible to compare results, because they examined correlationsbetween the frequency of aggressive acts directed by individualsin pairs toward one another in the two hierarchies rather thancomparing the ranks of individuals. With these techniques it ispossible to get a positive correlation and thus a ‘‘replication’’ ofan original hierarchy in situations in which several animalsactually change ranks from the first to the second hierarchies.
Table 1. Percentage of groups with different numbers of fishchanging ranks between first and second hierarchies (n ! 22)
No. of fish changing ranks Percentage of groups
0 27.32 36.43 18.24 18.2
Fig. 1. Transition patterns between ranks of fish in the first and secondhierarchies. Frequencies of experimental groups showing each pattern areindicated in parentheses. Open-headed arrows indicate transitions of rank.Solid-headed arrows show dominance relationships in intransitive triads; allthe fish in an intransitive triad share the same rank.
5746 ! www.pnas.org"cgi"doi"10.1073"pnas.082104199 Chase et al.
(one-sided binomial test: n ! 22, P " 0.001 and P " 0.03,respectively). In this light, 27% of the groups with identicalhierarchies is very small.
Discussion. When we rewound the tape of the fish to form newhierarchies, we usually did not get the same hierarchy twice. Thelinearity of the structures persisted and the individuals stayed thesame, but their ranks did not. Thus our results differ considerablyfrom those predicted by the prior attributes hypothesis. The factthat more identical hierarchies occurred than expected by chancealone supports the hypothesis that rank on prior attributesinfluences rank within hierarchies but not the hypothesis that
rank on prior attributes of itself creates the linear structure of thehierarchies. Although 50% of the fish changed ranks from onehierarchy to the other, almost all the hierarchies were linear instructure. Some factor other than differences in attributes seemsto have ensured high rates of linearity. In the next experiment,we tested to determine whether that factor might be socialdynamics.
It might seem possible that ‘‘noise,’’ random fluctuations inindividuals’ attributes or behaviors, could account for the ob-served differences between the first and second hierarchies.However, a careful consideration of the ways in which fluctua-tions might occur shows that this explanation is unlikely. Forexample, what if the differences were assumed to have occurredbecause some of the fish changed their ranks on attributes fromthe first to the second hierarchies? To account for our results,this assumption would require a mixture of stability and insta-bility in attribute ranks at just the right times and in just the rightproportion of groups. The rankings would have had to have beenstable for all the fish in all the groups for the day or two it tookthem to form their first hierarchies (or we would not have seenstable dominance relationships by our criterion). Then, in three-quarters of the groups (but not in the remaining one-quarter)various numbers of fish would have had to have swapped rankson attributes in the 2-week period of separation so as to haveproduced different second hierarchies. And finally, the rankingson attributes for all the fish in all the groups would have had tohave become stable once more for the day or two it took themto form their second hierarchies.
Alternatively, instead of attribute rank determining domi-nance rank as in the prior attribute model, dominance in pairsof fish might be considered to have been probabilistic, such thatat one meeting one might dominate, but at a second meetingthere was some chance that the other might dominate. Theproblem with this model is that earlier mathematical analysisdemonstrates that in situations in which one of each pair in agroup has even a small chance of dominating the other, theprobability of getting linear hierarchies is quite low (34). Andeven in a more restrictive model in which only pairs of fish thatare close in rank in the first hierarchies have modest probabilitiesof reversing their relationships, such as the level (0.25) weobserved in this experiment, the probability of getting as manylinear hierarchies as we observed is still very low (details areavailable from the authors).
We know of only one other study (47) in which researchersassembled groups to form initial hierarchies, separated theindividuals for a period, and then reassembled them to form asecond hierarchy (but see Guhl, ref. 48, for results in whichgroups had pairwise encounters between assembly and reassem-bly). Unfortunately, their techniques of analysis make it impos-sible to compare results, because they examined correlationsbetween the frequency of aggressive acts directed by individualsin pairs toward one another in the two hierarchies rather thancomparing the ranks of individuals. With these techniques it ispossible to get a positive correlation and thus a ‘‘replication’’ ofan original hierarchy in situations in which several animalsactually change ranks from the first to the second hierarchies.
Table 1. Percentage of groups with different numbers of fishchanging ranks between first and second hierarchies (n ! 22)
No. of fish changing ranks Percentage of groups
0 27.32 36.43 18.24 18.2
Fig. 1. Transition patterns between ranks of fish in the first and secondhierarchies. Frequencies of experimental groups showing each pattern areindicated in parentheses. Open-headed arrows indicate transitions of rank.Solid-headed arrows show dominance relationships in intransitive triads; allthe fish in an intransitive triad share the same rank.
5746 ! www.pnas.org"cgi"doi"10.1073"pnas.082104199 Chase et al.
I 22 observations: about 3/4 of the time, hierarchychanged
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 66/80
Music Lab Experiment
48 songs30,000 participants
multiple ‘worlds’Inter-world variability
I How probable is the world?I Can we estimate variability?I Superstars dominate but are unpredictable. Why?
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 67/80
Music Lab Experiment
Salganik et al. (2006) “An experimental study of inequalityand unpredictability in an artificial cultural market”
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 68/80
Music Lab Experiment
Experiment 1 Experiments 2–4
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 69/80
Music Lab Experiment
112243648
1
12
24
36
48
Experiment 1
Rank market share in indep. world
Ran
k m
arke
t sha
re in
influ
ence
wor
lds
112243648
1
12
24
36
48
Experiment 2
Rank market share in indep. worldR
ank
mar
ket s
hare
in in
fluen
ce w
orld
s
I Variability in final rank.
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 70/80
Music Lab Experiment
0 0.01 0.02 0.03 0.04 0.050
0.05
0.1
0.15
0.2Experiment 1
Market share in independent world
Mar
ket s
hare
in in
fluen
ce w
orld
s
0 0.01 0.02 0.03 0.04 0.050
0.05
0.1
0.15
0.2Experiment 2
Market share in independent world
Mar
ket s
hare
in in
fluen
ce w
orld
s
I Variability in final number of downloads.
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 71/80
Music Lab Experiment
0
0.2
0.4
0.6
Social Influence Indep.
Gin
i coe
ffici
ent G
Experiment 1
Social Influence Indep.
Experiment 2
I Inequality as measured by Gini coefficient:
G =1
(2Ns − 1)
Ns∑i=1
Ns∑j=1
|mi −mj |
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 72/80
Music Lab Experiment
0
0.005
0.01
0.015
SocialInfluence
Independent
Unp
redi
ctab
ility
U
Experiment 1
SocialInfluence
Independent
Experiment 2
I Unpredictability
U =1
Ns(Nw
2
) Ns∑i=1
Nw∑j=1
Nw∑k=j+1
|mi,j −mi,k |
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 73/80
Music Lab Experiment
Sensible result:I Stronger social signal leads to greater following and
greater inequality.
Peculiar result:I Stronger social signal leads to greater
unpredictability.
Very peculiar observation:
I The most unequal distributions would suggest thegreatest variation in underlying ‘quality.’
I But success may be due to social constructionthrough following...
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 74/80
Music Lab Experiment—Sneakiness
0 400 7520
100
200
300
400
500
Dow
nloa
ds
Exp. 3
Song 1
Song 481200 1600 2000 2400 2800
Exp. 4
Subjects
Song 1
Song 1
Song 1
Song 48
Song 48Song 48
Unchanged worldInverted worlds
0 400 7520
50
100
150
200
250
Dow
nloa
ds
Exp. 3
Song 2
Song 47
1200 1600 2000 2400 2800
Exp. 4
Subjects
Song 2
Song 2Song 2
Song 47
Song 47Song 47
Unchanged worldInverted worlds
I Inversion of download countI The ‘pretend rich’ get richer ...I ... but at a slower rate
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 75/80
References I
[1] M. Adler.Stardom and talent.American Economic Review, pages 208–212, 1985.pdf (�)
[2] S. Bikhchandani, D. Hirshleifer, and I. Welch.A theory of fads, fashion, custom, and culturalchange as informational cascades.J. Polit. Econ., 100:992–1026, 1992.
[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 (�)
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 76/80
References II
[4] J. Carlson and J. Doyle.Highly optimized tolerance: A mechanism for powerlaws in design systems.Phys. Rev. E, 60(2):1412–1427, 1999. pdf (�)
[5] J. Carlson and J. Doyle.Highly optimized tolerance: Robustness and designin complex systems.Phys. Rev. Lett., 84(11):2529–2532, 2000. pdf (�)
[6] I. D. Chase, C. Tovey, D. Spangler-Martin, andM. Manfredonia.Individual differences versus social dynamics in theformation of animal dominance hierarchies.Proc. Natl. Acad. Sci., 99(8):5744–5749, 2002.pdf (�)
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 77/80
References III
[7] M. Gladwell.The Tipping Point.Little, Brown and Company, New York, 2000.
[8] J. P. Gleeson.Cascades on correlated and modular randomnetworks.Phys. Rev. E, 77:046117, 2008. pdf (�)
[9] J. P. Gleeson and D. J. Cahalane.Seed size strongly affects cascades on randomnetworks.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 DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 78/80
References IV
[11] E. Hoffer.The Passionate State of Mind: And Other Aphorisms.Buccaneer Books, 1954.
[12] E. Katz and P. F. Lazarsfeld.Personal Influence.The Free Press, New York, 1955.
[13] T. Kuran.Now out of never: The element of surprise in the easteuropean revolution of 1989.World Politics, 44:7–48, 1991. pdf (�)
[14] T. Kuran.Private Truths, Public Lies: The SocialConsequences of Preference Falsification.Harvard University Press, Cambridge, MA, Reprintedition, 1997.
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 79/80
References V
[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 andunpredictability in an artificial cultural market.Science, 311:854–856, 2006. pdf (�)
[18] T. Schelling.Dynamic models of segregation.J. Math. Sociol., 1:143–186, 1971.
Contagion
Introduction
Simple DiseaseSpreading ModelsBackground
Prediction
Social ContagionModelsGranovetter’s model
Network version
Groups
Summary
Winning: it’s not foreveryoneSuperstars
Musiclab
References
Frame 80/80
References VI
[19] T. C. Schelling.Hockey helmets, concealed weapons, and daylightsaving: 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 randomnetworks.Proc. Natl. Acad. Sci., 99(9):5766–5771, 2002.pdf (�)