Social Contagion - Principles of Complex Systems CSYS/MATH ... · Social Contagion Social Contagion Models Background Granovetter’s model Network version Groups Chaos References

Social Contagion

Social ContagionModelsBackground

Granovetter’s model

Network version

Groups

Chaos

References

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Social ContagionPrinciples of Complex Systems

CSYS/MATH 300, Fall, 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.

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Outline

Social Contagion ModelsBackgroundGranovetter’s modelNetwork versionGroupsChaos

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Social Contagion

http://xkcd.com/610/ (�)

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http://xkcd.com/610/

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Social Contagion

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Social Contagion

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Social Contagion

Examples aboundI fashionI strikingI smoking (�) [6]

I residentialsegregation [16]

I ipodsI obesity (�) [5]

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

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http://content.nejm.org/cgi/content/short/358/21/2249

http://content.nejm.org/cgi/content/full/357/4/370

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Framingham heart study:

Evolving network stories (Christakis and Fowler):

I The spread of quitting smoking (�) [6]

I The spread of spreading (�) [5]

I Also: happiness (�) [8], loneliness, ...I The book: Connected: The Surprising Power of Our

Social Networks and How They Shape Our Lives (�)

Controversy:

I Are your friends making you fat? (�) (CliveThomspon, NY Times, September 10, 2009).

I Everything is contagious (�)—Doubts about thesocial plague stir in the human superorganism (DaveJohns, Slate, April 8, 2010).

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http://www.nejm.org/doi/full/10.1056/NEJMsa0706154

http://www.nejm.org/doi/full/10.1056/NEJMsa066082

http://www.bmj.com/content/337/bmj.a2338.full

http://www.amazon.com/Connected-Surprising-Power-Social-Networks/dp/0316036145

http://www.amazon.com/Connected-Surprising-Power-Social-Networks/dp/0316036145

http://www.nytimes.com/2009/09/13/magazine/13contagion-t.html?pagewanted=all

http://www.slate.com/id/2250102/

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Social Contagion

Two focuses for usI Widespread media influenceI Word-of-mouth influence

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Social Contagion

We need to understand influenceI Who influences whom? Very hard to measure...I What kinds of influence response functions are

there?I Are some individuals super influencers?

Highly popularized by Gladwell [9] as ‘connectors’I The infectious idea of opinion leaders (Katz and

Lazarsfeld) [13]

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The hypodermic model of influence

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The two step model of influence [13]

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The general model of influence

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Social Contagion

Why do things spread?

I Because of properties of special individuals?I Or system level properties?I Is the match that lights the fire important?I Yes. But only because we are narrative-making

machines...I We like to think things happened for reasons...I Reasons for success are usually ascribed to intrinsic

properties (e.g., Mona Lisa)I System/group properties harder to understandI Always good to examine what is said before and

after the fact...

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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, ...

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The completely unpredicted fall of EasternEurope

Timur Kuran: [14, 15] “Now Out of Never: The Element ofSurprise in the East European Revolution of 1989”

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The dismal predictive powers of editors...

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Social Contagion

Messing with social connections

I Ads based on message content(e.g., Google and email)

I Buzz mediaI Facebook’s advertising: Beacon (�)

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http://en.wikipedia.org/wiki/Beacon_(Facebook)

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Getting others to do things for you

A very good book: ‘Influence’ by Robert Cialdini [7]

Six modes of influence

1. Reciprocation: The Old Give and Take... and Take2. Commitment and Consistency: Hobgoblins of the

Mind3. Social Proof: Truths Are Us4. Liking: The Friendly Thief5. Authority: Directed Deference6. Scarcity: The Rule of the Few

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Examples

I Reciprocation: Free samples, Hare KrishnasI Commitment and Consistency: HazingI Social Proof: Catherine Genovese, JonestownI Liking: Separation into groups is enough to cause

problems.I Authority: Milgram’s obedience to authority

experiment.I Scarcity: Prohibition.

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Getting others to do things for you

I Cialdini’s modes are heuristics that help up us getthrough life.

I Useful but can be leveraged...

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Social Contagion

Other acts of influenceI Conspicuous Consumption (Veblen, 1912)I Conspicuous Destruction (Potlatch)

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Social Contagion

Some important models

I T ipping models—Schelling (1971) [16, 17, 18]

I Simulation on checker boardsI Idea of thresholdsI Explore the Netlogo (�) implementation [21]

I Threshold models—Granovetter (1978) [10]

I Herding models—Bikhchandani, Hirschleifer, Welch(1992) [1, 2]

I Social learning theory, Informational cascades,...

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http://ccl.northwestern.edu/netlogo/

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Social contagion models

ThresholdsI 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 can varyI Assumption: order of others’ adoption does not

matter... (unrealistic).I Assumption: level of influence per person is uniform

(unrealistic).

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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 externalitiesI Externalities = Effects on others not directly involved

in a transactionI Examples: telephones, fax machine, Facebook,

operating systemsI An individual’s utility increases with the adoption

level among peers and the population in general

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Granovetter’s Threshold model—definitionsI φ∗ = threshold of an individual.I f (φ∗) = distribution of thresholds in a population.I F (φ∗) = cumulative distribution =

∫ φ∗φ′∗=0 f (φ′∗)dφ′∗

I φt = fraction of people ‘rioting’ at time step t .

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Threshold models

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

φ

p

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

φp

I Example threshold influence response functions:deterministic and stochastic

I φ = fraction of contacts ‘on’ (e.g., rioting)I Two states: S and I.

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Threshold models

I At time t + 1, fraction rioting = fraction with φ∗ ≤ φt .I

φt+1 =

∫ φt

0f (φ∗)dφ∗ = F (φ∗)|φt

0 = F (φt)

I ⇒ Iterative maps of the unit interval [0, 1].

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Threshold models

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 Discrete time update (strong assumption!)I This is a Critical mass model

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Threshold models

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

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 Another example of critical mass model...

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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

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Threshold models

Implications for collective action theory:

1. Collective uniformity 6⇒ individual uniformity2. Small individual changes ⇒ large global changes

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Threshold models

Chaotic behavior possible [12, 11]

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

xn

F (

x n+1 )

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

xn

F (

x n+1 )

I Period doubling arises as map amplitude r isincreased.

I Synchronous update assumption is crucial

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Threshold model on a network

Many years after Granovetter and Soong’s work:

“A simple model of global cascades on random networks”D. J. Watts. Proc. Natl. Acad. Sci., 2002 [20]

I Mean field model → network modelI Individuals now have a limited view of the world

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I Interactions between individuals now represented bya network

I Network is sparseI Individual i has ki contactsI Influence on each link is reciprocal and of unit weightI Each individual i has a fixed threshold φi

I Individuals repeatedly poll contacts on networkI Synchronous, discrete time updatingI Individual i becomes active when

fraction of active contacts aiki≥ φi

I Individuals remain active when switched (norecovery = SI model)

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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.

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Snowballing

The Cascade Condition:

1. If one individual is initially activated, what is theprobability that an activation will spread over anetwork?

2. What features of a network determine whether acascade will occur or not?

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Snowballing

First study random networks:

I Start with N nodes with a degree distribution pk

I Nodes are randomly connected (carefully so)I Aim: Figure out when activation will propagateI Determine a cascade condition

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Snowballing

Follow active linksI An active link is a link connected to an activated

node.I If an infected link leads to at least 1 more infected

link, then activation spreads.I We need to understand which nodes can be

activated when only one of their neigbors becomesactive.

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The most gullible

Vulnerables:I We call individuals who can be activated by just one

contact being active vulnerablesI The vulnerability condition for node i :

1/ki ≥ φi

I Which means # contacts ki ≤ b1/φicI For global cascades on random networks, must have

a global cluster of vulnerables [20]

I Cluster of vulnerables = critical massI Network story: 1 node → critical mass → everyone.

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Cascade condition

Back to following a link:

I A randomly chosen link, traversed in a randomdirection, leads to a degree k node with probability∝ kPk .

I Follows from there being k ways to connect to anode with degree k .

I Normalization:∞∑

k=0

kPk = 〈k〉

I SoP(linked node has degree k) =

kPk

〈k〉

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Cascade condition

Next: Vulnerability of linked node

I Linked node is vulnerable with probability

βk =

∫ 1/k

φ′∗=0

f (φ′∗)dφ′∗

I If linked node is vulnerable, it produces k − 1 newoutgoing active links

I If linked node is not vulnerable, it produces no activelinks.

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Cascade condition

Putting things together:

I Expected number of active edges produced by anactive edge:

R =∞∑

k=1

(k − 1) · βk ·kPk

〈k〉︸︷︷︸success

+ 0 · (1− βk ) · kPk

〈k〉︸︷︷︸failure

=∞∑

k=1

(k − 1) · βk ·kPk

〈k〉

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Cascade condition

So... for random networks with fixed degree distributions,cacades take off when:

∞∑k=1

(k − 1) · βk ·kPk

〈k〉≥ 1.

I βk = probability a degree k node is vulnerable.I Pk = probability a node has degree k .

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Cascade condition

Two special cases:

I (1) Simple disease-like spreading succeeds: βk = β

β ·∞∑

k=1

(k − 1) · kPk

〈k〉≥ 1.

I (2) Giant component exists: β = 1

1 ·∞∑

k=1

(k − 1) · kPk

〈k〉≥ 1.

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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.

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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.

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Cascade window for random networks

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Cascade window—summary

For our simple model of a uniform threshold:

1. Low 〈k〉: No cascades in poorly connected networks.No global clusters of any kind.

2. High 〈k〉: Giant component exists but not enoughvulnerables.

3. Intermediate 〈k〉: Global cluster of vulnerables exists.Cascades are possible in “Cascade window.”

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All-to-all versus random networks

0

0.2

0.4

0.6

0.8

1

a0 a

t

F (

a t+1)

all−to−all networks

A

0

0.2

0.4

0.6

0.8

1

⟨ k ⟩

⟨ S ⟩

random networks

B

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

a0 a’

0at

F (

a t+1)

C

0 5 10 15 200

0.2

0.4

0.6

0.8

1

⟨ k ⟩

⟨ S ⟩

D

0 0.5 10

5

10

φ∗

f (φ ∗)

0 0.5 10

2

4

φ∗

f (φ ∗)

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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

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The multiplier effect:

1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

navg

S avg

A

1 2 3 4 5 60

1

2

3

4

navg

B

Gain

InfluenceInfluence

Averageindividuals

Top 10% individuals

Cas

cade

siz

e

Cascade size ratio

Degree ratio

I Fairly uniform levels of individual influence.I Multiplier effect is mostly below 1.

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The multiplier effect:

1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

navg

S avg

A

1 2 3 4 5 60

3

6

9

12

navg

B

Cas

cade

siz

e

InfluenceAverageIndividuals

Top 10% individuals Cascade size ratio

Degree ratio

Gain

I Skewed influence distribution example.

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Special subnetworks can act as triggers

i0

A

B

I φ = 1/3 for all nodes

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The power of groups...

despair.com

“A few harmless flakesworking together canunleash an avalancheof destruction.”

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despair.com

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Extensions

I Assumption of sparse interactions is goodI Degree distribution is (generally) key to a network’s

functionI Still, random networks don’t represent all networksI Major element missing: group structure

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Group structure—Ramified random networks

p = intergroup connection probabilityq = intragroup connection probability.

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Bipartite networks

c d ea b

2 3 41

a

b

c

d

e

contexts

individuals

unipartitenetwork

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Context distance

eca

high schoolteacher

occupation

health careeducation

nurse doctorteacherkindergarten

db

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Generalized affiliation model

100

eca b d

geography occupation age

0

(Blau & Schwartz, Simmel, Breiger)

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Generalized affiliation model networks withtriadic closure

I Connect nodes with probability ∝ exp−αd

whereα = homophily parameterandd = distance between nodes (height of lowestcommon ancestor)

I τ1 = intergroup probability of friend-of-friendconnection

I τ2 = intragroup probability of friend-of-friendconnection

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Cascade windows for group-based networksG

ener

alize

d Af

filiat

ion

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

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Multiplier effect for group-based networks:

4 8 12 16 200

0.2

0.4

0.6

0.8

1

navg

S avg

A

4 8 12 16 200

1

2

3

navg

B

0 4 8 12 160

0.2

0.4

0.6

0.8

1

navg

S avg

C

0 4 8 12 160

1

2

3

navg

D

Degree ratio

Gain

Cascadesize ratio

Cascadesize ratio < 1!

I Multiplier almost always below 1.

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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.

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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 Seems that cascade condition is a global one.I Most extreme/unexpected cascades occur in highly

connected networksI ‘Influentials’ are posterior constructs.I Many potential influentials exist.

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Social Contagion



Network version

Groups

Chaos

References

69 of 90

Social contagion

Implications

I Focus on the influential vulnerables.I Create entities that can be transmitted successfully

through many individuals rather than broadcast fromone ‘influential.’

I Only simple ideas can spread by word-of-mouth.(Idea of opinion leaders spreads well...)

I Want enough individuals who will adopt and display.I Displaying can be passive = free (yo-yo’s, fashion),

or active = harder to achieve (political messages).I Entities can be novel or designed to combine with

others, e.g. block another one.

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Social Contagion



Network version

Groups

Chaos

References

71 of 90

Chaotic contagion:

I What if individual response functions are notmonotonic?

I Consider a simple deterministic version:

I Node i has an ‘activation threshold’ φi,1

. . . and a ‘de-activation threshold’ φi,2

I Nodes like to imitate but only up to alimit—they don’t want to be likeeveryone else.

0 10

0.2

0.4

0.6

0.8

1

!i,1

"!i,2

"

A

!i,t

Pr(ai,t+1=1)

0 0.5 10

0.2

0.4

0.6

0.8

1B

!1

"

!2"

0 0.5 10

0.2

0.4

0.6

0.8

1

!1

"

!2"

C

0 0.5 10

0.5

1

0 0.5 10

0.5

1

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Social Contagion



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72 of 90

Two population examples:

0 10

0.2

0.4

0.6

0.8

1

φi,1∗ φ

i,2∗

A

φi,t

Pr(a

i,t+

1=1)

0 0.5 10

0.2

0.4

0.6

0.8

1B

φ1∗

φ 2∗

0 0.5 10

0.2

0.4

0.6

0.8

1

φ1∗

φ 2∗

C

0 0.5 10

0.5

1

0 0.5 10

0.5

1

I Randomly select (φi,1, φi,2) from gray regions shownin plots B and C.

I Insets show composite response function averagedover population.

I We’ll consider plot C’s example: the tent map.

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Social Contagion



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73 of 90

Chaotic contagion

Definition of the tent map:

F (x) =

{rx for 0 ≤ x ≤ 1

2 ,

r(1− x) for 12 ≤ x ≤ 1.

I The usual business: look at how F iteratively mapsthe unit interval [0, 1].

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Social Contagion



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74 of 90

The tent map

Effect of increasing r from 1 to 2.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

xn

F (

x n+1 )

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

xn

F (

x n+1 )

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

xn

F (

x n+1 )

Orbit diagram:Chaotic behavior increasesas map slope r is increased.

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Social Contagion



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75 of 90

Chaotic behavior

Take r = 2 case:

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

xn

F (

x n+1 )

I What happens if nodes have limited information?I As before, allow interactions to take place on a

sparse random network.I Vary average degree z = 〈k〉, a measure of

information

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76 of 90

Invariant densities—stochastic responsefunctions

0 500 1000 1500 20000

0.2

0.4

0.6

0.8

1

t

s

z = 5

0 0.5 10

20

40

60

sP(

s)

z = 5

activation time series activation density

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Social Contagion



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0 500 1000 1500 20000

0.2

0.4

0.6

0.8

1

t

s

z = 5

0 0.5 10

20

40

60

s

P(s)

z = 5

0 500 1000 1500 20000

0.2

0.4

0.6

0.8

1

t

s

z = 10

0 0.5 10

10

20

30

s

P(s)

z = 10

0 500 1000 1500 20000

0.2

0.4

0.6

0.8

1

t

s

z = 18

0 0.5 10

5

10

15

20

s

P(s)

z = 18

0 500 1000 1500 20000

0.2

0.4

0.6

0.8

1

t

s

z = 22

0 0.5 10

2

4

6

8

10

s

P(s)

z = 22

0 500 1000 1500 20000

0.2

0.4

0.6

0.8

1

t

s

z = 24

0 0.5 10

2

4

6

8

s

P(s)

z = 24

0 500 1000 1500 20000

0.2

0.4

0.6

0.8

1

t

s

z = 30

0 0.5 10

2

4

6

sP(

s)

z = 30

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Social Contagion



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78 of 90

Invariant densities—deterministic responsefunctions for one specific network with〈k〉 = 18

0 500 1000 1500 20000

0.2

0.4

0.6

0.8

1

t

s

z = 18

0 0.5 10

10

20

30

s

P(s)

z = 18

0 500 1000 1500 20000

0.2

0.4

0.6

0.8

1

t

s

z = 18

0 0.5 10

5

10

15

20

25

s

P(s)

z = 18

0 500 1000 1500 20000

0.2

0.4

0.6

0.8

1

t

s

z = 18

0 0.5 10

10

20

30

40

50

s

P(s)

z = 18

0 500 1000 1500 20000

0.2

0.4

0.6

0.8

1

t

s

z = 18

0 0.5 10

10

20

30

40

50

s

P(s)

z = 18

0 500 1000 1500 20000

0.2

0.4

0.6

0.8

1

t

s

z = 18

0 0.5 10

2

4

6

8

s

P(s)

z = 18

0 500 1000 1500 20000

0.2

0.4

0.6

0.8

1

t

s

z = 18

0 0.5 10

10

20

30

s

P(s)

z = 18

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Social Contagion



Network version

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References

79 of 90


0 500 1000 1500 20000

0.2

0.4

0.6

0.8

1

t

s

z = 100

0 0.5 10

1

2

3

4

s

P(s)

z = 100

0 500 1000 1500 20000

0.2

0.4

0.6

0.8

1

t

s

z = 1000

0 0.5 10

0.5

1

1.5

2

2.5

s

P(s)

z = 1000

Trying out higher values of 〈k〉. . .

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Social Contagion



Network version

Groups

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References

80 of 90

Invariant densities—deterministic responsefunctions

0 500 1000 1500 20000

0.2

0.4

0.6

0.8

1

t

s

z = 100

0 0.5 10

1

2

3

4

s

P(s)

z = 100

0 500 1000 1500 20000

0.2

0.4

0.6

0.8

1

t

s

z = 1000

0 0.5 10

0.5

1

1.5

2

2.5

s

P(s)

z = 1000

0 5000 100000

0.2

0.4

0.6

0.8

1

t

s

z = 100

0 0.5 10

10

20

30

s

P(s)

z = 100

Trying out higher values of 〈k〉. . .

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Social Contagion



Network version

Groups

Chaos

References

81 of 90

Connectivity leads to chaos:

Stochastic response functions:

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Social Contagion



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82 of 90

Chaotic behavior in coupled systems

Coupled maps are well explored(Kaneko/Kuramoto):

xi,n+1 = f (xi,n) +∑j∈Ni

δi,j f (xj,n)

I Ni = neighborhood of node i

1. Node states are continuous2. Increase δ and neighborhood size |N |

⇒ synchronization

But for contagion model:

1. Node states are binary2. Asynchrony remains as connectivity increases

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Social Contagion



Network version

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References

83 of 90

Bifurcation diagram: Asynchronous updating

0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

α

P(s

| r)/

max

P(s

| r)

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Social Contagion



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84 of 90

References I

[1] 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.

[2] 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 (�)

[3] J. M. Carlson and J. Doyle.Highly optimized tolerance: A mechanism for powerlaws in design systems.Phys. Rev. E, 60(2):1412–1427, 1999. pdf (�)

http://www.uvm.edu


http://www.uvm.edu/~pdodds/research/papers/others/1998/bikhchandani1998a.pdf

http://www.uvm.edu/~pdodds/research/papers/others/1999/carlson1999a.pdf

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85 of 90

References II

[4] J. M. Carlson and J. Doyle.Highly optimized tolerance: Robustness and designin complex systems.Phys. Rev. Lett., 84(11):2529–2532, 2000. pdf (�)

[5] N. A. Christakis and J. H. Fowler.The spread of obesity in a large social network over32 years.New England Journal of Medicine, 357:370–379,2007. pdf (�)

[6] N. A. Christakis and J. H. Fowler.The collective dynamics of smoking in a large socialnetwork.New England Journal of Medicine, 358:2249–2258,2008. pdf (�)

http://www.uvm.edu


http://www.uvm.edu/~pdodds/research/papers/others/2000/carlson2000a.pdf

http://www.uvm.edu/~pdodds/research/papers/others/2007/christakis2007a.pdf

http://www.uvm.edu/~pdodds/research/papers/others/2008/christakis2008a.pdf

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86 of 90

References III

[7] R. B. Cialdini.Influence: Science and Practice.Allyn and Bacon, Boston, MA, 4th edition, 2000.

[8] J. H. Fowler and N. A. Christakis.Dynamic spread of happiness in a large socialnetwork: longitudinal analysis over 20 years in theFramingham Heart Study.BMJ, 337:article #2338, 2008. pdf (�)

[9] M. Gladwell.The Tipping Point.Little, Brown and Company, New York, 2000.

[10] M. Granovetter.Threshold models of collective behavior.Am. J. Sociol., 83(6):1420–1443, 1978. pdf (�)

http://www.uvm.edu


http://www.uvm.edu/~pdodds/research/papers/others/2008/fowler2008a.pdf

http://www.uvm.edu/~pdodds/research/papers/others/1978/granovetter1978a.pdf

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87 of 90

References IV

[11] M. Granovetter and R. Soong.Threshold models of diversity: Chinese restaurants,residential segregation, and the spiral of silence.Sociological Methodology, 18:69–104, 1988. pdf (�)

[12] M. S. Granovetter and R. Soong.Threshold models of interpersonal effects inconsumer demand.Journal of Economic Behavior & Organization,7:83–99, 1986.Formulates threshold as function of price, andintroduces exogenous supply curve. pdf (�)

[13] E. Katz and P. F. Lazarsfeld.Personal Influence.The Free Press, New York, 1955.

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88 of 90

References V

[14] T. Kuran.Now out of never: The element of surprise in theeast european revolution of 1989.World Politics, 44:7–48, 1991. pdf (�)

[15] T. Kuran.Private Truths, Public Lies: The SocialConsequences of Preference Falsification.Harvard University Press, Cambridge, MA, Reprintedition, 1997.

[16] T. Schelling.Dynamic models of segregation.J. Math. Sociol., 1:143–186, 1971.

http://www.uvm.edu


http://www.uvm.edu/~pdodds/research/papers/others/1991/kuran1991a.pdf

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89 of 90

References VI

[17] T. C. Schelling.Hockey helmets, concealed weapons, and daylightsaving: A study of binary choices with externalities.J. Conflict Resolut., 17:381–428, 1973. pdf (�)

[18] T. C. Schelling.Micromotives and Macrobehavior.Norton, New York, 1978.

[19] D. Sornette.Critical Phenomena in Natural Sciences.Springer-Verlag, Berlin, 2nd edition, 2003.

[20] D. J. Watts.A simple model of global cascades on randomnetworks.Proc. Natl. Acad. Sci., 99(9):5766–5771, 2002.pdf (�)

http://www.uvm.edu


http://www.uvm.edu/~pdodds/research/papers/others/1973/schelling1973a.pdf

http://www.uvm.edu/~pdodds/research/papers/others/2002/watts2002a.pdf

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90 of 90

References VII

[21] U. Wilensky.Netlogo segregation model.http://ccl.northwestern.edu/netlogo/models/Segregation. Center for ConnectedLearning and Computer-Based Modeling,Northwestern University, Evanston, IL., 1998.

http://www.uvm.edu


http://ccl.northwestern.edu/netlogo/models/Segregation

http://ccl.northwestern.edu/netlogo/models/Segregation

Documents

Social Contagion - Principles of Complex Systems CSYS/MATH ... · Social Contagion Social Contagion Models Background Granovetter’s model Network version Groups Chaos References