Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center

Social Contagion

Social ContagionModelsBackground

Granovetter’s model

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Social ContagionPrinciples of Complex Systems

CSYS/MATH 300, Fall, 2011

Prof. Peter Dodds

Department of Mathematics & Statistics | Center for Complex Systems |Vermont Advanced Computing Center | University 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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Social Contagion

Examples abound

I 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 possibleI 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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Examples abound







: dieting

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Social Contagion

Examples abound







: dieting

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Examples abound






SIR and SIRS contagion possibleI Classes of behavior versus specific behavior: dieting

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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? (�) (Clive

Thomspon, NY Times, September 10, 2009).I Everything is contagious (�)—Doubts about the

social 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.bmj.com/content/337/bmj.a2338.full

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

We need to understand influenceI Who influences whom?

Very hard to measure...

I What kinds of influence response functions arethere?

I Are some individuals super influencers?

Highly popularized by Gladwell [9] as ‘connectors’

I The infectious idea of opinion leaders (Katz andLazarsfeld) [13]

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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?



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

Lazarsfeld) [13]

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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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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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after the fact...

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after the fact...

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after the fact...

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after the fact...

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after the fact...

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after the fact...

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after the fact...

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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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parody, ...

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parody, ...

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The Mona Lisa


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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Messing with social connectionsI Ads based on message content

(e.g., Google and email)

I BzzAgent (�)I Facebook’s advertising: Beacon (�)

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http://about.bzzagent.com/

http://en.wikipedia.org/wiki/Facebook_Beacon

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(e.g., Google and email)

I BzzAgent (�)I Facebook’s advertising: Beacon (�)

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(e.g., Google and email)I BzzAgent (�)I Facebook’s advertising: Beacon (�)

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Getting others to do things for you

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

Six modes of influence1. Reciprocation: The Old Give and Take... and Take

e.g., Free samples, Hare Krishnas.2. Commitment and Consistency: Hobgoblins of the

Minde.g., Hazing.

3. Social Proof: Truths Are Use.g., Catherine Genovese, Jonestown

4. Liking: The Friendly ThiefSeparation into groups is enough to cause problems.

5. Authority: Directed DeferenceMilgram’s obedience to authority experiment.

6. Scarcity: The Rule of the FewProhibition.

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http://en.wikipedia.org/wiki/Robert_Cialdini

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Minde.g., Hazing.





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Minde.g., Hazing.





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Minde.g., Hazing.





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Minde.g., Hazing.





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

I Useful but can be leveraged...

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

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Some important modelsI 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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matter...

(unrealistic).


(unrealistic).

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matter...

(unrealistic).


(unrealistic).

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matter...

(unrealistic).


(unrealistic).

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matter...

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(unrealistic).

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matter...

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(unrealistic).

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matter...

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(unrealistic).

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

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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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∫ φ∗φ′∗=0 f (φ′∗)dφ′∗


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∫ φ∗φ′∗=0 f (φ′∗)dφ′∗


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∫ φ∗φ′∗=0 f (φ′∗)dφ′∗


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∫ φ∗φ′∗=0 f (φ′∗)dφ′∗


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Threshold models

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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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φt+1 =

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φt+1 =

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Action based on perceived behavior of others.

0 10

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f (φ∗ )

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F (

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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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γ

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I Another example of critical mass model...

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f(γ)

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I Example of single stable state model

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Implications for collective action theory:1. Collective uniformity 6⇒ individual uniformity2. Small individual changes ⇒ large global changes

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Chaotic behavior possible [12, 11]

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I Period doubling arises as map amplitude r isincreased.

I Synchronous update assumption is crucial

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References

32 of 88

Threshold models


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 )



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Social Contagion



Network version

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References

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Outline


References

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Social Contagion



Network version

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References

34 of 88

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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Network version

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References

34 of 88





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Social Contagion



Network version

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References

34 of 88





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Social Contagion



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35 of 88


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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Network version

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References

35 of 88







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Social Contagion



Network version

Groups

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References

35 of 88







http://www.uvm.edu


Social Contagion



Network version

Groups

Chaos

References

35 of 88







http://www.uvm.edu


Social Contagion



Network version

Groups

Chaos

References

35 of 88







http://www.uvm.edu


Social Contagion



Network version

Groups

Chaos

References

35 of 88







http://www.uvm.edu


Social Contagion



Network version

Groups

Chaos

References

35 of 88







http://www.uvm.edu


Social Contagion



Network version

Groups

Chaos

References

35 of 88







http://www.uvm.edu


Social Contagion



Network version

Groups

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References

35 of 88







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Social Contagion



Network version

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References

36 of 88


t=1

bc

e

a

d

I All nodes have threshold φ = 0.2.

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Social Contagion



Network version

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Chaos

References

36 of 88


t=1 t=2

c

a

b

a

bc

ee

d d


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Social Contagion



Network version

Groups

Chaos

References

36 of 88


t=1 t=2 t=3

c

a

bc

e

a

b

e

a

bc

e

d dd


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Social Contagion



Network version

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References

37 of 88

Snowballing

The Cascade Condition:1. If one individual is initially activated, what is the

probability that an activation will spread over anetwork?

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

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References

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Snowballing

The Cascade Condition:1. If one individual is initially activated, what is the

probability 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



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Social Contagion



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Groups

Chaos

References

38 of 88

Snowballing



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Social Contagion



Network version

Groups

Chaos

References

38 of 88

Snowballing



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Social Contagion



Network version

Groups

Chaos

References

38 of 88

Snowballing



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Social Contagion



Network version

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Chaos

References

39 of 88

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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References

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Snowballing





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Social Contagion



Network version

Groups

Chaos

References

39 of 88

Snowballing





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Social Contagion



Network version

Groups

Chaos

References

39 of 88

Snowballing





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Social Contagion



Network version

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Chaos

References

40 of 88

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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Social Contagion



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References

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The most gullible



1/ki ≥ φi




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Social Contagion



Network version

Groups

Chaos

References

40 of 88

The most gullible



1/ki ≥ φi




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Social Contagion



Network version

Groups

Chaos

References

40 of 88

The most gullible



1/ki ≥ φi




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Social Contagion



Network version

Groups

Chaos

References

40 of 88

The most gullible



1/ki ≥ φi




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Social Contagion



Network version

Groups

Chaos

References

40 of 88

The most gullible



1/ki ≥ φi




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Social Contagion



Network version

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Chaos

References

40 of 88

The most gullible



1/ki ≥ φi




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References

41 of 88

Cascade condition

Back to following a link:I A randomly chosen link, traversed in a random

direction, 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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References

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Cascade condition





k=0

kPk = 〈k〉


kPk

〈k〉

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Network version

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Chaos

References

41 of 88

Cascade condition





k=0

kPk = 〈k〉


kPk

〈k〉

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Chaos

References

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Cascade condition





k=0

kPk = 〈k〉


kPk

〈k〉

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Chaos

References

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Cascade condition





k=0

kPk = 〈k〉


kPk

〈k〉

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Chaos

References

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Cascade condition

Next: Vulnerability of linked nodeI 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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References

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Cascade condition


βk =

∫ 1/k

φ′∗=0

f (φ′∗)dφ′∗



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References

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Cascade condition


βk =

∫ 1/k

φ′∗=0

f (φ′∗)dφ′∗



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References

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Cascade condition


βk =

∫ 1/k

φ′∗=0

f (φ′∗)dφ′∗



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Social Contagion



Network version

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Chaos

References

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Cascade condition

Putting things together:I Expected number of active edges produced by an

active 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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Network version

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Chaos

References

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Cascade condition


active edge:

R =∞∑

k=1

(k − 1) · βk ·kPk


+

0 · (1− βk ) · kPk


=∞∑

k=1

(k − 1) · βk ·kPk

〈k〉

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Social Contagion



Network version

Groups

Chaos

References

43 of 88

Cascade condition


active edge:

R =∞∑

k=1

(k − 1) · βk ·kPk


+ 0 · (1− βk ) · kPk


=∞∑

k=1

(k − 1) · βk ·kPk

〈k〉

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Network version

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Chaos

References

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Cascade condition


active edge:

R =∞∑

k=1

(k − 1) · βk ·kPk


+ 0 · (1− βk ) · kPk


=∞∑

k=1

(k − 1) · βk ·kPk

〈k〉

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Network version

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Chaos

References

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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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References

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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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Chaos

References

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Cascade condition


β ·∞∑

k=1

(k − 1) · kPk

〈k〉≥ 1.


1 ·∞∑

k=1

(k − 1) · kPk

〈k〉≥ 1.

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Social Contagion



Network version

Groups

Chaos

References

45 of 88

Cascade condition


β ·∞∑

k=1

(k − 1) · kPk

〈k〉≥ 1.


1 ·∞∑

k=1

(k − 1) · kPk

〈k〉≥ 1.

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Social Contagion



Network version

Groups

Chaos

References

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Cascade condition


β ·∞∑

k=1

(k − 1) · kPk

〈k〉≥ 1.


1 ·∞∑

k=1

(k − 1) · kPk

〈k〉≥ 1.

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Social Contagion



Network version

Groups

Chaos

References

45 of 88

Cascade condition


β ·∞∑

k=1

(k − 1) · kPk

〈k〉≥ 1.


1 ·∞∑

k=1

(k − 1) · kPk

〈k〉≥ 1.

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Social Contagion



Network version

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Chaos

References

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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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Network version

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Chaos

References

46 of 88


1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

z

⟨ S

⟩

Example networks

PossibleNo

Cascades

Low influence


cascade sizeFinal

CascadesNo Cascades

CascadesNo

High influence




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Social Contagion



Network version

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Chaos

References

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1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

z

⟨ S

⟩

Example networks

PossibleNo

Cascades

Low influence


cascade sizeFinal

CascadesNo Cascades

CascadesNo

High influence




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Social Contagion



Network version

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Chaos

References

47 of 88

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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Network version

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References

48 of 88

Cascade window for random networks

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References

49 of 88

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 enough

vulnerables.3. Intermediate 〈k〉: Global cluster of vulnerables exists.

Cascades are possible in “Cascade window.”

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Network version

Groups

Chaos

References

49 of 88






http://www.uvm.edu


Social Contagion



Network version

Groups

Chaos

References

49 of 88






http://www.uvm.edu


Social Contagion



Network version

Groups

Chaos

References

49 of 88






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Social Contagion



Network version

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Chaos

References

50 of 88

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 0.5 10

0.2

0.4

0.6

0.8

1

a0 a’

0a

t

F (

a t+1)

C

0 10 200

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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Network version

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Chaos

References

51 of 88

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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Network version

Groups

Chaos

References

51 of 88


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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Social Contagion



Network version

Groups

Chaos

References

51 of 88


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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Social Contagion



Network version

Groups

Chaos

References

51 of 88


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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Social Contagion



Network version

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Chaos

References

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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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Social Contagion



Network version

Groups

Chaos

References

53 of 88

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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Social Contagion



Network version

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Chaos

References

54 of 88

Special subnetworks can act as triggers

i0

A

B

I φ = 1/3 for all nodes

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Social Contagion



Network version

Groups

Chaos

References

55 of 88

Outline


References

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Social Contagion



Network version

Groups

Chaos

References

56 of 88

The power of groups...

despair.com

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

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despair.com

Social Contagion



Network version

Groups

Chaos

References

57 of 88

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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Social Contagion



Network version

Groups

Chaos

References

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Extensions



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Social Contagion



Network version

Groups

Chaos

References

57 of 88

Extensions



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Social Contagion



Network version

Groups

Chaos

References

57 of 88

Extensions



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Social Contagion



Network version

Groups

Chaos

References

58 of 88

Group structure—Ramified random networks

p = intergroup connection probabilityq = intragroup connection probability.

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Network version

Groups

Chaos

References

59 of 88

Bipartite networks

c d ea b

2 3 41

a

b

c

d

e

contexts

individuals

unipartitenetwork

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Network version

Groups

Chaos

References

60 of 88

Context distance

eca

high schoolteacher

occupation

health careeducation

nurse doctorteacherkindergarten

db

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Social Contagion



Network version

Groups

Chaos

References

61 of 88

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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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

SummaryI ‘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



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Social contagion



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Social contagion



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Social contagion



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Social contagion



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Social contagion



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Social contagion



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Social contagion

ImplicationsI 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







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Social contagion







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Social contagion







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Social contagion







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Outline


References

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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.

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Chaotic contagion:






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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Chaotic contagion:






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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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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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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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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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 )

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The tent map


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 )

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The tent map


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 )

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The tent map


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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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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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 )



information

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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 )



information

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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 )



information

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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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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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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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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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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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Connectivity leads to chaos:

Stochastic response functions:

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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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δi,j f (xj,n)



⇒ synchronization


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δi,j f (xj,n)



⇒ synchronization


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References

80 of 88




δi,j f (xj,n)



⇒ synchronization


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Social Contagion



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References

80 of 88




δi,j f (xj,n)



⇒ synchronization


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δi,j f (xj,n)



⇒ synchronization


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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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References

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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 designed 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

Social Contagion



Network version

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References

83 of 88

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

Social Contagion



Network version

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References

84 of 88

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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References

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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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References

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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. C. 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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References

87 of 88

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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References

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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

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Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center