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Social Contagion Social Contagion Social Contagion Models Principles of Complex Systems Background Granovetters model Course 300, Fall, 2008 Network version Groups Chaos References Prof. Peter Dodds Department of Mathematics &

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Social Contagion Social Contagion Models

Background Granovetter’s model Network version Groups Chaos

References Frame 1/86

Social Contagion

Principles of Complex Systems Course 300, Fall, 2008

  • Prof. Peter Dodds

Department of Mathematics & Statistics University of Vermont

Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.

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Social Contagion Social Contagion Models

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Outline

Social Contagion Models Background Granovetter’s model Network version Groups Chaos References

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Social Contagion Social Contagion Models

Background Granovetter’s model Network version Groups Chaos

References Frame 4/86

Social Contagion

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Social Contagion Social Contagion Models

Background Granovetter’s model Network version Groups Chaos

References Frame 5/86

Social Contagion

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Social Contagion Social Contagion Models

Background Granovetter’s model Network version Groups Chaos

References Frame 6/86

Social Contagion

Examples abound

◮ fashion ◮ striking ◮ smoking (⊞) [6] ◮ residential

segregation [15]

◮ ipods ◮ obesity (⊞) [5] ◮ Harry Potter ◮ voting ◮ gossip ◮ Rubik’s cube ◮ religious beliefs ◮ leaving lectures

SIR and SIRS contagion possible

◮ Classes of behavior versus specific behavior: dieting

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Social Contagion Social Contagion Models

Background Granovetter’s model Network version Groups Chaos

References Frame 7/86

Framingham heart study:

Evolving network stories:

◮ The spread of quitting smoking (⊞) [6] ◮ The spread of spreading (⊞) [5]

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Social Contagion Social Contagion Models

Background Granovetter’s model Network version Groups Chaos

References Frame 8/86

Social Contagion

Two focuses for us

◮ Widespread media influence ◮ Word-of-mouth influence

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Social Contagion Social Contagion Models

Background Granovetter’s model Network version Groups Chaos

References Frame 9/86

Social Contagion

We need to understand influence

◮ Who influences whom? Very hard to measure... ◮ What kinds of influence response functions are

there?

◮ Are some individuals super influencers?

Highly popularized by Gladwell [8] as ‘connectors’

◮ The infectious idea of opinion leaders (Katz and

Lazarsfeld) [12]

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Social Contagion Social Contagion Models

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References Frame 10/86

The hypodermic model of influence

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Social Contagion Social Contagion Models

Background Granovetter’s model Network version Groups Chaos

References Frame 11/86

The two step model of influence [12]

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Social Contagion Social Contagion Models

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References Frame 12/86

The general model of influence

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Social Contagion Social Contagion Models

Background Granovetter’s model Network version Groups Chaos

References Frame 13/86

Social Contagion

Why do things spread?

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

machines...

◮ We like to think things happened for reasons... ◮ System/group properties harder to understand ◮ Always good to examine what is said before and

after the fact...

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Social Contagion Social Contagion Models

Background Granovetter’s model Network version Groups Chaos

References Frame 14/86

The Mona Lisa

◮ “Becoming Mona Lisa: The Making of a Global

Icon”—David Sassoon

◮ Not the world’s greatest painting from the start... ◮ Escalation through theft, vandalism, parody, ...

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References Frame 15/86

The completely unpredicted fall of Eastern Europe

Timur Kuran: [13, 14] “Now Out of Never: The Element of Surprise in the East European Revolution of 1989”

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Social Contagion Social Contagion Models

Background Granovetter’s model Network version Groups Chaos

References Frame 16/86

The dismal predictive powers of editors...

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Social Contagion Social Contagion Models

Background Granovetter’s model Network version Groups Chaos

References Frame 17/86

Social Contagion

Messing with social connections

◮ Ads based on message content

(e.g., Google and email)

◮ Buzz media ◮ Facebook’s advertising: Beacon (⊞)

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Social Contagion Social Contagion Models

Background Granovetter’s model Network version Groups Chaos

References Frame 18/86

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 Take
  • 2. Commitment and Consistency: Hobgoblins of the

Mind

  • 3. Social Proof: Truths Are Us
  • 4. Liking: The Friendly Thief
  • 5. Authority: Directed Deference
  • 6. Scarcity: The Rule of the Few
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References Frame 19/86

Examples

◮ Reciprocation: Free samples, Hare Krishnas ◮ Commitment and Consistency: Hazing ◮ Social Proof: Catherine Genovese, Jonestown ◮ Liking: Separation into groups is enough to cause

problems.

◮ Authority: Milgram’s obedience to authority

experiment.

◮ Scarcity: Prohibition.

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References Frame 20/86

Getting others to do things for you

◮ Cialdini’s modes are heuristics that help up us get

through life.

◮ Useful but can be leveraged...

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Social Contagion Social Contagion Models

Background Granovetter’s model Network version Groups Chaos

References Frame 21/86

Social Contagion

Other acts of influence

◮ Conspicuous Consumption (Veblen, 1912) ◮ Conspicuous Destruction (Potlatch)

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Background Granovetter’s model Network version Groups Chaos

References Frame 22/86

Social Contagion

Some important models

◮ Tipping models—Schelling (1971) [15, 16, 17]

◮ Simulation on checker boards ◮ Idea of thresholds ◮ Fun with Netlogo and Schelling’s model [20]...

◮ Threshold models—Granovetter (1978) [9] ◮ Herding models—Bikhchandani, Hirschleifer, Welch

(1992) [1, 2]

◮ Social learning theory, Informational cascades,...

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Social Contagion Social Contagion Models

Background Granovetter’s model Network version Groups Chaos

References Frame 23/86

Social contagion models

Thresholds

◮ Basic idea: individuals adopt a behavior when a

certain fraction of others have adopted

◮ ‘Others’ may be everyone in a population, an

individual’s close friends, any reference group.

◮ Response can be probabilistic or deterministic. ◮ Individual thresholds can vary ◮ Assumption: order of others’ adoption does not

matter... (unrealistic).

◮ Assumption: level of influence per person is uniform

(unrealistic).

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References Frame 24/86

Social Contagion

Some possible origins of thresholds:

◮ Desire to coordinate, to conform. ◮ Lack of information: impute the worth of a good or

behavior based on degree of adoption (social proof)

◮ Economics: Network effects or network externalities ◮ Externalities = Effects on others not directly involved

in a transaction

◮ Examples: telephones, fax machine, Facebook,

  • perating systems

◮ An individual’s utility increases with the adoption

level among peers and the population in general

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References Frame 26/86

Social Contagion

Granovetter’s Threshold model—definitions

◮ φ∗ = threshold of an individual. ◮ f(φ∗) = distribution of thresholds in a population. ◮ F(φ∗) = cumulative distribution =

φ∗

φ′

∗=0 f(φ′

∗)dφ′ ∗ ◮ φt = fraction of people ‘rioting’ at time step t.

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Social Contagion Social Contagion Models

Background Granovetter’s model Network version Groups Chaos

References Frame 27/86

Threshold models

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

φ p

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

φ p

◮ Example threshold influence response functions:

deterministic and stochastic

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

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References Frame 28/86

Threshold models

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

φt+1 = φt f(φ∗)dφ∗ = F(φ∗)|φt

0 = F(φt) ◮ ⇒ Iterative maps of the unit interval [0, 1].

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References Frame 29/86

Threshold models

Action based on perceived behavior of others.

1 0.2 0.4 0.6 0.8 1 φi

∗

A

φi,t Pr(ai,t+1=1)

0.5 1 0.5 1 1.5 2 2.5 B

φ∗ f (φ∗)

0.5 1 0.2 0.4 0.6 0.8 1

φt φt+1 = F (φt)

C

◮ Two states: S and I. ◮ φ = fraction of contacts ‘on’ (e.g., rioting) ◮ Discrete time update (strong assumption!) ◮ This is a Critical mass model

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Social Contagion Social Contagion Models

Background Granovetter’s model Network version Groups Chaos

References Frame 30/86

Threshold models

0.2 0.4 0.6 0.8 1 0.5 1 1.5 2

γ f(γ)

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

φt φt+1

◮ Another example of critical mass model...

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References Frame 31/86

Threshold models

0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 2.5 3

γ f(γ)

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

φt φt+1

◮ Example of single stable state model

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References Frame 32/86

Threshold models

Implications for collective action theory:

  • 1. Collective uniformity ⇒ individual uniformity
  • 2. Small individual changes ⇒ large global changes
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Social Contagion Social Contagion Models

Background Granovetter’s model Network version Groups Chaos

References Frame 33/86

Threshold models

Chaotic behavior possible [11, 10]

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

xn F ( xn+1 )

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

xn F ( xn+1 )

◮ Period doubling arises as map amplitude r is

increased.

◮ Synchronous update assumption is crucial

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Background Granovetter’s model Network version Groups Chaos

References Frame 35/86

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 [19]

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

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Background Granovetter’s model Network version Groups Chaos

References Frame 36/86

Threshold model on a network

◮ Interactions between individuals now represented by

a network

◮ Network is sparse ◮ Individual i has ki contacts ◮ Influence on each link is reciprocal and of unit weight ◮ Each individual i has a fixed threshold φi ◮ Individuals repeatedly poll contacts on network ◮ Synchronous, discrete time updating ◮ Individual i becomes active when

fraction of active contacts ai ≥ φiki

◮ Individuals remain active when switched (no

recovery = SI model)

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References Frame 37/86

Threshold model on a network

t=1

b c e a d

t=1

c a b e d

◮ All nodes have threshold φ = 0.2.

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References Frame 38/86

Snowballing

The Cascade Condition: If one individual is initially activated, what is the probability that an activation will spread over a network? What features of a network determine whether a cascade will occur or not?

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References Frame 39/86

Snowballing

First study random networks:

◮ Start with N nodes with a degree distribution pk ◮ Nodes are randomly connected (carefully so) ◮ Aim: Figure out when activation will propagate ◮ Determine a cascade condition

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References Frame 40/86

Snowballing

Follow active links

◮ An active link is a link connected to an activated

node.

◮ If an infected link leads to at least 1 more infected

link, then activation spreads.

◮ We need to understand which nodes can be

activated when only one of their neigbors becomes active.

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Social Contagion Social Contagion Models

Background Granovetter’s model Network version Groups Chaos

References Frame 41/86

The most gullible

Vulnerables:

◮ We call individuals who can be activated by just one

contact being active vulnerables

◮ The vulnerability condition for node i:

1/ki ≥ φi

◮ Which means # contacts ki ≤ ⌊1/φi⌋ ◮ For global cascades on random networks, must have

a global cluster of vulnerables [19]

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

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References Frame 42/86

Cascade condition

Back to following a link:

◮ Link from leads to a node with probability ∝ kPk. ◮ Follows from links being random + having k chances

to connect to a node with degree k.

◮ Normalization: ∞

  • k=0

kPk = k = z

◮ So

P(linked node has degree k) = kPk k

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Social Contagion Social Contagion Models

Background Granovetter’s model Network version Groups Chaos

References Frame 43/86

Cascade condition

Next: Vulnerability of linked node

◮ Linked node is vulnerable with probability

βk = 1/k

φ′

∗=0

f(φ′

∗)dφ′ ∗ ◮ If linked node is vulnerable, it produces k − 1 new

  • utgoing active links

◮ If linked node is not vulnerable, it produces no active

links.

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Background Granovetter’s model Network version Groups Chaos

References Frame 44/86

Cascade condition

Putting things together:

◮ Expected number of active edges produced by an

active edge =

◮ ∞

  • k=1

(k − 1)βk kPk z

  • success

+ 0(1 − βk)kPk z

  • failure

◮

=

∞

  • k=1

(k − 1)kβkPk/z

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References Frame 45/86

Cascade condition

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

∞

  • k=1

k(k − 1)βkPk/z ≥ 1.

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

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Background Granovetter’s model Network version Groups Chaos

References Frame 46/86

Cascade condition

Two special cases:

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

β

∞

  • k=1

k(k − 1)Pk/z ≥ 1.

◮ (2) Giant component exists: β = 1 ◮ ∞

  • k=1

k(k − 1)Pk/z ≥ 1.

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Social Contagion Social Contagion Models

Background Granovetter’s model Network version Groups Chaos

References Frame 47/86

Cascades on random networks

1 2 3 4 5 6 7 0.2 0.4 0.6 0.8 1

z 〈 S 〉

Example networks Possible No Cascades Low influence

Fraction of Vulnerables cascade size Final

Cascades No Cascades Cascades No High influence

◮ Cascades occur

  • nly if size of max

vulnerable cluster > 0.

◮ System may be

‘robust-yet-fragile’.

◮ ‘Ignorance’

facilitates spreading.

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Social Contagion Social Contagion Models

Background Granovetter’s model Network version Groups Chaos

References Frame 48/86

Cascade window for random networks

1 2 3 4 5 6 7 0.2 0.4 0.6 0.8 1

z 〈 S 〉

0.05 0.1 0.15 0.2 0.25 5 10 15 20 25 30

φ z

cascades no cascades

influence = uniform individual threshold

◮ ‘Cascade window’ widens as threshold φ decreases. ◮ Lower thresholds enable spreading.

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Social Contagion Social Contagion Models

Background Granovetter’s model Network version Groups Chaos

References Frame 49/86

Cascade window for random networks

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Background Granovetter’s model Network version Groups Chaos

References Frame 50/86

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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Social Contagion Social Contagion Models

Background Granovetter’s model Network version Groups Chaos

References Frame 51/86

All-to-all versus random networks

0.2 0.4 0.6 0.8 1

a0

at F (at+1) all−to−all networks A

0.2 0.4 0.6 0.8 1

〈 k 〉 〈 S 〉 random networks B

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

a0 a’0

at F (at+1) C

5 10 15 20 0.2 0.4 0.6 0.8 1

〈 k 〉 〈 S 〉 D

0.5 1 5 10 φ∗ f (φ∗) 0.5 1 2 4 φ∗ f (φ∗)

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Background Granovetter’s model Network version Groups Chaos

References Frame 52/86

Early adopters—degree distributions

t = 0 t = 1 t = 2 t = 3

5 10 15 20 0.05 0.1 0.15 0.2

t = 0

5 10 15 20 0.2 0.4 0.6 0.8

t = 1

5 10 15 20 0.2 0.4 0.6 0.8

t = 2

5 10 15 20 0.2 0.4 0.6 0.8

t = 3

t = 4 t = 6 t = 8 t = 10

5 10 15 20 0.1 0.2 0.3 0.4 0.5

t = 4

5 10 15 20 0.1 0.2 0.3 0.4 0.5

t = 6

5 10 15 20 0.1 0.2 0.3 0.4

t = 8

5 10 15 20 0.1 0.2 0.3 0.4

t = 10

t = 12 t = 14 t = 16 t = 18

5 10 15 20 0.05 0.1 0.15 0.2

t = 12

5 10 15 20 0.05 0.1 0.15 0.2

t = 14

5 10 15 20 0.05 0.1 0.15 0.2

t = 16

5 10 15 20 0.05 0.1 0.15 0.2

t = 18

Pk,t versus k

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References Frame 53/86

The multiplier effect:

1 2 3 4 5 6 0.2 0.4 0.6 0.8 1

navg Savg A

1 2 3 4 5 6 1 2 3 4

navg B

Gain Influence Influence Average individuals Top 10% individuals Cascade size Cascade size ratio Degree ratio ◮ Fairly uniform levels of individual influence. ◮ Multiplier effect is mostly below 1.

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References Frame 54/86

The multiplier effect:

1 2 3 4 5 6 0.2 0.4 0.6 0.8 1

navg Savg A

1 2 3 4 5 6 3 6 9 12

navg B

Cascade size Influence Average Individuals Top 10% individuals Cascade size ratio Degree ratio Gain ◮ Skewed influence distribution example.

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Background Granovetter’s model Network version Groups Chaos

References Frame 55/86

Special subnetworks can act as triggers

i0 A B

◮ φ = 1/3 for all nodes

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References Frame 57/86

The power of groups...

despair.com

“A few harmless flakes working together can unleash an avalanche

  • f destruction.”
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References Frame 58/86

Extensions

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

function

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

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References Frame 59/86

Group structure—Ramified random networks

p = intergroup connection probability q = intragroup connection probability.

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References Frame 60/86

Bipartite networks

c d e a b 2 3 4 1 a b c d e contexts individuals unipartite network

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References Frame 61/86

Context distance

e c a high school teacher

  • ccupation

health care education nurse doctor teacher kindergarten d b

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References Frame 62/86

Generalized affiliation model

100

e c a b d geography

  • ccupation

age

(Blau & Schwartz, Simmel, Breiger)

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Background Granovetter’s model Network version Groups Chaos

References Frame 63/86

Generalized affiliation model networks with triadic closure

◮ Connect nodes with probability ∝ exp−αd

where α = homophily parameter and d = distance between nodes (height of lowest common ancestor)

◮ τ1 = intergroup probability of friend-of-friend

connection

◮ τ2 = intragroup probability of friend-of-friend

connection

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References Frame 64/86

Cascade windows for group-based networks

Generalized Affiliation A Group networks Single seed Coherent group seed Model networks Random set seed Random F C D E B

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Background Granovetter’s model Network version Groups Chaos

References Frame 65/86

Multiplier effect for group-based networks:

4 8 12 16 20 0.2 0.4 0.6 0.8 1

navg Savg A

4 8 12 16 20 1 2 3

navg B

4 8 12 16 0.2 0.4 0.6 0.8 1

navg Savg C

4 8 12 16 1 2 3

navg D

Degree ratio Gain Cascade size ratio Cascade size ratio < 1!

◮ Multiplier almost always below 1.

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Background Granovetter’s model Network version Groups Chaos

References Frame 66/86

Assortativity in group-based networks

5 10 15 20 0.2 0.4 0.6 0.8

k

4 8 12 0.5 1 k

Average Cascade size Local influence Degree distribution for initially infected node

◮ The most connected nodes aren’t always the most

‘influential.’

◮ Degree assortativity is the reason.

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Background Granovetter’s model Network version Groups Chaos

References Frame 67/86

Social contagion

Summary

◮ ‘Influential vulnerables’ are key to spread. ◮ Early adopters are mostly vulnerables. ◮ Vulnerable nodes important but not necessary. ◮ Groups may greatly facilitate spread. ◮ Seems that cascade condition is a global one. ◮ Most extreme/unexpected cascades occur in highly

connected networks

◮ ‘Influentials’ are posterior constructs. ◮ Many potential influentials exist.

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Background Granovetter’s model Network version Groups Chaos

References Frame 68/86

Social contagion

Implications

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

through many individuals rather than broadcast from

  • ne ‘influential.’

◮ Only simple ideas can spread by word-of-mouth.

(Idea of opinion leaders spreads well...)

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

  • r active = harder to achieve (political messages).

◮ Entities can be novel or designed to combine with

  • thers, e.g. block another one.
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Background Granovetter’s model Network version Groups Chaos

References Frame 70/86

Chaotic contagion:

◮ What if individual response functions are not

monotonic?

◮ Consider a simple deterministic version:

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

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

◮ Nodes like to imitate but only up to a

limit—they don’t want to be like everyone else.

1 0.2 0.4 0.6 0.8 1 i,1

  • i,2
  • A

i,t Pr(ai,t+1=1)

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Social Contagion Social Contagion Models

Background Granovetter’s model Network version Groups Chaos

References Frame 71/86

Two population examples:

1 0.2 0.4 0.6 0.8 1 φi,1

∗

φi,2

∗

A

φi,t Pr(ai,t+1=1)

0.5 1 0.2 0.4 0.6 0.8 1 B

φ1

∗

φ2

∗ 0.5 1 0.2 0.4 0.6 0.8 1

φ1

∗

φ2

∗ C

0.5 1 0.5 1 0.5 1 0.5 1

◮ Randomly select (φi,1, φi,2) from gray regions shown

in plots B and C.

◮ Insets show composite response function averaged

  • ver population.

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

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Background Granovetter’s model Network version Groups Chaos

References Frame 72/86

Chaotic contagion

Definition of the tent map:

F(x) = rx for 0 ≤ x ≤ 1

2,

r(1 − x) for 1

2 ≤ x ≤ 1. ◮ The usual business: look at how F iteratively maps

the unit interval [0, 1].

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Background Granovetter’s model Network version Groups Chaos

References Frame 73/86

The tent map

Effect of increasing r from 1 to 2.

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

xn F ( xn+1 )

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

xn F ( xn+1 )

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

xn F ( xn+1 )

Orbit diagram:

Chaotic behavior increases as map slope r is increased.

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Social Contagion Social Contagion Models

Background Granovetter’s model Network version Groups Chaos

References Frame 74/86

Chaotic behavior

Take r = 2 case:

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

xn F ( xn+1 )

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

sparse random network.

◮ Vary average degree z = k, a measure of

information

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Social Contagion Social Contagion Models

Background Granovetter’s model Network version Groups Chaos

References Frame 75/86

Invariant densities—stochastic response functions

500 1000 1500 2000 0.2 0.4 0.6 0.8 1

t s

z = 5

0.5 1 20 40 60

s P(s)

z = 5

activation time series activation density

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Social Contagion Social Contagion Models

Background Granovetter’s model Network version Groups Chaos

References Frame 76/86

Invariant densities—stochastic response functions

500 1000 1500 2000 0.2 0.4 0.6 0.8 1

t s

z = 5

0.5 1 20 40 60

s P(s)

z = 5 500 1000 1500 2000 0.2 0.4 0.6 0.8 1

t s

z = 10

0.5 1 10 20 30

s P(s)

z = 10 500 1000 1500 2000 0.2 0.4 0.6 0.8 1

t s

z = 18

0.5 1 5 10 15 20

s P(s)

z = 18 500 1000 1500 2000 0.2 0.4 0.6 0.8 1

t s

z = 22

0.5 1 2 4 6 8 10

s P(s)

z = 22 500 1000 1500 2000 0.2 0.4 0.6 0.8 1

t s

z = 24

0.5 1 2 4 6 8

s P(s)

z = 24 500 1000 1500 2000 0.2 0.4 0.6 0.8 1

t s

z = 30

0.5 1 2 4 6

s P(s)

z = 30

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Background Granovetter’s model Network version Groups Chaos

References Frame 77/86

Invariant densities—deterministic response functions for one specific network with k = 18

500 1000 1500 2000 0.2 0.4 0.6 0.8 1

t s

z = 18

0.5 1 10 20 30

s P(s)

z = 18 500 1000 1500 2000 0.2 0.4 0.6 0.8 1

t s

z = 18

0.5 1 5 10 15 20 25

s P(s)

z = 18 500 1000 1500 2000 0.2 0.4 0.6 0.8 1

t s

z = 18

0.5 1 10 20 30 40 50

s P(s)

z = 18 500 1000 1500 2000 0.2 0.4 0.6 0.8 1

t s

z = 18

0.5 1 10 20 30 40 50

s P(s)

z = 18 500 1000 1500 2000 0.2 0.4 0.6 0.8 1

t s

z = 18

0.5 1 2 4 6 8

s P(s)

z = 18 500 1000 1500 2000 0.2 0.4 0.6 0.8 1

t s

z = 18

0.5 1 10 20 30

s P(s)

z = 18

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Social Contagion Social Contagion Models

Background Granovetter’s model Network version Groups Chaos

References Frame 78/86

Invariant densities—stochastic response functions

500 1000 1500 2000 0.2 0.4 0.6 0.8 1

t s

z = 100

0.5 1 1 2 3 4

s P(s)

z = 100 500 1000 1500 2000 0.2 0.4 0.6 0.8 1

t s

z = 1000

0.5 1 0.5 1 1.5 2 2.5

s P(s)

z = 1000

Trying out higher values of k. . .

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SLIDE 74

Social Contagion Social Contagion Models

Background Granovetter’s model Network version Groups Chaos

References Frame 79/86

Invariant densities—deterministic response functions

500 1000 1500 2000 0.2 0.4 0.6 0.8 1

t s

z = 100

0.5 1 1 2 3 4

s P(s)

z = 100 500 1000 1500 2000 0.2 0.4 0.6 0.8 1

t s

z = 1000

0.5 1 0.5 1 1.5 2 2.5

s P(s)

z = 1000

5000 10000 0.2 0.4 0.6 0.8 1

t s

z = 100

0.5 1 10 20 30

s P(s)

z = 100

Trying out higher values of k. . .

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SLIDE 75

Social Contagion Social Contagion Models

Background Granovetter’s model Network version Groups Chaos

References Frame 80/86

Connectivity leads to chaos:

Stochastic response functions:

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SLIDE 76

Social Contagion Social Contagion Models

Background Granovetter’s model Network version Groups Chaos

References Frame 81/86

Chaotic behavior in coupled systems

Coupled maps are well explored (Kaneko/Kuramoto):

xi,n+1 = f(xi,n) +

  • j∈Ni

δi,jf(xj,n)

◮ Ni = neighborhood of node i

  • 1. Node states are continuous
  • 2. Increase δ and neighborhood size |N|

⇒ synchronization

But for contagion model:

  • 1. Node states are binary
  • 2. Asynchrony remains as connectivity increases
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Background Granovetter’s model Network version Groups Chaos

References Frame 82/86

Bifurcation diagram: Asynchronous updating

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

α

P(s | r)/max P(s | r)

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Background Granovetter’s model Network version Groups Chaos

References Frame 83/86

References I

  • S. Bikhchandani, D. Hirshleifer, and I. Welch.

A theory of fads, fashion, custom, and cultural change as informational cascades.

  • J. Polit. Econ., 100:992–1026, 1992.
  • 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 (⊞)
  • J. Carlson and J. Doyle.

Highly optimized tolerance: A mechanism for power laws in design systems.

  • Phys. Rev. Lett., 60(2):1412–1427, 1999. pdf (⊞)
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Background Granovetter’s model Network version Groups Chaos

References Frame 84/86

References II

  • J. Carlson and J. Doyle.

Highly optimized tolerance: Robustness and design in complex systems.

  • Phys. Rev. Lett., 84(11):2529–2532, 2000. pdf (⊞)
  • N. A. Christakis and J. H. Fowler.

The spread of obesity in a large social network over 32 years. New England Journal of Medicine, 357:370–379,

  • 2007. pdf (⊞)
  • N. A. Christakis and J. H. Fowler.

The collective dynamics of smoking in a large social network. New England Journal of Medicine, 358:2249–2258,

  • 2008. pdf (⊞)
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SLIDE 80

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Background Granovetter’s model Network version Groups Chaos

References Frame 85/86

References III

  • R. B. Cialdini.

Influence: Science and Practice. Allyn and Bacon, Boston, MA, 4th edition, 2000.

  • M. Gladwell.

The Tipping Point. Little, Brown and Company, New York, 2000.

  • M. Granovetter.

Threshold models of collective behavior.

  • Am. J. Sociol., 83(6):1420–1443, 1978. pdf (⊞)
  • 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 (⊞)

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SLIDE 81

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Background Granovetter’s model Network version Groups Chaos

References Frame 86/86

References IV

  • M. S. Granovetter and R. Soong.

Threshold models of interpersonal effects in consumer demand. Journal of Economic Behavior & Organization, 7:83–99, 1986. Formulates threshold as function of price, and introduces exogenous supply curve. pdf (⊞)

  • E. Katz and P

. F . Lazarsfeld. Personal Influence. The Free Press, New York, 1955.

  • T. Kuran.

Now out of never: The element of surprise in the east european revolution of 1989. World Politics, 44:7–48, 1991.

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Background Granovetter’s model Network version Groups Chaos

References Frame 87/86

References V

  • T. Kuran.

Private Truths, Public Lies: The Social Consequences of Preference Falsification. Harvard University Press, Cambridge, MA, Reprint edition, 1997.

  • T. Schelling.

Dynamic models of segregation.

  • J. Math. Sociol., 1:143–186, 1971.
  • T. C. Schelling.

Hockey helmets, concealed weapons, and daylight saving: A study of binary choices with externalities.

  • J. Conflict Resolut., 17:381–428, 1973.
  • T. C. Schelling.

Micromotives and Macrobehavior. Norton, New York, 1978.

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Background Granovetter’s model Network version Groups Chaos

References Frame 88/86

References VI

  • D. Sornette.

Critical Phenomena in Natural Sciences. Springer-Verlag, Berlin, 2nd edition, 2003.

  • D. J. Watts.

A simple model of global cascades on random networks.

  • Proc. Natl. Acad. Sci., 99(9):5766–5771, 2002.

pdf (⊞)

  • U. Wilensky.

Netlogo segregation model. http://ccl.northwestern.edu/netlogo/ models/Segregation. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL., 1998.