contagion - complex networks, sfi summer school, june, 2010 · 2010-06-07 · contagion...

Contagion

Introduction

Simple DiseaseSpreading ModelsBackground

Prediction

Social ContagionModelsGranovetter’s model

Network version

Groups

Summary

Winning: it’s not foreveryoneSuperstars

Musiclab

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ContagionComplex Networks, SFI Summer School, June, 2010

Prof. Peter Dodds

Department of Mathematics & StatisticsCenter for Complex Systems

Vermont Advanced Computing CenterUniversity of Vermont

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

Contagion

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Outline

Introduction

Simple Disease Spreading ModelsBackgroundPrediction

Social Contagion ModelsGranovetter’s modelNetwork versionGroupsSummary

Winning: it’s not for everyoneSuperstarsMusiclab

References

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Contagion

Definition:I (1) The spreading of a quality or quantity between

individuals in a population.I (2) A disease itself:

the plague, a blight, the dreaded lurgi, ...

Two main classes of contagion:

1. Infectious diseases:tuberculosis, HIV, ebola, SARS, influenza, ...

2. Social contagion:fashion, word usage, rumors, riots, religion, ...

Contagion

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Contagion models

Some large questions concerning networkcontagion:

1. For a given spreading mechanism on a givennetwork, what’s the probability that there will beglobal spreading?

2. If spreading does take off, how far will it go?3. How do the details of the network affect the

outcome?4. How do the details of the spreading mechanism

affect the outcome?5. What if the seed is one or many nodes?

http://www.uvm.edu//~pdodds/teaching/courses/2010-06SFI-SFI Summer School/

http://www.uvm.edu/~pdodds

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http://www.uvm.edu/

http://www.uvm.edu/~pdodds

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Mathematical Epidemiology

The standard SIR model:I Three states:

I S = SusceptibleI I = InfectedI R = Recovered

I S(t) + I(t) + R(t) = 1I Presumes random

interactions

Discrete time example:

I

R

SβI

1 − ρ

ρ

1 − βI

r1 − r

Transition Probabilities:

β for being infected givencontact with infectedr for recoveryρ for loss of immunity

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Independent Interaction models

Reproduction Number R0:

I R0 = expected number of infected individualsresulting from a single initial infective.

I Epidemic threshold: If R0 > 1, ‘epidemic’ occurs.I Example:

0 1 2 3 40

0.2

0.4

0.6

0.8

1

R0

Frac

tion

infe

cted

I Continuous phasetransition.

I Fine idea from asimple model.

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Disease spreading models

For ‘novel’ diseases:

1. Can we predict the size of an epidemic?2. How important/useful is the reproduction number R0?3. What is the population size N?

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R0 and variation in epidemic sizes

R0 approximately the same for all of the following:

I 1918-19 “Spanish Flu” ∼ 500,000 deaths in USI 1957-58 “Asian Flu” ∼ 70,000 deaths in USI 1968-69 “Hong Kong Flu” ∼ 34,000 deaths in USI 2003 “SARS Epidemic” ∼ 800 deaths world-wide

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Size distributions

Elsewhere, event size distributions are important:

I earthquakes (Gutenberg-Richter law)I city sizes, forest fires, war fatalitiesI wealth distributionsI ‘popularity’ (books, music, websites, ideas)I What about Epidemics?

Power laws distributions are common but not obligatory...

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Feeling icky in Iceland

Caseload recorded monthly for range of diseases inIceland, 1888-1990

1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 19900

0.01

0.02

0.03

Date

Freq

uenc

y

Iceland: measlesnormalized count

Treat outbreaks separated in time as ‘novel’ diseases.

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Measles

0 0.025 0.05 0.075 0.10

1

2

3

4

5

75

N (

ψ)

ψ

A

10−5

10−4

10−3

10−2

10−1

100

101

102

ψ

N>(ψ

)

Insert plots:Complementary cumulativefrequency distributions:

N>(Ψ) ∝ Ψ−γ+1

Ψ = fractional epidemic size

Measured values of γ:

I measles: 1.40 (low Ψ) and 1.13 (high Ψ)I Expect 2 ≤ γ < 3 (finite mean, infinite variance)I Distribution is rather flat...

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Resurgence—example of SARS

D

Date of onset

# N

ew c

ases

Nov 16, ’02 Dec 16, ’02 Jan 15, ’03 Feb 14, ’03 Mar 16, ’03 Apr 15, ’03 May 15, ’03 Jun 14, ’03

160

120

80

40

0

I Epidemic discovers new ‘pools’ of susceptibles:Resurgence.

I Importance of rare, stochastic events.

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A challenge

So... can a simple model produce

1. broad epidemic distributionsand

2. resurgence ?

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Size distributions

0 0.25 0.5 0.75 10

500

1000

1500

2000A

ψ

N(ψ

)

R0=3

Simple modelstypically producebimodal or unimodalsize distributions.

I This includes network models:random, small-world, scale-free, ...

I Some exceptions:1. Forest fire models2. Sophisticated metapopulation models

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A toy agent-based model

Geography: allow people to move between contexts:

b=2

i j

x ij =2l=3

n=8

I P = probability of travelI Movement distance: Pr(d) ∝ exp(−d/ξ)

I ξ = typical travel distance

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Example model output: size distributions

0 0.25 0.5 0.75 10

100

200

300

400

1942

ψ

N(ψ

)

R0=3

0 0.25 0.5 0.75 10

100

200

300

400

683

ψ

N(ψ

)

R0=12

I Flat distributions are possible for certain ξ and P.I Different R0’s may produce similar distributionsI Same epidemic sizes may arise from different R0’s

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Standard model:

0 500 1000 15000

2000

4000

6000

t

# N

ew c

ases D R

0=3

Standard model with transport: Resurgence

0 500 1000 15000

200

400

t

# N

ew c

ases G R

0=3

I Disease spread highly sensitive to populationstructure

I Rare events may matter enormously

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Simple disease spreading models

Attempts to use beyond disease:

I Adoption of ideas/beliefs (Goffman & Newell, 1964)I Spread of rumors (Daley & Kendall, 1965)I Diffusion of innovations (Bass, 1969)I Spread of fanatical behavior (Castillo-Chávez &

Song, 2003)

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Social Contagion Contagion

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Social Contagion

Examples abound:

I being polite/rudeI strikesI innovationI residential segregationI ipodsI obesity

I Harry PotterI votingI gossip

I Rubik’s cubeI religious beliefsI leaving lectures

SIR and SIRS contagion possible

I Classes of behavior versus specific behavior: dieting

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Social Contagion

Two focuses for us:I Widespread media influenceI Word-of-mouth influence

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The hypodermic model of influence:

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The two step model of influence: Contagion

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The general model of influence:

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Social Contagion

Why do things spread?

I Because of system level properties?I Or properties of special individuals?I Is the match that lights the forest fire the key?

(Katz and Lazarsfeld; Gladwell)I Yes. But only because we are narrative-making

machines...I System/group properties harder to understandI Always good to examine what is said before and

after the fact...

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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: “Now Out of Never: The Element ofSurprise in the East European Revolution of 1989”

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Social Contagion

Some important models:

I Tipping models—Schelling (1971)I Simulation on checker boardsI Idea of thresholds

I Threshold models—Granovetter (1978)I Herding models—Bikhchandani, Hirschleifer, Welch

(1992)I Social learning theory, Informational cascades,...

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Social contagion models

Thresholds:I Basic idea: individuals adopt a behavior when a

certain fraction of others have adoptedI ‘Others’ may be everyone in a population, an

individual’s close friends, any reference group.I Response can be probabilistic or deterministic.I Individual thresholds vary.

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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 externalities

I Telephones, Facebook, operating systems, ...

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Imitation

despair.com

“When people are freeto do as they please,they usually imitateeach other.”

—Eric Hoffer“The Passionate Stateof Mind” [11]

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Granovetter’s threshold model:

Action based on perceived behavior of others:

0 10

0.2

0.4

0.6

0.8

1

φi∗

A

φi,t

Pr(a

i,t+

1=1)

0 0.5 10

0.5

1

1.5

2

2.5B

φ∗

f (φ∗ )

0 0.5 10

0.2

0.4

0.6

0.8

1

φt

φ t+1 =

F (

φ t)

C

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

φt+1 =

∫ φt

0f (γ)dγ = F (γ)|φt

0 = F (φt)

I This is a Critical Mass model

despair.com

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Social Sciences: Threshold models

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

γ

f(γ)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

φt

φ t+1

I Example of single stable state model

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Social Sciences—Threshold models

Implications for collective action theory:

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

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Threshold model on a network

t=1 t=2 t=3

c

a

bc

e

a

b

e

a

bc

e

d dd

I All nodes have threshold φ = 0.2.I “A simple model of global cascades on random

networks”D. J. Watts. Proc. Natl. Acad. Sci., 2002

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Snowballing

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

probability that an activation will spread over anetwork?

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

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The most gullible

Vulnerables:I = Individuals who can be activated by just one

‘infected’ contactI For global cascades on random networks, must have

a global cluster of vulnerablesI Cluster of vulnerables = critical massI Network story: 1 node → critical mass → everyone.

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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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Analytic work

I Threshold model completely solved (by 2008):I Cascade condition: [22]

∞∑k=1

k(k − 1)βkPk/z ≥ 1.

where βk = probability a degree k node is vulnerable.I Final size of spread figured out by Gleeson and

Calahane [9, 8].I Solution involves finding fixed points of an iterative

map of the interval.I Spreading takes off: expansionI Spreading reaches a particular node: contraction

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Expected size of spread

i

ϕ = 1/3

t=4= active at t=0

= active at t=1

= active at t=2

= active at t=3

= active at t=4

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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 power of groups...

despair.com

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

despair.com

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Group structure—Ramified random networks

p = intergroup connection probabilityq = intragroup connection probability.

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Generalized affiliation model

100

eca b d

geography occupation age

0

(Blau & Schwartz, Simmel, Breiger)

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Cascade windows for group-based networks

Gen

eral

ized

Affili

atio

n

A

Gro

up n

etwo

rks

Single seed Coherent group seed

Mod

el n

etwo

rks

Random set seed

Rand

om

F

C

D E

B

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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 Extreme/unexpected cascades may occur in highly

connected networksI Many potential ‘influentials’ exist.I Average individuals may be more influential

system-wise than locally influential individuals.I ‘Influentials’ are posterior constructs.

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Social contagion

Implications:

I Focus on the influential vulnerables.I Create entities that many individuals ‘out in the wild’

will adopt and display rather than broadcast from afew ‘influentials.’

I Displaying can be passive = free (yo-yo’s, fashion),or active = harder to achieve (political messages).

I Accept that movement of entities will be out oforiginator’s control.

I Possibly only simple ideas can spread byword-of-mouth.

(Idea of opinion leaders has spread well...)

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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)

Arguably not always a good idea...

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The collective...

despair.com

“Never Underestimatethe Power of StupidPeople in LargeGroups.”

despair.com

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Where do superstars come from?

Rosen (1981): “The Economics of Superstars”

Examples:

I Full-time Comedians (≈ 200)I Soloists in Classical MusicI Economic Textbooks (the usual myopic example)

I Highly skewed distributions again...

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Superstars

Rosen’s theory:

I Individual quality q maps to reward R(q)

I R(q) is ‘convex’ (d2R/dq2 > 0)I Two reasons:

1. Imperfect substitution:A very good surgeon is worth many mediocre ones

2. Technology:Media spreads & technology reduces cost ofreproduction of books, songs, etc.

I No social element—success follows ‘inherent quality’

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Superstars

Adler (1985): “Stardom and Talent”

I Assumes extreme case of equal ‘inherent quality’I Argues desire for coordination in knowledge and

culture leads to differential successI Success is then purely a social construction

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Dominance hierarchies

Chase et al. (2002): “Individual differences versus socialdynamics in the formation of animal dominancehierarchies”The aggressive female Metriaclima zebra (�):

Pecking orders for fish...

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

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Dominance hierarchies

I Fish forget—changing of dominance hierarchies:

(one-sided binomial test: n ! 22, P " 0.001 and P " 0.03,respectively). In this light, 27% of the groups with identicalhierarchies is very small.

Discussion. When we rewound the tape of the fish to form newhierarchies, we usually did not get the same hierarchy twice. Thelinearity of the structures persisted and the individuals stayed thesame, but their ranks did not. Thus our results differ considerablyfrom those predicted by the prior attributes hypothesis. The factthat more identical hierarchies occurred than expected by chancealone supports the hypothesis that rank on prior attributesinfluences rank within hierarchies but not the hypothesis that

rank on prior attributes of itself creates the linear structure of thehierarchies. Although 50% of the fish changed ranks from onehierarchy to the other, almost all the hierarchies were linear instructure. Some factor other than differences in attributes seemsto have ensured high rates of linearity. In the next experiment,we tested to determine whether that factor might be socialdynamics.

It might seem possible that ‘‘noise,’’ random fluctuations inindividuals’ attributes or behaviors, could account for the ob-served differences between the first and second hierarchies.However, a careful consideration of the ways in which fluctua-tions might occur shows that this explanation is unlikely. Forexample, what if the differences were assumed to have occurredbecause some of the fish changed their ranks on attributes fromthe first to the second hierarchies? To account for our results,this assumption would require a mixture of stability and insta-bility in attribute ranks at just the right times and in just the rightproportion of groups. The rankings would have had to have beenstable for all the fish in all the groups for the day or two it tookthem to form their first hierarchies (or we would not have seenstable dominance relationships by our criterion). Then, in three-quarters of the groups (but not in the remaining one-quarter)various numbers of fish would have had to have swapped rankson attributes in the 2-week period of separation so as to haveproduced different second hierarchies. And finally, the rankingson attributes for all the fish in all the groups would have had tohave become stable once more for the day or two it took themto form their second hierarchies.

Alternatively, instead of attribute rank determining domi-nance rank as in the prior attribute model, dominance in pairsof fish might be considered to have been probabilistic, such thatat one meeting one might dominate, but at a second meetingthere was some chance that the other might dominate. Theproblem with this model is that earlier mathematical analysisdemonstrates that in situations in which one of each pair in agroup has even a small chance of dominating the other, theprobability of getting linear hierarchies is quite low (34). Andeven in a more restrictive model in which only pairs of fish thatare close in rank in the first hierarchies have modest probabilitiesof reversing their relationships, such as the level (0.25) weobserved in this experiment, the probability of getting as manylinear hierarchies as we observed is still very low (details areavailable from the authors).

We know of only one other study (47) in which researchersassembled groups to form initial hierarchies, separated theindividuals for a period, and then reassembled them to form asecond hierarchy (but see Guhl, ref. 48, for results in whichgroups had pairwise encounters between assembly and reassem-bly). Unfortunately, their techniques of analysis make it impos-sible to compare results, because they examined correlationsbetween the frequency of aggressive acts directed by individualsin pairs toward one another in the two hierarchies rather thancomparing the ranks of individuals. With these techniques it ispossible to get a positive correlation and thus a ‘‘replication’’ ofan original hierarchy in situations in which several animalsactually change ranks from the first to the second hierarchies.

Table 1. Percentage of groups with different numbers of fishchanging ranks between first and second hierarchies (n ! 22)

No. of fish changing ranks Percentage of groups

0 27.32 36.43 18.24 18.2

Fig. 1. Transition patterns between ranks of fish in the first and secondhierarchies. Frequencies of experimental groups showing each pattern areindicated in parentheses. Open-headed arrows indicate transitions of rank.Solid-headed arrows show dominance relationships in intransitive triads; allthe fish in an intransitive triad share the same rank.

5746 ! www.pnas.org"cgi"doi"10.1073"pnas.082104199 Chase et al.

(one-sided binomial test: n ! 22, P " 0.001 and P " 0.03,respectively). In this light, 27% of the groups with identicalhierarchies is very small.

Discussion. When we rewound the tape of the fish to form newhierarchies, we usually did not get the same hierarchy twice. Thelinearity of the structures persisted and the individuals stayed thesame, but their ranks did not. Thus our results differ considerablyfrom those predicted by the prior attributes hypothesis. The factthat more identical hierarchies occurred than expected by chancealone supports the hypothesis that rank on prior attributesinfluences rank within hierarchies but not the hypothesis that

rank on prior attributes of itself creates the linear structure of thehierarchies. Although 50% of the fish changed ranks from onehierarchy to the other, almost all the hierarchies were linear instructure. Some factor other than differences in attributes seemsto have ensured high rates of linearity. In the next experiment,we tested to determine whether that factor might be socialdynamics.

It might seem possible that ‘‘noise,’’ random fluctuations inindividuals’ attributes or behaviors, could account for the ob-served differences between the first and second hierarchies.However, a careful consideration of the ways in which fluctua-tions might occur shows that this explanation is unlikely. Forexample, what if the differences were assumed to have occurredbecause some of the fish changed their ranks on attributes fromthe first to the second hierarchies? To account for our results,this assumption would require a mixture of stability and insta-bility in attribute ranks at just the right times and in just the rightproportion of groups. The rankings would have had to have beenstable for all the fish in all the groups for the day or two it tookthem to form their first hierarchies (or we would not have seenstable dominance relationships by our criterion). Then, in three-quarters of the groups (but not in the remaining one-quarter)various numbers of fish would have had to have swapped rankson attributes in the 2-week period of separation so as to haveproduced different second hierarchies. And finally, the rankingson attributes for all the fish in all the groups would have had tohave become stable once more for the day or two it took themto form their second hierarchies.

Alternatively, instead of attribute rank determining domi-nance rank as in the prior attribute model, dominance in pairsof fish might be considered to have been probabilistic, such thatat one meeting one might dominate, but at a second meetingthere was some chance that the other might dominate. Theproblem with this model is that earlier mathematical analysisdemonstrates that in situations in which one of each pair in agroup has even a small chance of dominating the other, theprobability of getting linear hierarchies is quite low (34). Andeven in a more restrictive model in which only pairs of fish thatare close in rank in the first hierarchies have modest probabilitiesof reversing their relationships, such as the level (0.25) weobserved in this experiment, the probability of getting as manylinear hierarchies as we observed is still very low (details areavailable from the authors).

We know of only one other study (47) in which researchersassembled groups to form initial hierarchies, separated theindividuals for a period, and then reassembled them to form asecond hierarchy (but see Guhl, ref. 48, for results in whichgroups had pairwise encounters between assembly and reassem-bly). Unfortunately, their techniques of analysis make it impos-sible to compare results, because they examined correlationsbetween the frequency of aggressive acts directed by individualsin pairs toward one another in the two hierarchies rather thancomparing the ranks of individuals. With these techniques it ispossible to get a positive correlation and thus a ‘‘replication’’ ofan original hierarchy in situations in which several animalsactually change ranks from the first to the second hierarchies.

Table 1. Percentage of groups with different numbers of fishchanging ranks between first and second hierarchies (n ! 22)

No. of fish changing ranks Percentage of groups

0 27.32 36.43 18.24 18.2

Fig. 1. Transition patterns between ranks of fish in the first and secondhierarchies. Frequencies of experimental groups showing each pattern areindicated in parentheses. Open-headed arrows indicate transitions of rank.Solid-headed arrows show dominance relationships in intransitive triads; allthe fish in an intransitive triad share the same rank.

5746 ! www.pnas.org"cgi"doi"10.1073"pnas.082104199 Chase et al.

I 22 observations: about 3/4 of the time, hierarchychanged

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Music Lab Experiment

48 songs30,000 participants

multiple ‘worlds’Inter-world variability

I How probable is the world?I Can we estimate variability?I Superstars dominate but are unpredictable. Why?

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Salganik et al. (2006) “An experimental study of inequalityand unpredictability in an artificial cultural market”

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Experiment 1 Experiments 2–4

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0 0.01 0.02 0.03 0.04 0.050

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Sensible result:I Stronger social signal leads to greater following and

greater inequality.

Peculiar result:I Stronger social signal leads to greater

unpredictability.

Very peculiar observation:

I The most unequal distributions would suggest thegreatest variation in underlying ‘quality.’

I But success may be due to social constructionthrough following...

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Music Lab Experiment—Sneakiness

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I Inversion of download countI The ‘pretend rich’ get richer ...I ... but at a slower rate

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References I

[1] M. Adler.Stardom and talent.American Economic Review, pages 208–212, 1985.pdf (�)

[2] S. Bikhchandani, D. Hirshleifer, and I. Welch.A theory of fads, fashion, custom, and culturalchange as informational cascades.J. Polit. Econ., 100:992–1026, 1992.

[3] S. Bikhchandani, D. Hirshleifer, and I. Welch.Learning from the behavior of others: Conformity,fads, and informational cascades.J. Econ. Perspect., 12(3):151–170, 1998. pdf (�)

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References II

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

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

[6] I. D. Chase, C. Tovey, D. Spangler-Martin, andM. Manfredonia.Individual differences versus social dynamics in theformation of animal dominance hierarchies.Proc. Natl. Acad. Sci., 99(8):5744–5749, 2002.pdf (�)

http://www.uvm.edu/~pdodds/research/papers/others/1985/adler1985a.pdf

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

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

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

http://www.uvm.edu/~pdodds/research/papers/others/2002/chase2002a.pdf

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References III

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

[8] J. P. Gleeson.Cascades on correlated and modular randomnetworks.Phys. Rev. E, 77:046117, 2008. pdf (�)

[9] J. P. Gleeson and D. J. Cahalane.Seed size strongly affects cascades on randomnetworks.Phys. Rev. E, 75:056103, 2007. pdf (�)

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

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References IV

[11] E. Hoffer.The Passionate State of Mind: And Other Aphorisms.Buccaneer Books, 1954.

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

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

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

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References V

[15] J. D. Murray.Mathematical Biology.Springer, New York, Third edition, 2002.

[16] S. Rosen.The economics of superstars.Am. Econ. Rev., 71:845–858, 1981. pdf (�)

[17] M. J. Salganik, P. S. Dodds, and D. J. Watts.An experimental study of inequality andunpredictability in an artificial cultural market.Science, 311:854–856, 2006. pdf (�)

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

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References VI

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

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

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

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

http://www.uvm.edu/~pdodds/research/papers/others/2008/gleeson2008a.pdf

http://www.uvm.edu/~pdodds/research/papers/others/2007/gleeson2007a.pdf

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

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

http://www.uvm.edu/~pdodds/research/papers/others/1981/rosen1981a.pdf

http://www.uvm.edu/~pdodds/research/papers/others/2006/salganik2006a.pdf

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

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

contagion - complex networks, sfi summer school, june, 2010 · 2010-06-07 · contagion...

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