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Social ContagionPrinciples of Complex Systems
CSYS/MATH 300, Spring, 2013 | #SpringPoCS2013
Prof. Peter Dodds@peterdodds
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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These slides brought to you by:
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Outline
Social Contagion ModelsBackgroundGranovetter’s modelNetwork versionFinal sizeSpreading successGroups
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Things that spread well:
buzzfeed.com ():
+ News ...
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LOL + cute + fail + wtf:
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The whole lolcats thing:
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Some things really stick:
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wtf + geeky + omg:
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Social Contagion
http://xkcd.com/610/ ()
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Social Contagion
Examples abound
I fashionI strikingI smoking () [7]
I residentialsegregation [19]
I ipodsI obesity () [6]
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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Framingham heart study:
Evolving network stories (Christakis and Fowler):I The spread of quitting smoking () [7]
I The spread of spreading () [6]
I Also: happiness () [9], 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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Social Contagion
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 are
there?I Are some individuals super influencers?
Highly popularized by Gladwell [10] as ‘connectors’I The infectious idea of opinion leaders (Katz and
Lazarsfeld) [16]
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The hypodermic model of influence
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The two step model of influence [16]
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The general model of influence
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Social Contagion
Why do things spread?I Because of properties of special individuals?I Or system level properties?I Is the match that lights the fire important?I Yes. But only because we are narrative-making
machines...I We like to think things happened for reasons...I Reasons for success are usually ascribed to intrinsic
properties (e.g., Mona Lisa)I System/group properties harder to understandI Always good to examine what is said before and
after the fact...
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The Mona Lisa
I “Becoming Mona Lisa: The Making of a GlobalIcon”—David Sassoon
I Not the world’s greatest painting from the start...I Escalation through theft, vandalism, parody, ...
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The completely unpredicted fall of EasternEurope
Timur Kuran: [17, 18] “Now Out of Never: The Element ofSurprise in the East European Revolution of 1989”
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The dismal predictive powers of editors...
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Social Contagion
Messing with social connectionsI Ads based on message content
(e.g., Google and email)I BzzAgent ()I One of Facebook’s early advertising attempts:
Beacon ()I All of Facebook’s advertising attempts.
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Getting others to do things for youA very good book: ‘Influence’ [8] by Robert Cialdini ()
Six modes of influence:1. Reciprocation: The Old Give and Take... and Take;
e.g., Free samples, Hare Krishnas.2. Commitment and Consistency: Hobgoblins of the
Mind ; e.g., Hazing.3. Social Proof: Truths Are Us;
e.g., Jonestown (),Kitty Genovese () (contested).
4. Liking: The Friendly Thief ; e.g., Separation intogroups is enough to cause problems.
5. Authority: Directed Deference;e.g., Milgram’s obedience to authorityexperiment. ()
6. Scarcity: The Rule of the Few ; e.g., Prohibition.
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Social contagion
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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Social Contagion
Some important models:I Tipping models—Schelling (1971) [19, 20, 21]
I Simulation on checker boardsI Idea of thresholdsI Explore the Netlogo () online
implementation () [26]
I Threshold models—Granovetter (1978) [13]
I Herding models—Bikhchandani, Hirschleifer, Welch(1992) [2, 3]
I Social learning theory, Informational cascades,...
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Social contagion models
ThresholdsI Basic idea: individuals adopt a behavior when a
certain fraction of others have adoptedI ‘Others’ may be everyone in a population, an
individual’s close friends, any reference group.I Response can be probabilistic or deterministic.I Individual thresholds can varyI Assumption: order of others’ adoption does not
matter... (unrealistic).I Assumption: level of influence per person is uniform
(unrealistic).
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Social Contagion
Some possible origins of thresholds:I Inherent, evolution-devised inclination to coordinate,
to conform, to imitate. [1]
I Lack of information: impute the worth of a good orbehavior 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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Threshold models—response functions
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
φ
p
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
φ
p
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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Social Contagion
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 .
I At time t + 1, fraction rioting = fraction with φ∗ ≤ φt .I
φt+1 =
∫ φt
0f (φ∗)dφ∗ = F (φ∗)|φt
0 = F (φt )
I ⇒ Iterative maps of the unit interval [0,1].
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Threshold models
Action based on perceived behavior of others:
0 10
0.2
0.4
0.6
0.8
1
φi∗
A
φi,t
Pr(a
i,t+
1=1)
0 0.5 10
0.5
1
1.5
2
2.5B
φ∗
f (φ∗ )
0 0.5 10
0.2
0.4
0.6
0.8
1
φt
φ t+1 =
F (
φ t)
C
I Two states: S and I.I φ = fraction of contacts ‘on’ (e.g., rioting)I Discrete time update (strong assumption!)I This is a Critical mass model
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Threshold models
Another example of critical mass model:
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
γ
f(γ)
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
φt
φ t+1
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Threshold models
Example of single stable state model:
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
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Threshold models
Chaotic behavior possible [15, 14]
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
xn
F (
x n+1 )
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
xn
F (
x n+1 )
I Period doubling arises as map amplitude r isincreased.
I Synchronous update assumption is crucial
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Threshold models—Nutshell
Implications for collective action theory:1. Collective uniformity 6⇒ individual uniformity2. Small individual changes⇒ large global changes
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“A simple model of global cascades on randomnetworks”I Many years after Granovetter and Soong’s work: D.
J. Watts. Proc. Natl. Acad. Sci., 2002 [23]
I Mean field model→ network modelI Individuals now have a limited view of the world
We’ll also explore:I “Seed size strongly affects cascades on random
networks” [12]
Gleeson and Cahalane, Phys. Rev. E, 2007.I “Influentials, Networks, and Public Opinion
Formation” [24]
Watts and Dodds, J. Cons. Res., 2007.I “Threshold models of Social Influence” [25]
Watts and Dodds, The Oxford Handbook ofAnalytical Sociology, 2009.
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Threshold model on a network
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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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.
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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
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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Example random network structure:
I Ωcrit = Ωvuln =critical mass =globalvulnerablecomponent
I Ωtrig =triggeringcomponent
I Ωfinal =potential extentof spread
I Ω = entirenetwork
Ωcrit ⊂ Ωtrig; Ωcrit ⊂ Ωfinal; and Ωtrig,Ωfinal ⊂ Ω.
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Snowballing
Follow active linksI An active link is a link connected to an activated
node.I If an infected link leads to at least 1 more infected
link, then activation spreads.I We need to understand which nodes can be
activated when only one of their neigbors becomesactive.
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The most gullible
Vulnerables:I We call individuals who can be activated by just one
contact being active vulnerablesI The vulnerability condition for node i :
1/ki ≥ φi
I Which means # contacts ki ≤ b1/φicI For global cascades on random networks, must have
a global cluster of vulnerables [23]
I Cluster of vulnerables = critical massI Network story: 1 node→ critical mass→ everyone.
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Cascade condition
Back to following a link:I A randomly chosen link, traversed in a 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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Cascade condition
Next: Vulnerability of linked nodeI Linked node is vulnerable with probability
βk =
∫ 1/k
φ′∗=0f (φ′∗)dφ′∗
I If linked node is vulnerable, it produces k − 1 newoutgoing active links
I If linked node is not vulnerable, it produces no activelinks.
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Cascade condition
Putting things together:I Expected number of active edges produced by 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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Cascade condition
So... for random networks with fixed degree distributions,cacades take off when:
∞∑k=1
(k − 1) · βk ·kPk
〈k〉≥ 1.
I βk = probability a degree k node is vulnerable.I Pk = probability a node has degree k .
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Cascade condition
Two special cases:I (1) Simple disease-like spreading succeeds: βk = β
β ·∞∑
k=1
(k − 1) · kPk
〈k〉≥ 1.
I (2) Giant component exists: β = 1
1 ·∞∑
k=1
(k − 1) · kPk
〈k〉≥ 1.
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Final size
Spreading success
Groups
References
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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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.
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Final size
Spreading success
Groups
References
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
51 of 94
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.
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Final size
Spreading success
Groups
References
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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Cascade window for random networks
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Final size
Spreading success
Groups
References
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
53 of 94
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 ∗)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Final size
Spreading success
Groups
References
i
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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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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.”
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Final size
Spreading success
Groups
References
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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Threshold contagion on random networks
I Next: Find expected fractional size of spread.I Not obvious even for uniform threshold problem.I Difficulty is in figuring out if and when nodes that
need ≥ 2 hits switch on.I Problem solved for infinite seed case by Gleeson and
Cahalane:“Seed size strongly affects cascades on randomnetworks,” Phys. Rev. E, 2007. [12]
I Developed further by Gleeson in “Cascades oncorrelated and modular random networks,” Phys.Rev. E, 2008. [11]
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Final size
Spreading success
Groups
References
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
57 of 94
Expected size of spread
Idea:I Randomly turn on a fraction φ0 of nodes at time t = 0I Capitalize on local branching network structure of
random networks (again)I Now think about what must happen for a specific
node i to become active at time t :• t = 0: i is one of the seeds (prob = φ0)• t = 1: i was not a seed but enough of i ’s friends
switched on at time t = 0 so that i ’s threshold is nowexceeded.• t = 2: enough of i ’s friends and friends-of-friends
switched on at time t = 0 so that i ’s threshold is nowexceeded.• t = n: enough nodes within n hops of i switched on
at t = 0 and their effects have propagated to reach i .
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Final size
Spreading success
Groups
References
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
58 of 94
Expected size of spread
i
ϕ = 1/3
t=0= active,
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Final size
Spreading success
Groups
References
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
59 of 94
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
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Final size
Spreading success
Groups
References
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
60 of 94
Expected size of spread
Notes:I Calculations are possible if nodes do not become
inactive (strong restriction).I Not just for threshold model—works for a wide range
of contagion processes.I We can analytically determine the entire time
evolution, not just the final size.I We can in fact determine
Pr(node of degree k switching on at time t).I Asynchronous updating can be handled too.
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Final size
Spreading success
Groups
References
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
61 of 94
Expected size of spread
Pleasantness:I Taking off from a single seed story is about
expansion away from a node.I Extent of spreading story is about contraction at a
node.
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Final size
Spreading success
Groups
References
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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Expected size of spreadI Notation:φk ,t = Pr(a degree k node is active at time t).
I Notation: Bkj = Pr (a degree k node becomes activeif j neighbors are active).
I Our starting point: φk ,0 = φ0.
I(k
j
)φ j
0(1− φ0)k−j = Pr (j of a degree k node’sneighbors were seeded at time t = 0).
I Probability a degree k node was a seed at t = 0 is φ0(as above).
I Probability a degree k node was not a seed at t = 0is (1− φ0).
I Combining everything, we have:
φk ,1 = φ0 + (1− φ0)k∑
j=0
(kj
)φ j
0(1− φ0)k−jBkj .
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Final size
Spreading success
Groups
References
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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Expected size of spread
I For general t , we need to know the probability anedge coming into a degree k node at time t is active.
I Notation: call this probability θt .I We already know θ0 = φ0.I Story analogous to t = 1 case. For node i :
φi,t+1 = φ0 + (1− φ0)
ki∑j=0
(ki
j
)θ j
t (1− θt )ki−jBki j .
I Average over all nodes to obtain expression for φt+1:
φt+1 = φ0 + (1− φ0)∞∑
k=0
Pk
k∑j=0
(kj
)θ j
t (1− θt )k−jBkj .
I So we need to compute θt ... massive excitement...
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Final size
Spreading success
Groups
References
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
64 of 94
Expected size of spread
First connect θ0 to θ1:I θ1 = φ0+
(1− φ0)∞∑
k=1
kPk
〈k〉
k−1∑j=0
(k − 1
j
)θ j
0 (1− θ0)k−1−jBkj
I kPk〈k〉 = Rk = Pr (edge connects to a degree k node).
I∑k−1
j=0 piece gives Pr(degree node k activates) of itsneighbors k − 1 incoming neighbors are active.
I φ0 and (1− φ0) terms account for state of node attime t = 0.
I See this all generalizes to give θt+1 in terms of θt ...
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Final size
Spreading success
Groups
References
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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Expected size of spreadTwo pieces: edges first, and then nodes
1. θt+1 = φ0︸︷︷︸exogenous
+(1− φ0)∞∑
k=1
kPk
〈k〉
k−1∑j=0
(k − 1
j
)θ j
t (1− θt )k−1−jBkj︸ ︷︷ ︸
social effects
with θ0 = φ0.2. φt+1 =
φ0︸︷︷︸exogenous
+(1− φ0)∞∑
k=0
Pk
k∑j=0
(kj
)θ j
t (1− θt )k−jBkj︸ ︷︷ ︸
social effects
.
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Final size
Spreading success
Groups
References
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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Expected size of spread:
I Retrieve cascade condition for spreading from asingle seed in limit φ0 → 0.
I Depends on map θt+1 = G(θt ;φ0).I First: if self-starters are present, some activation is
assured:
G(0;φ0) =∞∑
k=1
kPk
〈k〉• Bk0 > 0.
meaning Bk0 > 0 for at least one value of k ≥ 1.I If θ = 0 is a fixed point of G (i.e., G(0;φ0) = 0) then
spreading occurs if
G′(0;φ0) =∞∑
k=0
kPk
〈k〉• (k − 1) • Bk1 > 1.
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Final size
Spreading success
Groups
References
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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Expected size of spread:
In words:I If G(0;φ0) > 0, spreading must occur because some
nodes turn on for free.I If G has an unstable fixed point at θ = 0, then
cascades are also always possible.
Non-vanishing seed case:I Cascade condition is more complicated for φ0 > 0.I If G has a stable fixed point at θ = 0, and an unstable
fixed point for some 0 < θ∗ < 1, then for θ0 > θ∗,spreading takes off.
I Tricky point: G depends on φ0, so as we change φ0,we also change G.
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Final size
Spreading success
Groups
References
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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General fixed point story:
00
1
1θt
θ t+1=G(θ
t;φ0)
00
1
1θt
θ t+1=G(θ
t;φ0)
00
1
1θt
θ t+1=G(θ
t;φ0)
I Given θ0(= φ0), θ∞ will be the nearest stable fixedpoint, either above or below.
I n.b., adjacent fixed points must have oppositestability types.
I Important: Actual form of G depends on φ0.I So choice of φ0 dictates both G and starting
point—can’t start anywhere for a given G.
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Final size
Spreading success
Groups
References
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
70 of 94
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
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Final size
Spreading success
Groups
References
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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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.
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Final size
Spreading success
Groups
References
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
72 of 94
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.
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Final size
Spreading success
Groups
References
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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Special subnetworks can act as triggers
i0
A
B
I φ = 1/3 for all nodes
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Final size
Spreading success
Groups
References
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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The power of groups...
despair.com
“A few harmless flakesworking together canunleash an avalancheof destruction.”
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Final size
Spreading success
Groups
References
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
76 of 94
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
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Final size
Spreading success
Groups
References
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
77 of 94
Group structure—Ramified random networks
p = intergroup connection probabilityq = intragroup connection probability.
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Final size
Spreading success
Groups
References
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
78 of 94
Bipartite networks
c d ea b
2 3 41
a
b
c
d
e
contexts
individuals
unipartitenetwork
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Final size
Spreading success
Groups
References
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
79 of 94
Context distance
eca
high schoolteacher
occupation
health careeducation
nurse doctorteacherkindergarten
db
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Final size
Spreading success
Groups
References
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
80 of 94
Generalized affiliation model
100
eca b d
geography occupation age
0
(Blau & Schwartz, Simmel, Breiger)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Final size
Spreading success
Groups
References
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
81 of 94
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
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Final size
Spreading success
Groups
References
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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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
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Final size
Spreading success
Groups
References
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
83 of 94
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.
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Final size
Spreading success
Groups
References
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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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.
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Final size
Spreading success
Groups
References
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
85 of 94
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.
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Final size
Spreading success
Groups
References
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
86 of 94
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.
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Final size
Spreading success
Groups
References
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
87 of 94
References I
[1] A. Bentley, M. Earls, and M. J. O’Brien.I’ll Have What She’s Having: Mapping SocialBehavior.MIT Press, Cambridge, MA, 2011.
[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 ()
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Final size
Spreading success
Groups
References
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
88 of 94
References II
[4] 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 ()
[5] J. M. Carlson and J. Doyle.Highly optimized tolerance: Robustness and designin complex systems.Phys. Rev. Lett., 84(11):2529–2532, 2000. pdf ()
[6] 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 ()
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Final size
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t=4 = active at t=0
= active at t=1
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Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Final size
Spreading success
Groups
References
i
ϕ = 1/3
t=4 = active at t=0
= active at t=1
= active at t=2
= active at t=3
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Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Final size
Spreading success
Groups
References
i
ϕ = 1/3
t=4 = active at t=0
= active at t=1
= active at t=2
= active at t=3
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Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Final size
Spreading success
Groups
References
i
ϕ = 1/3
t=4 = active at t=0
= active at t=1
= active at t=2
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Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Final size
Spreading success
Groups
References
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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Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Final size
Spreading success
Groups
References
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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