cascading behavior in networksbettina.berendt/teaching/... · 2011-05-13 · cascading behavior in...
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Cascading Behavior in Networks
Lesley Van der Wee
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Goal
Modelling the spread of the word!
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Overview
• Introduction• Basic idea• The limit of cascades• Extension• Advanced
- Cascade capacity- Bilingual option (co-presentation)
• Conclusion• Q&A
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Introduction
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Introduction
Networks visualized as graphs
(e.g. social networks)
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Basic idea
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Basic idea
Diffusion in a network- New behavior- Adopt if certain amount of neighbors do
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Basic idea
Simple Game- Possible behaviors: A and B- Players: 2 nodes (v and w)
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Basic idea
Simple game- Link between v and w
=> motive to adopt same behavior
- Both adopt A: payoff = a > 0- Both adopt B: payoff = b > 0- Opposite behaviors: payoff = 0
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Basic idea
• Each node- plays the game with each neighbor- Payoff = sum of payoffs of each link
• Consider node v- Some neighbors adopt A- Some neighbors adopt B
=> what should v do to maximize its payoff?
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Basic idea
Example
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Basic idea
Example
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Basic idea
What should v do to maximize its payoff?
A node v should adopt A if at least a certain fraction of its neighbors follow A.
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Basic idea
d = # neighbors
p : fraction of neighbors which adopt A
(0 ≤ p ≤ 1)
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Basic idea
Decision ruleA is the better choice if
pda ≥ (1 – p)db
<=> p ≥ = q
q : threshold for which a node should adopt A
bab
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Basic idea
TerminologyThe nodes that adopt the behavior A first are called the “initial adopters”.
RemarkAssumption:
If a node has adopted behavior A, it will never switch back to B.
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Basic idea
DefinitionA cascade of adoptions of A is a chain reaction of switches from B to A.
2 possibilities- Every node switches to A (complete cascade)- Or, the cascade runs, but stops after a while
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Basic idea
Cascading behavior
some examples...
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Basic idea – Example 1
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Basic idea – Example 1
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Basic idea – Example 1
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Basic idea – Example 1
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Basic idea – Example 2
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Basic idea – Example 2
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Basic idea – Example 2
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The limit of cascades
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The limit of cascades
QuestionWhat causes the spread of A to stop?
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The limit of cascades
Intuitive answer- Network structure- Choice of initial adopters- Values of a and b
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The limit of cascades
Question (revisited)What causes the spread of A to stop?
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The limit of cascades
Technical answerDense clusters
Definitions- A cluster of density p is a set of nodes such that
each node in the set has at least a p fraction of its neighbors in the set.
- A dense cluster, in a network with threshold q, is a cluster of density greater than 1 – q.
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The limit of cascades
Example- Density p = 2/3- Suppose threshold q = 3/5
=> both clusters are dense
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The limit of cascades
ClaimThe initial adopters of A (with threshold q) will not cause a complete cascade
if and only if
the network contains a cluster of density greater than 1 – q.
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The limit of cascades
Part 1Dense clusters are obstacles to cascades.
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The limit of cascades
ExplanationSuppose v is the first node in the cluster that adopts behavior A.
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The limit of cascades
ExplanationSince v is the first, all neighbors of v using A are located outside the cluster.
Density is greater than 1 – q, hence less than qd neighbors are outside the cluster.Contradiction!
=> v could never have adopted A
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The limit of cascades
Part 2Dense clusters are the only obstacles to cascades.
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The limit of cascades
Explanation
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The limit of cascades
Explanation- Let S be the set of nodes still using B.
- For every node w in S:w doesn't want to switch to A=> less than qd neighbors are using A=> more than (1 – q)d neighbors are using B=> fraction of neighbors in S is greater than (1 – q)
=> S is a dense cluster
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Extension of the model
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Extension
v and w value behaviors A and B differentlyav vs aw and bv vs bw
=> adopt behavior w.r.t. personal threshold
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Extension
Example
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Extension
Example
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Advanced material
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Advanced - Cascade capacity
DefinitionThe cascade capacity of a network is the maximum threshold for which a “small” set of initial adopters (i.e. a finite set) will cause a complete cascade.
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Advanced - Cascade capacity
Example 1
cascade capacity = 1/2
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Advanced - Cascade capacity
Example 2
cascade capacity = 3/8
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Advanced - Cascade capacity
ClaimThere is no network for which the cascade capacity is bigger than 1/2.
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Advanced – Bilingual option
Co-presentation...
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Conclusion
• Model to simulate/predict/visualize the spread of a new behavior
• Real-life applications- Telephone and fax machines- New/compatible technologies- Youtube video vs political ideas- Limit on communication by government
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Q&A
Thanks for your attention!
ReferencesNetworks, Crowds, and Markets: Reasoning about a Highly Connected World (ch.19), David Easley and Jon Kleinberg, Cambridge University Press, 2010