social contagion - principles of complex systems …pdodds/teaching/courses/2011-08uvm...csys/math...
TRANSCRIPT
![Page 1: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/1.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
1 of 88
Social ContagionPrinciples of Complex Systems
CSYS/MATH 300, Fall, 2011
Prof. Peter Dodds
Department of Mathematics & Statistics | Center for Complex Systems |Vermont Advanced Computing Center | University of Vermont
Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.
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Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
2 of 88
Outline
Social Contagion ModelsBackgroundGranovetter’s modelNetwork versionGroupsChaos
References
![Page 3: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/3.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
3 of 88
Outline
Social Contagion ModelsBackgroundGranovetter’s modelNetwork versionGroupsChaos
References
![Page 4: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/4.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
4 of 88
Social Contagion
http://xkcd.com/610/ (�)
![Page 5: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/5.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
5 of 88
Social Contagion
![Page 6: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/6.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
6 of 88
Social Contagion
![Page 7: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/7.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
7 of 88
Social Contagion
Examples abound
I fashionI strikingI smoking (�) [6]
I residentialsegregation [16]
I ipodsI obesity (�) [5]
I Harry PotterI votingI gossip
I Rubik’s cubeI religious beliefsI leaving lectures
SIR and SIRS contagion possibleI Classes of behavior versus specific behavior
: dieting
![Page 8: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/8.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
7 of 88
Social Contagion
Examples abound
I fashionI strikingI smoking (�) [6]
I residentialsegregation [16]
I ipodsI obesity (�) [5]
I Harry PotterI votingI gossip
I Rubik’s cubeI religious beliefsI leaving lectures
SIR and SIRS contagion possibleI Classes of behavior versus specific behavior
: dieting
![Page 9: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/9.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
7 of 88
Social Contagion
Examples abound
I fashionI strikingI smoking (�) [6]
I residentialsegregation [16]
I ipodsI obesity (�) [5]
I Harry PotterI votingI gossip
I Rubik’s cubeI religious beliefsI leaving lectures
SIR and SIRS contagion possibleI Classes of behavior versus specific behavior
: dieting
![Page 10: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/10.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
7 of 88
Social Contagion
Examples abound
I fashionI strikingI smoking (�) [6]
I residentialsegregation [16]
I ipodsI obesity (�) [5]
I Harry PotterI votingI gossip
I Rubik’s cubeI religious beliefsI leaving lectures
SIR and SIRS contagion possibleI Classes of behavior versus specific behavior: dieting
![Page 11: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/11.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
8 of 88
Framingham heart study:
Evolving network stories (Christakis and Fowler):I The spread of quitting smoking (�) [6]
I The spread of spreading (�) [5]
I Also: happiness (�) [8], loneliness, ...I The book: Connected: The Surprising Power of Our
Social Networks and How They Shape Our Lives (�)
Controversy:I Are your friends making you fat? (�) (Clive
Thomspon, NY Times, September 10, 2009).I Everything is contagious (�)—Doubts about the
social plague stir in the human superorganism (DaveJohns, Slate, April 8, 2010).
![Page 12: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/12.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
8 of 88
Framingham heart study:
Evolving network stories (Christakis and Fowler):I The spread of quitting smoking (�) [6]
I The spread of spreading (�) [5]
I Also: happiness (�) [8], loneliness, ...I The book: Connected: The Surprising Power of Our
Social Networks and How They Shape Our Lives (�)
Controversy:I Are your friends making you fat? (�) (Clive
Thomspon, NY Times, September 10, 2009).I Everything is contagious (�)—Doubts about the
social plague stir in the human superorganism (DaveJohns, Slate, April 8, 2010).
![Page 13: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/13.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
8 of 88
Framingham heart study:
Evolving network stories (Christakis and Fowler):I The spread of quitting smoking (�) [6]
I The spread of spreading (�) [5]
I Also: happiness (�) [8], loneliness, ...I The book: Connected: The Surprising Power of Our
Social Networks and How They Shape Our Lives (�)
Controversy:I Are your friends making you fat? (�) (Clive
Thomspon, NY Times, September 10, 2009).I Everything is contagious (�)—Doubts about the
social plague stir in the human superorganism (DaveJohns, Slate, April 8, 2010).
![Page 14: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/14.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
8 of 88
Framingham heart study:
Evolving network stories (Christakis and Fowler):I The spread of quitting smoking (�) [6]
I The spread of spreading (�) [5]
I Also: happiness (�) [8], loneliness, ...I The book: Connected: The Surprising Power of Our
Social Networks and How They Shape Our Lives (�)
Controversy:I Are your friends making you fat? (�) (Clive
Thomspon, NY Times, September 10, 2009).I Everything is contagious (�)—Doubts about the
social plague stir in the human superorganism (DaveJohns, Slate, April 8, 2010).
![Page 15: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/15.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
8 of 88
Framingham heart study:
Evolving network stories (Christakis and Fowler):I The spread of quitting smoking (�) [6]
I The spread of spreading (�) [5]
I Also: happiness (�) [8], loneliness, ...I The book: Connected: The Surprising Power of Our
Social Networks and How They Shape Our Lives (�)
Controversy:I Are your friends making you fat? (�) (Clive
Thomspon, NY Times, September 10, 2009).I Everything is contagious (�)—Doubts about the
social plague stir in the human superorganism (DaveJohns, Slate, April 8, 2010).
![Page 16: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/16.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
9 of 88
Social Contagion
![Page 17: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/17.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
10 of 88
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 arethere?
I Are some individuals super influencers?
Highly popularized by Gladwell [9] as ‘connectors’
I The infectious idea of opinion leaders (Katz andLazarsfeld) [13]
![Page 18: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/18.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
10 of 88
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 arethere?
I Are some individuals super influencers?
Highly popularized by Gladwell [9] as ‘connectors’
I The infectious idea of opinion leaders (Katz andLazarsfeld) [13]
![Page 19: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/19.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
10 of 88
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 arethere?
I Are some individuals super influencers?
Highly popularized by Gladwell [9] as ‘connectors’
I The infectious idea of opinion leaders (Katz andLazarsfeld) [13]
![Page 20: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/20.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
10 of 88
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 arethere?
I Are some individuals super influencers?
Highly popularized by Gladwell [9] as ‘connectors’
I The infectious idea of opinion leaders (Katz andLazarsfeld) [13]
![Page 21: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/21.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
10 of 88
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 arethere?
I Are some individuals super influencers?
Highly popularized by Gladwell [9] as ‘connectors’
I The infectious idea of opinion leaders (Katz andLazarsfeld) [13]
![Page 22: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/22.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
10 of 88
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 [9] as ‘connectors’
I The infectious idea of opinion leaders (Katz andLazarsfeld) [13]
![Page 23: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/23.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
10 of 88
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 [9] as ‘connectors’
I The infectious idea of opinion leaders (Katz andLazarsfeld) [13]
![Page 24: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/24.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
10 of 88
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 [9] as ‘connectors’
I The infectious idea of opinion leaders (Katz andLazarsfeld) [13]
![Page 25: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/25.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
10 of 88
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 [9] as ‘connectors’I The infectious idea of opinion leaders (Katz and
Lazarsfeld) [13]
![Page 26: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/26.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
10 of 88
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 [9] as ‘connectors’I The infectious idea of opinion leaders (Katz and
Lazarsfeld) [13]
![Page 27: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/27.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
11 of 88
The hypodermic model of influence
![Page 28: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/28.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
12 of 88
The two step model of influence [13]
![Page 29: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/29.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
13 of 88
The general model of influence
![Page 30: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/30.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
14 of 88
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...
![Page 31: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/31.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
14 of 88
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...
![Page 32: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/32.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
14 of 88
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...
![Page 33: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/33.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
14 of 88
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...
![Page 34: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/34.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
14 of 88
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...
![Page 35: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/35.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
14 of 88
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...
![Page 36: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/36.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
14 of 88
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...
![Page 37: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/37.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
14 of 88
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...
![Page 38: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/38.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
14 of 88
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...
![Page 39: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/39.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
15 of 88
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, ...
![Page 40: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/40.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
15 of 88
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, ...
![Page 41: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/41.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
15 of 88
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, ...
![Page 42: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/42.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
15 of 88
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, ...
![Page 43: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/43.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
15 of 88
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, ...
![Page 44: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/44.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
15 of 88
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, ...
![Page 45: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/45.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
16 of 88
The completely unpredicted fall of EasternEurope
Timur Kuran: [14, 15] “Now Out of Never: The Element ofSurprise in the East European Revolution of 1989”
![Page 46: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/46.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
17 of 88
The dismal predictive powers of editors...
![Page 47: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/47.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
18 of 88
Social Contagion
Messing with social connectionsI Ads based on message content
(e.g., Google and email)
I BzzAgent (�)I Facebook’s advertising: Beacon (�)
![Page 48: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/48.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
18 of 88
Social Contagion
Messing with social connectionsI Ads based on message content
(e.g., Google and email)
I BzzAgent (�)I Facebook’s advertising: Beacon (�)
![Page 49: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/49.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
18 of 88
Social Contagion
Messing with social connectionsI Ads based on message content
(e.g., Google and email)I BzzAgent (�)I Facebook’s advertising: Beacon (�)
![Page 50: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/50.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
18 of 88
Social Contagion
Messing with social connectionsI Ads based on message content
(e.g., Google and email)I BzzAgent (�)I Facebook’s advertising: Beacon (�)
![Page 51: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/51.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
18 of 88
Social Contagion
Messing with social connectionsI Ads based on message content
(e.g., Google and email)I BzzAgent (�)I Facebook’s advertising: Beacon (�)
![Page 52: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/52.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
19 of 88
Getting others to do things for you
A very good book: ‘Influence’ [7] by Robert Cialdini (�)
Six modes of influence1. Reciprocation: The Old Give and Take... and Take
e.g., Free samples, Hare Krishnas.2. Commitment and Consistency: Hobgoblins of the
Minde.g., Hazing.
3. Social Proof: Truths Are Use.g., Catherine Genovese, Jonestown
4. Liking: The Friendly ThiefSeparation into groups is enough to cause problems.
5. Authority: Directed DeferenceMilgram’s obedience to authority experiment.
6. Scarcity: The Rule of the FewProhibition.
![Page 53: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/53.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
19 of 88
Getting others to do things for you
A very good book: ‘Influence’ [7] by Robert Cialdini (�)
Six modes of influence1. Reciprocation: The Old Give and Take... and Take
e.g., Free samples, Hare Krishnas.2. Commitment and Consistency: Hobgoblins of the
Minde.g., Hazing.
3. Social Proof: Truths Are Use.g., Catherine Genovese, Jonestown
4. Liking: The Friendly ThiefSeparation into groups is enough to cause problems.
5. Authority: Directed DeferenceMilgram’s obedience to authority experiment.
6. Scarcity: The Rule of the FewProhibition.
![Page 54: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/54.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
19 of 88
Getting others to do things for you
A very good book: ‘Influence’ [7] by Robert Cialdini (�)
Six modes of influence1. Reciprocation: The Old Give and Take... and Take
e.g., Free samples, Hare Krishnas.2. Commitment and Consistency: Hobgoblins of the
Minde.g., Hazing.
3. Social Proof: Truths Are Use.g., Catherine Genovese, Jonestown
4. Liking: The Friendly ThiefSeparation into groups is enough to cause problems.
5. Authority: Directed DeferenceMilgram’s obedience to authority experiment.
6. Scarcity: The Rule of the FewProhibition.
![Page 55: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/55.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
19 of 88
Getting others to do things for you
A very good book: ‘Influence’ [7] by Robert Cialdini (�)
Six modes of influence1. Reciprocation: The Old Give and Take... and Take
e.g., Free samples, Hare Krishnas.2. Commitment and Consistency: Hobgoblins of the
Minde.g., Hazing.
3. Social Proof: Truths Are Use.g., Catherine Genovese, Jonestown
4. Liking: The Friendly ThiefSeparation into groups is enough to cause problems.
5. Authority: Directed DeferenceMilgram’s obedience to authority experiment.
6. Scarcity: The Rule of the FewProhibition.
![Page 56: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/56.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
19 of 88
Getting others to do things for you
A very good book: ‘Influence’ [7] by Robert Cialdini (�)
Six modes of influence1. Reciprocation: The Old Give and Take... and Take
e.g., Free samples, Hare Krishnas.2. Commitment and Consistency: Hobgoblins of the
Minde.g., Hazing.
3. Social Proof: Truths Are Use.g., Catherine Genovese, Jonestown
4. Liking: The Friendly ThiefSeparation into groups is enough to cause problems.
5. Authority: Directed DeferenceMilgram’s obedience to authority experiment.
6. Scarcity: The Rule of the FewProhibition.
![Page 57: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/57.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
19 of 88
Getting others to do things for you
A very good book: ‘Influence’ [7] by Robert Cialdini (�)
Six modes of influence1. Reciprocation: The Old Give and Take... and Take
e.g., Free samples, Hare Krishnas.2. Commitment and Consistency: Hobgoblins of the
Minde.g., Hazing.
3. Social Proof: Truths Are Use.g., Catherine Genovese, Jonestown
4. Liking: The Friendly ThiefSeparation into groups is enough to cause problems.
5. Authority: Directed DeferenceMilgram’s obedience to authority experiment.
6. Scarcity: The Rule of the FewProhibition.
![Page 58: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/58.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
19 of 88
Getting others to do things for you
A very good book: ‘Influence’ [7] by Robert Cialdini (�)
Six modes of influence1. Reciprocation: The Old Give and Take... and Take
e.g., Free samples, Hare Krishnas.2. Commitment and Consistency: Hobgoblins of the
Minde.g., Hazing.
3. Social Proof: Truths Are Use.g., Catherine Genovese, Jonestown
4. Liking: The Friendly ThiefSeparation into groups is enough to cause problems.
5. Authority: Directed DeferenceMilgram’s obedience to authority experiment.
6. Scarcity: The Rule of the FewProhibition.
![Page 59: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/59.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
19 of 88
Getting others to do things for you
A very good book: ‘Influence’ [7] by Robert Cialdini (�)
Six modes of influence1. Reciprocation: The Old Give and Take... and Take
e.g., Free samples, Hare Krishnas.2. Commitment and Consistency: Hobgoblins of the
Minde.g., Hazing.
3. Social Proof: Truths Are Use.g., Catherine Genovese, Jonestown
4. Liking: The Friendly ThiefSeparation into groups is enough to cause problems.
5. Authority: Directed DeferenceMilgram’s obedience to authority experiment.
6. Scarcity: The Rule of the FewProhibition.
![Page 60: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/60.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
20 of 88
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)
![Page 61: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/61.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
20 of 88
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)
![Page 62: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/62.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
20 of 88
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)
![Page 63: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/63.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
20 of 88
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)
![Page 64: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/64.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
20 of 88
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)
![Page 65: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/65.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
20 of 88
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)
![Page 66: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/66.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
21 of 88
Social Contagion
Some important modelsI T ipping models—Schelling (1971) [16, 17, 18]
I Simulation on checker boardsI Idea of thresholdsI Explore the Netlogo (�) implementation [21]
I Threshold models—Granovetter (1978) [10]
I Herding models—Bikhchandani, Hirschleifer, Welch(1992) [1, 2]
I Social learning theory, Informational cascades,...
![Page 67: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/67.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
21 of 88
Social Contagion
Some important modelsI T ipping models—Schelling (1971) [16, 17, 18]
I Simulation on checker boardsI Idea of thresholdsI Explore the Netlogo (�) implementation [21]
I Threshold models—Granovetter (1978) [10]
I Herding models—Bikhchandani, Hirschleifer, Welch(1992) [1, 2]
I Social learning theory, Informational cascades,...
![Page 68: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/68.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
21 of 88
Social Contagion
Some important modelsI T ipping models—Schelling (1971) [16, 17, 18]
I Simulation on checker boardsI Idea of thresholdsI Explore the Netlogo (�) implementation [21]
I Threshold models—Granovetter (1978) [10]
I Herding models—Bikhchandani, Hirschleifer, Welch(1992) [1, 2]
I Social learning theory, Informational cascades,...
![Page 69: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/69.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
21 of 88
Social Contagion
Some important modelsI T ipping models—Schelling (1971) [16, 17, 18]
I Simulation on checker boardsI Idea of thresholdsI Explore the Netlogo (�) implementation [21]
I Threshold models—Granovetter (1978) [10]
I Herding models—Bikhchandani, Hirschleifer, Welch(1992) [1, 2]
I Social learning theory, Informational cascades,...
![Page 70: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/70.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
21 of 88
Social Contagion
Some important modelsI T ipping models—Schelling (1971) [16, 17, 18]
I Simulation on checker boardsI Idea of thresholdsI Explore the Netlogo (�) implementation [21]
I Threshold models—Granovetter (1978) [10]
I Herding models—Bikhchandani, Hirschleifer, Welch(1992) [1, 2]
I Social learning theory, Informational cascades,...
![Page 71: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/71.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
21 of 88
Social Contagion
Some important modelsI T ipping models—Schelling (1971) [16, 17, 18]
I Simulation on checker boardsI Idea of thresholdsI Explore the Netlogo (�) implementation [21]
I Threshold models—Granovetter (1978) [10]
I Herding models—Bikhchandani, Hirschleifer, Welch(1992) [1, 2]
I Social learning theory, Informational cascades,...
![Page 72: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/72.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
21 of 88
Social Contagion
Some important modelsI T ipping models—Schelling (1971) [16, 17, 18]
I Simulation on checker boardsI Idea of thresholdsI Explore the Netlogo (�) implementation [21]
I Threshold models—Granovetter (1978) [10]
I Herding models—Bikhchandani, Hirschleifer, Welch(1992) [1, 2]
I Social learning theory, Informational cascades,...
![Page 73: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/73.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
22 of 88
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).
![Page 74: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/74.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
22 of 88
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).
![Page 75: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/75.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
22 of 88
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).
![Page 76: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/76.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
22 of 88
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).
![Page 77: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/77.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
22 of 88
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).
![Page 78: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/78.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
22 of 88
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).
![Page 79: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/79.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
22 of 88
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).
![Page 80: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/80.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
22 of 88
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).
![Page 81: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/81.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
23 of 88
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 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
![Page 82: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/82.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
23 of 88
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 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
![Page 83: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/83.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
23 of 88
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 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
![Page 84: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/84.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
23 of 88
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 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
![Page 85: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/85.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
23 of 88
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 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
![Page 86: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/86.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
23 of 88
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 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
![Page 87: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/87.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
23 of 88
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 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
![Page 88: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/88.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
24 of 88
Outline
Social Contagion ModelsBackgroundGranovetter’s modelNetwork versionGroupsChaos
References
![Page 89: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/89.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
25 of 88
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 .
![Page 90: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/90.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
25 of 88
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 .
![Page 91: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/91.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
25 of 88
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 .
![Page 92: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/92.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
25 of 88
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 .
![Page 93: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/93.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
25 of 88
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 .
![Page 94: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/94.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
26 of 88
Threshold models
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
φp
I Example threshold influence response functions:deterministic and stochastic
I φ = fraction of contacts ‘on’ (e.g., rioting)I Two states: S and I.
![Page 95: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/95.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
26 of 88
Threshold models
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
φp
I Example threshold influence response functions:deterministic and stochastic
I φ = fraction of contacts ‘on’ (e.g., rioting)I Two states: S and I.
![Page 96: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/96.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
26 of 88
Threshold models
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
φp
I Example threshold influence response functions:deterministic and stochastic
I φ = fraction of contacts ‘on’ (e.g., rioting)I Two states: S and I.
![Page 97: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/97.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
27 of 88
Threshold models
I At time t + 1, fraction rioting = fraction with φ∗ ≤ φt .I
φt+1 =
∫ φt
0f (φ∗)dφ∗ = F (φ∗)|φt
0 = F (φt)
I ⇒ Iterative maps of the unit interval [0, 1].
![Page 98: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/98.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
27 of 88
Threshold models
I At time t + 1, fraction rioting = fraction with φ∗ ≤ φt .I
φt+1 =
∫ φt
0f (φ∗)dφ∗ = F (φ∗)|φt
0 = F (φt)
I ⇒ Iterative maps of the unit interval [0, 1].
![Page 99: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/99.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
27 of 88
Threshold models
I At time t + 1, fraction rioting = fraction with φ∗ ≤ φt .I
φt+1 =
∫ φt
0f (φ∗)dφ∗ = F (φ∗)|φt
0 = F (φt)
I ⇒ Iterative maps of the unit interval [0, 1].
![Page 100: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/100.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
28 of 88
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
![Page 101: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/101.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
28 of 88
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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Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
28 of 88
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
![Page 103: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/103.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
29 of 88
Threshold models
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
I Another example of critical mass model...
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Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
30 of 88
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 Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
31 of 88
Threshold models
Implications for collective action theory:1. Collective uniformity 6⇒ individual uniformity2. Small individual changes ⇒ large global changes
![Page 106: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/106.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
31 of 88
Threshold models
Implications for collective action theory:1. Collective uniformity 6⇒ individual uniformity2. Small individual changes ⇒ large global changes
![Page 107: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/107.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
31 of 88
Threshold models
Implications for collective action theory:1. Collective uniformity 6⇒ individual uniformity2. Small individual changes ⇒ large global changes
![Page 108: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/108.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
32 of 88
Threshold models
Chaotic behavior possible [12, 11]
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
![Page 109: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/109.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
32 of 88
Threshold models
Chaotic behavior possible [12, 11]
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
![Page 110: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/110.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
32 of 88
Threshold models
Chaotic behavior possible [12, 11]
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
![Page 111: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/111.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
33 of 88
Outline
Social Contagion ModelsBackgroundGranovetter’s modelNetwork versionGroupsChaos
References
![Page 112: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/112.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
34 of 88
Threshold model on a network
Many years after Granovetter and Soong’s work:
“A simple model of global cascades on random networks”D. J. Watts. Proc. Natl. Acad. Sci., 2002 [20]
I Mean field model → network modelI Individuals now have a limited view of the world
![Page 113: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/113.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
34 of 88
Threshold model on a network
Many years after Granovetter and Soong’s work:
“A simple model of global cascades on random networks”D. J. Watts. Proc. Natl. Acad. Sci., 2002 [20]
I Mean field model → network modelI Individuals now have a limited view of the world
![Page 114: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/114.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
34 of 88
Threshold model on a network
Many years after Granovetter and Soong’s work:
“A simple model of global cascades on random networks”D. J. Watts. Proc. Natl. Acad. Sci., 2002 [20]
I Mean field model → network modelI Individuals now have a limited view of the world
![Page 115: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/115.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
35 of 88
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)
![Page 116: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/116.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
35 of 88
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)
![Page 117: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/117.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
35 of 88
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)
![Page 118: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/118.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
35 of 88
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)
![Page 119: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/119.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
35 of 88
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)
![Page 120: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/120.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
35 of 88
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)
![Page 121: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/121.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
35 of 88
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)
![Page 122: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/122.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
35 of 88
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)
![Page 123: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/123.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
35 of 88
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)
![Page 124: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/124.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
36 of 88
Threshold model on a network
t=1
bc
e
a
d
I All nodes have threshold φ = 0.2.
![Page 125: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/125.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
36 of 88
Threshold model on a network
t=1 t=2
c
a
b
a
bc
ee
d d
I All nodes have threshold φ = 0.2.
![Page 126: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/126.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
36 of 88
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.
![Page 127: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/127.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
37 of 88
Snowballing
The Cascade Condition:1. If one individual is initially activated, what is the
probability that an activation will spread over anetwork?
2. What features of a network determine whether acascade will occur or not?
![Page 128: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/128.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
37 of 88
Snowballing
The Cascade Condition:1. If one individual is initially activated, what is the
probability that an activation will spread over anetwork?
2. What features of a network determine whether acascade will occur or not?
![Page 129: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/129.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
38 of 88
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
![Page 130: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/130.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
38 of 88
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
![Page 131: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/131.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
38 of 88
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
![Page 132: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/132.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
38 of 88
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
![Page 133: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/133.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
38 of 88
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
![Page 134: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/134.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
39 of 88
Snowballing
Follow active linksI An active link is a link connected to an activated
node.I If an infected link leads to at least 1 more infected
link, then activation spreads.I We need to understand which nodes can be
activated when only one of their neigbors becomesactive.
![Page 135: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/135.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
39 of 88
Snowballing
Follow active linksI An active link is a link connected to an activated
node.I If an infected link leads to at least 1 more infected
link, then activation spreads.I We need to understand which nodes can be
activated when only one of their neigbors becomesactive.
![Page 136: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/136.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
39 of 88
Snowballing
Follow active linksI An active link is a link connected to an activated
node.I If an infected link leads to at least 1 more infected
link, then activation spreads.I We need to understand which nodes can be
activated when only one of their neigbors becomesactive.
![Page 137: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/137.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
39 of 88
Snowballing
Follow active linksI An active link is a link connected to an activated
node.I If an infected link leads to at least 1 more infected
link, then activation spreads.I We need to understand which nodes can be
activated when only one of their neigbors becomesactive.
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Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
40 of 88
The most gullible
Vulnerables:I We call individuals who can be activated by just one
contact being active vulnerablesI The vulnerability condition for node i :
1/ki ≥ φi
I Which means # contacts ki ≤ b1/φicI For global cascades on random networks, must have
a global cluster of vulnerables [20]
I Cluster of vulnerables = critical massI Network story: 1 node → critical mass → everyone.
![Page 139: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/139.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
40 of 88
The most gullible
Vulnerables:I We call individuals who can be activated by just one
contact being active vulnerablesI The vulnerability condition for node i :
1/ki ≥ φi
I Which means # contacts ki ≤ b1/φicI For global cascades on random networks, must have
a global cluster of vulnerables [20]
I Cluster of vulnerables = critical massI Network story: 1 node → critical mass → everyone.
![Page 140: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/140.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
40 of 88
The most gullible
Vulnerables:I We call individuals who can be activated by just one
contact being active vulnerablesI The vulnerability condition for node i :
1/ki ≥ φi
I Which means # contacts ki ≤ b1/φicI For global cascades on random networks, must have
a global cluster of vulnerables [20]
I Cluster of vulnerables = critical massI Network story: 1 node → critical mass → everyone.
![Page 141: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/141.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
40 of 88
The most gullible
Vulnerables:I We call individuals who can be activated by just one
contact being active vulnerablesI The vulnerability condition for node i :
1/ki ≥ φi
I Which means # contacts ki ≤ b1/φicI For global cascades on random networks, must have
a global cluster of vulnerables [20]
I Cluster of vulnerables = critical massI Network story: 1 node → critical mass → everyone.
![Page 142: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/142.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
40 of 88
The most gullible
Vulnerables:I We call individuals who can be activated by just one
contact being active vulnerablesI The vulnerability condition for node i :
1/ki ≥ φi
I Which means # contacts ki ≤ b1/φicI For global cascades on random networks, must have
a global cluster of vulnerables [20]
I Cluster of vulnerables = critical massI Network story: 1 node → critical mass → everyone.
![Page 143: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/143.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
40 of 88
The most gullible
Vulnerables:I We call individuals who can be activated by just one
contact being active vulnerablesI The vulnerability condition for node i :
1/ki ≥ φi
I Which means # contacts ki ≤ b1/φicI For global cascades on random networks, must have
a global cluster of vulnerables [20]
I Cluster of vulnerables = critical massI Network story: 1 node → critical mass → everyone.
![Page 144: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/144.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
40 of 88
The most gullible
Vulnerables:I We call individuals who can be activated by just one
contact being active vulnerablesI The vulnerability condition for node i :
1/ki ≥ φi
I Which means # contacts ki ≤ b1/φicI For global cascades on random networks, must have
a global cluster of vulnerables [20]
I Cluster of vulnerables = critical massI Network story: 1 node → critical mass → everyone.
![Page 145: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/145.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
41 of 88
Cascade condition
Back to following a link:I A randomly chosen link, traversed in a random
direction, leads to a degree k node with probability∝ kPk .
I Follows from there being k ways to connect to anode with degree k .
I Normalization:∞∑
k=0
kPk = 〈k〉
I SoP(linked node has degree k) =
kPk
〈k〉
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Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
41 of 88
Cascade condition
Back to following a link:I A randomly chosen link, traversed in a random
direction, leads to a degree k node with probability∝ kPk .
I Follows from there being k ways to connect to anode with degree k .
I Normalization:∞∑
k=0
kPk = 〈k〉
I SoP(linked node has degree k) =
kPk
〈k〉
![Page 147: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/147.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
41 of 88
Cascade condition
Back to following a link:I A randomly chosen link, traversed in a random
direction, leads to a degree k node with probability∝ kPk .
I Follows from there being k ways to connect to anode with degree k .
I Normalization:∞∑
k=0
kPk = 〈k〉
I SoP(linked node has degree k) =
kPk
〈k〉
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Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
41 of 88
Cascade condition
Back to following a link:I A randomly chosen link, traversed in a random
direction, leads to a degree k node with probability∝ kPk .
I Follows from there being k ways to connect to anode with degree k .
I Normalization:∞∑
k=0
kPk = 〈k〉
I SoP(linked node has degree k) =
kPk
〈k〉
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Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
41 of 88
Cascade condition
Back to following a link:I A randomly chosen link, traversed in a random
direction, leads to a degree k node with probability∝ kPk .
I Follows from there being k ways to connect to anode with degree k .
I Normalization:∞∑
k=0
kPk = 〈k〉
I SoP(linked node has degree k) =
kPk
〈k〉
![Page 150: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/150.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
42 of 88
Cascade condition
Next: Vulnerability of linked nodeI Linked node is vulnerable with probability
βk =
∫ 1/k
φ′∗=0
f (φ′∗)dφ′∗
I If linked node is vulnerable, it produces k − 1 newoutgoing active links
I If linked node is not vulnerable, it produces no activelinks.
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Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
42 of 88
Cascade condition
Next: Vulnerability of linked nodeI Linked node is vulnerable with probability
βk =
∫ 1/k
φ′∗=0
f (φ′∗)dφ′∗
I If linked node is vulnerable, it produces k − 1 newoutgoing active links
I If linked node is not vulnerable, it produces no activelinks.
![Page 152: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/152.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
42 of 88
Cascade condition
Next: Vulnerability of linked nodeI Linked node is vulnerable with probability
βk =
∫ 1/k
φ′∗=0
f (φ′∗)dφ′∗
I If linked node is vulnerable, it produces k − 1 newoutgoing active links
I If linked node is not vulnerable, it produces no activelinks.
![Page 153: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/153.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
42 of 88
Cascade condition
Next: Vulnerability of linked nodeI Linked node is vulnerable with probability
βk =
∫ 1/k
φ′∗=0
f (φ′∗)dφ′∗
I If linked node is vulnerable, it produces k − 1 newoutgoing active links
I If linked node is not vulnerable, it produces no activelinks.
![Page 154: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/154.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
43 of 88
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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Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
43 of 88
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〉
![Page 156: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/156.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
43 of 88
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〉
![Page 157: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/157.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
43 of 88
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〉
![Page 158: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/158.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
44 of 88
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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Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
45 of 88
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.
![Page 160: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/160.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
45 of 88
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.
![Page 161: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/161.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
45 of 88
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.
![Page 162: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/162.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
45 of 88
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.
![Page 163: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/163.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
45 of 88
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.
![Page 164: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/164.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
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Cascades on random networks
1 2 3 4 5 6 70
0.2
0.4
0.6
0.8
1
z
⟨ S
⟩
Example networks
PossibleNo
Cascades
Low influence
Fraction ofVulnerables
cascade sizeFinal
CascadesNo Cascades
CascadesNo
High influence
I Cascades occuronly if size of maxvulnerable cluster> 0.
I System may be‘robust-yet-fragile’.
I ‘Ignorance’facilitatesspreading.
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Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
46 of 88
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.
![Page 166: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/166.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
46 of 88
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.
![Page 167: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/167.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
47 of 88
Cascade window for random networks
1 2 3 4 5 6 70
0.2
0.4
0.6
0.8
1
z
⟨ S
⟩
0.05 0.1 0.15 0.2 0.250
5
10
15
20
25
30
φ
z
cascades
no cascades
infl
uenc
e
= uniform individual threshold
I ‘Cascade window’ widens as threshold φ decreases.I Lower thresholds enable spreading.
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Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
48 of 88
Cascade window for random networks
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Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
49 of 88
Cascade window—summary
For our simple model of a uniform threshold:1. Low 〈k〉: No cascades in poorly connected networks.
No global clusters of any kind.2. High 〈k〉: Giant component exists but not enough
vulnerables.3. Intermediate 〈k〉: Global cluster of vulnerables exists.
Cascades are possible in “Cascade window.”
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Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
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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.”
![Page 171: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/171.jpg)
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Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
49 of 88
Cascade window—summary
For our simple model of a uniform threshold:1. Low 〈k〉: No cascades in poorly connected networks.
No global clusters of any kind.2. High 〈k〉: Giant component exists but not enough
vulnerables.3. Intermediate 〈k〉: Global cluster of vulnerables exists.
Cascades are possible in “Cascade window.”
![Page 172: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/172.jpg)
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Granovetter’s model
Network version
Groups
Chaos
References
49 of 88
Cascade window—summary
For our simple model of a uniform threshold:1. Low 〈k〉: No cascades in poorly connected networks.
No global clusters of any kind.2. High 〈k〉: Giant component exists but not enough
vulnerables.3. Intermediate 〈k〉: Global cluster of vulnerables exists.
Cascades are possible in “Cascade window.”
![Page 173: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/173.jpg)
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Network version
Groups
Chaos
References
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All-to-all versus random networks
0
0.2
0.4
0.6
0.8
1
a0 a
t
F (
a t+1)
all−to−all networks
A
0
0.2
0.4
0.6
0.8
1
⟨ k ⟩⟨ S
⟩
random networks
B
0 0.5 10
0.2
0.4
0.6
0.8
1
a0 a’
0a
t
F (
a t+1)
C
0 10 200
0.2
0.4
0.6
0.8
1
⟨ k ⟩
⟨ S ⟩
D
0 0.5 10
5
10
φ∗
f (φ ∗)
0 0.5 10
2
4
φ∗
f (φ ∗)
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Groups
Chaos
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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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Network version
Groups
Chaos
References
51 of 88
Early adopters—degree distributions
t = 0 t = 1 t = 2 t = 3
0 5 10 15 200
0.05
0.1
0.15
0.2
t = 0
0 5 10 15 200
0.2
0.4
0.6
0.8
t = 1
0 5 10 15 200
0.2
0.4
0.6
0.8
t = 2
0 5 10 15 200
0.2
0.4
0.6
0.8
t = 3
t = 4 t = 6 t = 8 t = 10
0 5 10 15 200
0.1
0.2
0.3
0.4
0.5
t = 4
0 5 10 15 200
0.1
0.2
0.3
0.4
0.5
t = 6
0 5 10 15 200
0.1
0.2
0.3
0.4
t = 8
0 5 10 15 200
0.1
0.2
0.3
0.4
t = 10
t = 12 t = 14 t = 16 t = 18
0 5 10 15 200
0.05
0.1
0.15
0.2
t = 12
0 5 10 15 200
0.05
0.1
0.15
0.2
t = 14
0 5 10 15 200
0.05
0.1
0.15
0.2
t = 16
0 5 10 15 200
0.05
0.1
0.15
0.2
t = 18
Pk ,t versus k
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Network version
Groups
Chaos
References
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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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Granovetter’s model
Network version
Groups
Chaos
References
51 of 88
Early adopters—degree distributions
t = 0 t = 1 t = 2 t = 3
0 5 10 15 200
0.05
0.1
0.15
0.2
t = 0
0 5 10 15 200
0.2
0.4
0.6
0.8
t = 1
0 5 10 15 200
0.2
0.4
0.6
0.8
t = 2
0 5 10 15 200
0.2
0.4
0.6
0.8
t = 3
t = 4 t = 6 t = 8 t = 10
0 5 10 15 200
0.1
0.2
0.3
0.4
0.5
t = 4
0 5 10 15 200
0.1
0.2
0.3
0.4
0.5
t = 6
0 5 10 15 200
0.1
0.2
0.3
0.4
t = 8
0 5 10 15 200
0.1
0.2
0.3
0.4
t = 10
t = 12 t = 14 t = 16 t = 18
0 5 10 15 200
0.05
0.1
0.15
0.2
t = 12
0 5 10 15 200
0.05
0.1
0.15
0.2
t = 14
0 5 10 15 200
0.05
0.1
0.15
0.2
t = 16
0 5 10 15 200
0.05
0.1
0.15
0.2
t = 18
Pk ,t versus k
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The multiplier effect:
1 2 3 4 5 60
0.2
0.4
0.6
0.8
1
navg
S avg
A
1 2 3 4 5 60
1
2
3
4
navg
B
Gain
InfluenceInfluence
Averageindividuals
Top 10% individuals
Cas
cade
siz
e
Cascade size ratio
Degree ratio
I Fairly uniform levels of individual influence.I Multiplier effect is mostly below 1.
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Network version
Groups
Chaos
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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
3
6
9
12
navg
B
Cas
cade
siz
e
InfluenceAverageIndividuals
Top 10% individuals Cascade size ratio
Degree ratio
Gain
I Skewed influence distribution example.
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Groups
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Special subnetworks can act as triggers
i0
A
B
I φ = 1/3 for all nodes
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Outline
Social Contagion ModelsBackgroundGranovetter’s modelNetwork versionGroupsChaos
References
![Page 182: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/182.jpg)
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The power of groups...
despair.com
“A few harmless flakesworking together canunleash an avalancheof destruction.”
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Network version
Groups
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Extensions
I Assumption of sparse interactions is goodI Degree distribution is (generally) key to a network’s
functionI Still, random networks don’t represent all networksI Major element missing: group structure
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Network version
Groups
Chaos
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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
![Page 185: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/185.jpg)
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Granovetter’s model
Network version
Groups
Chaos
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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
![Page 186: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/186.jpg)
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Network version
Groups
Chaos
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Extensions
I Assumption of sparse interactions is goodI Degree distribution is (generally) key to a network’s
functionI Still, random networks don’t represent all networksI Major element missing: group structure
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Group structure—Ramified random networks
p = intergroup connection probabilityq = intragroup connection probability.
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Bipartite networks
c d ea b
2 3 41
a
b
c
d
e
contexts
individuals
unipartitenetwork
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Context distance
eca
high schoolteacher
occupation
health careeducation
nurse doctorteacherkindergarten
db
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Generalized affiliation model
100
eca b d
geography occupation age
0
(Blau & Schwartz, Simmel, Breiger)
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Generalized affiliation model networks withtriadic closure
I Connect nodes with probability ∝ exp−αd
whereα = homophily parameterandd = distance between nodes (height of lowestcommon ancestor)
I τ1 = intergroup probability of friend-of-friendconnection
I τ2 = intragroup probability of friend-of-friendconnection
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Generalized affiliation model networks withtriadic closure
I Connect nodes with probability ∝ exp−αd
whereα = homophily parameterandd = distance between nodes (height of lowestcommon ancestor)
I τ1 = intergroup probability of friend-of-friendconnection
I τ2 = intragroup probability of friend-of-friendconnection
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Groups
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Generalized affiliation model networks withtriadic closure
I Connect nodes with probability ∝ exp−αd
whereα = homophily parameterandd = distance between nodes (height of lowestcommon ancestor)
I τ1 = intergroup probability of friend-of-friendconnection
I τ2 = intragroup probability of friend-of-friendconnection
![Page 194: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/194.jpg)
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Cascade windows for group-based networksG
ener
alize
d Af
filiat
ion
A
Gro
up n
etwo
rks
Single seed Coherent group seed
Mod
el n
etwo
rks
Random set seed
Rand
om
F
C
D E
B
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Multiplier effect for group-based networks:
4 8 12 16 200
0.2
0.4
0.6
0.8
1
navg
S avg
A
4 8 12 16 200
1
2
3
navg
B
0 4 8 12 160
0.2
0.4
0.6
0.8
1
navg
S avg
C
0 4 8 12 160
1
2
3
navg
D
Degree ratio
Gain
Cascadesize ratio
Cascadesize ratio < 1!
I Multiplier almost always below 1.
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Groups
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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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Groups
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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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Groups
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Social contagion
SummaryI ‘Influential vulnerables’ are key to spread.I Early adopters are mostly vulnerables.I Vulnerable nodes important but not necessary.I Groups may greatly facilitate spread.I Seems that cascade condition is a global one.I Most extreme/unexpected cascades occur in highly
connected networksI ‘Influentials’ are posterior constructs.I Many potential influentials exist.
![Page 199: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/199.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
66 of 88
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.
![Page 200: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/200.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
66 of 88
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.
![Page 201: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/201.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
66 of 88
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.
![Page 202: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/202.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
66 of 88
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.
![Page 203: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/203.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
66 of 88
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.
![Page 204: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/204.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
66 of 88
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.
![Page 205: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/205.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
66 of 88
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.
![Page 206: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/206.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
67 of 88
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.
![Page 207: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/207.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
67 of 88
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.
![Page 208: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/208.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
67 of 88
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.
![Page 209: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/209.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
67 of 88
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.
![Page 210: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/210.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
67 of 88
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.
![Page 211: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/211.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
67 of 88
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.
![Page 212: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/212.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
68 of 88
Outline
Social Contagion ModelsBackgroundGranovetter’s modelNetwork versionGroupsChaos
References
![Page 213: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/213.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
69 of 88
Chaotic contagion:
I What if individual response functions are notmonotonic?
I Consider a simple deterministic version:
I Node i has an ‘activation threshold’ φi,1
. . . and a ‘de-activation threshold’ φi,2
I Nodes like to imitate but only up to alimit—they don’t want to be likeeveryone else.
![Page 214: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/214.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
69 of 88
Chaotic contagion:
I What if individual response functions are notmonotonic?
I Consider a simple deterministic version:
I Node i has an ‘activation threshold’ φi,1
. . . and a ‘de-activation threshold’ φi,2
I Nodes like to imitate but only up to alimit—they don’t want to be likeeveryone else.
0 10
0.2
0.4
0.6
0.8
1
!i,1
"!i,2
"
A
!i,t
Pr(ai,t+1=1)
0 0.5 10
0.2
0.4
0.6
0.8
1B
!1
"
!2"
0 0.5 10
0.2
0.4
0.6
0.8
1
!1
"
!2"
C
0 0.5 10
0.5
1
0 0.5 10
0.5
1
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Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
69 of 88
Chaotic contagion:
I What if individual response functions are notmonotonic?
I Consider a simple deterministic version:
I Node i has an ‘activation threshold’ φi,1
. . . and a ‘de-activation threshold’ φi,2
I Nodes like to imitate but only up to alimit—they don’t want to be likeeveryone else.
0 10
0.2
0.4
0.6
0.8
1
!i,1
"!i,2
"
A
!i,t
Pr(ai,t+1=1)
0 0.5 10
0.2
0.4
0.6
0.8
1B
!1
"
!2"
0 0.5 10
0.2
0.4
0.6
0.8
1
!1
"
!2"
C
0 0.5 10
0.5
1
0 0.5 10
0.5
1
![Page 216: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/216.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
70 of 88
Two population examples:
0 10
0.2
0.4
0.6
0.8
1
φi,1∗ φ
i,2∗
A
φi,t
Pr(a
i,t+
1=1)
0 0.5 10
0.2
0.4
0.6
0.8
1B
φ1∗
φ 2∗
0 0.5 10
0.2
0.4
0.6
0.8
1
φ1∗
φ 2∗
C
0 0.5 10
0.5
1
0 0.5 10
0.5
1
I Randomly select (φi,1, φi,2) from gray regions shownin plots B and C.
I Insets show composite response function averagedover population.
I We’ll consider plot C’s example: the tent map.
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Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
70 of 88
Two population examples:
0 10
0.2
0.4
0.6
0.8
1
φi,1∗ φ
i,2∗
A
φi,t
Pr(a
i,t+
1=1)
0 0.5 10
0.2
0.4
0.6
0.8
1B
φ1∗
φ 2∗
0 0.5 10
0.2
0.4
0.6
0.8
1
φ1∗
φ 2∗
C
0 0.5 10
0.5
1
0 0.5 10
0.5
1
I Randomly select (φi,1, φi,2) from gray regions shownin plots B and C.
I Insets show composite response function averagedover population.
I We’ll consider plot C’s example: the tent map.
![Page 218: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/218.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
71 of 88
Chaotic contagion
Definition of the tent map:
F (x) =
{rx for 0 ≤ x ≤ 1
2 ,
r(1− x) for 12 ≤ x ≤ 1.
I The usual business: look at how F iteratively mapsthe unit interval [0, 1].
![Page 219: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/219.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
72 of 88
The tent map
Effect of increasing r from 1 to 2.
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
xn
F (
x n+1 )
![Page 220: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/220.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
72 of 88
The tent map
Effect of increasing r from 1 to 2.
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
xn
F (
x n+1 )
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
xn
F (
x n+1 )
![Page 221: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/221.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
72 of 88
The tent map
Effect of increasing r from 1 to 2.
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
xn
F (
x n+1 )
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 )
![Page 222: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/222.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
72 of 88
The tent map
Effect of increasing r from 1 to 2.
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
xn
F (
x n+1 )
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
xn
F (
x n+1 )
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
xn
F (
x n+1 )
Orbit diagram:Chaotic behavior increasesas map slope r is increased.
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Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
73 of 88
Chaotic behavior
Take r = 2 case:
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
xn
F (
x n+1 )
I What happens if nodes have limited information?I As before, allow interactions to take place on a
sparse random network.I Vary average degree z = 〈k〉, a measure of
information
![Page 224: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/224.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
73 of 88
Chaotic behavior
Take r = 2 case:
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
xn
F (
x n+1 )
I What happens if nodes have limited information?I As before, allow interactions to take place on a
sparse random network.I Vary average degree z = 〈k〉, a measure of
information
![Page 225: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/225.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
73 of 88
Chaotic behavior
Take r = 2 case:
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
xn
F (
x n+1 )
I What happens if nodes have limited information?I As before, allow interactions to take place on a
sparse random network.I Vary average degree z = 〈k〉, a measure of
information
![Page 226: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/226.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
73 of 88
Chaotic behavior
Take r = 2 case:
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
xn
F (
x n+1 )
I What happens if nodes have limited information?I As before, allow interactions to take place on a
sparse random network.I Vary average degree z = 〈k〉, a measure of
information
![Page 227: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/227.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
74 of 88
Invariant densities—stochastic responsefunctions
0 500 1000 1500 20000
0.2
0.4
0.6
0.8
1
t
s
z = 5
0 0.5 10
20
40
60
sP(
s)
z = 5
activation time series activation density
![Page 228: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/228.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
75 of 88
Invariant densities—stochastic responsefunctions
0 500 1000 1500 20000
0.2
0.4
0.6
0.8
1
t
s
z = 5
0 0.5 10
20
40
60
s
P(s)
z = 5
0 500 1000 1500 20000
0.2
0.4
0.6
0.8
1
t
s
z = 10
0 0.5 10
10
20
30
s
P(s)
z = 10
0 500 1000 1500 20000
0.2
0.4
0.6
0.8
1
t
s
z = 18
0 0.5 10
5
10
15
20
s
P(s)
z = 18
0 500 1000 1500 20000
0.2
0.4
0.6
0.8
1
t
s
z = 22
0 0.5 10
2
4
6
8
10
s
P(s)
z = 22
0 500 1000 1500 20000
0.2
0.4
0.6
0.8
1
t
s
z = 24
0 0.5 10
2
4
6
8
s
P(s)
z = 24
0 500 1000 1500 20000
0.2
0.4
0.6
0.8
1
t
s
z = 30
0 0.5 10
2
4
6
sP(
s)
z = 30
![Page 229: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/229.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
76 of 88
Invariant densities—deterministic responsefunctions for one specific network with〈k〉 = 18
0 500 1000 1500 20000
0.2
0.4
0.6
0.8
1
t
s
z = 18
0 0.5 10
10
20
30
s
P(s)
z = 18
0 500 1000 1500 20000
0.2
0.4
0.6
0.8
1
t
s
z = 18
0 0.5 10
5
10
15
20
25
s
P(s)
z = 18
0 500 1000 1500 20000
0.2
0.4
0.6
0.8
1
t
s
z = 18
0 0.5 10
10
20
30
40
50
s
P(s)
z = 18
0 500 1000 1500 20000
0.2
0.4
0.6
0.8
1
t
s
z = 18
0 0.5 10
10
20
30
40
50
s
P(s)
z = 18
0 500 1000 1500 20000
0.2
0.4
0.6
0.8
1
t
s
z = 18
0 0.5 10
2
4
6
8
s
P(s)
z = 18
0 500 1000 1500 20000
0.2
0.4
0.6
0.8
1
t
s
z = 18
0 0.5 10
10
20
30
s
P(s)
z = 18
![Page 230: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/230.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
77 of 88
Invariant densities—stochastic responsefunctions
0 500 1000 1500 20000
0.2
0.4
0.6
0.8
1
t
s
z = 100
0 0.5 10
1
2
3
4
s
P(s)
z = 100
0 500 1000 1500 20000
0.2
0.4
0.6
0.8
1
t
s
z = 1000
0 0.5 10
0.5
1
1.5
2
2.5
s
P(s)
z = 1000
Trying out higher values of 〈k〉. . .
![Page 231: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/231.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
78 of 88
Invariant densities—deterministic responsefunctions
0 500 1000 1500 20000
0.2
0.4
0.6
0.8
1
t
s
z = 100
0 0.5 10
1
2
3
4
s
P(s)
z = 100
0 500 1000 1500 20000
0.2
0.4
0.6
0.8
1
t
s
z = 1000
0 0.5 10
0.5
1
1.5
2
2.5
s
P(s)
z = 1000
0 5000 100000
0.2
0.4
0.6
0.8
1
t
s
z = 100
0 0.5 10
10
20
30
s
P(s)
z = 100
Trying out higher values of 〈k〉. . .
![Page 232: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/232.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
79 of 88
Connectivity leads to chaos:
Stochastic response functions:
![Page 233: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/233.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
80 of 88
Chaotic behavior in coupled systems
Coupled maps are well explored(Kaneko/Kuramoto):
xi,n+1 = f (xi,n) +∑j∈Ni
δi,j f (xj,n)
I Ni = neighborhood of node i
1. Node states are continuous2. Increase δ and neighborhood size |N |
⇒ synchronization
But for contagion model:1. Node states are binary2. Asynchrony remains as connectivity increases
![Page 234: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/234.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
80 of 88
Chaotic behavior in coupled systems
Coupled maps are well explored(Kaneko/Kuramoto):
xi,n+1 = f (xi,n) +∑j∈Ni
δi,j f (xj,n)
I Ni = neighborhood of node i
1. Node states are continuous2. Increase δ and neighborhood size |N |
⇒ synchronization
But for contagion model:1. Node states are binary2. Asynchrony remains as connectivity increases
![Page 235: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/235.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
80 of 88
Chaotic behavior in coupled systems
Coupled maps are well explored(Kaneko/Kuramoto):
xi,n+1 = f (xi,n) +∑j∈Ni
δi,j f (xj,n)
I Ni = neighborhood of node i
1. Node states are continuous2. Increase δ and neighborhood size |N |
⇒ synchronization
But for contagion model:1. Node states are binary2. Asynchrony remains as connectivity increases
![Page 236: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/236.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
80 of 88
Chaotic behavior in coupled systems
Coupled maps are well explored(Kaneko/Kuramoto):
xi,n+1 = f (xi,n) +∑j∈Ni
δi,j f (xj,n)
I Ni = neighborhood of node i
1. Node states are continuous2. Increase δ and neighborhood size |N |
⇒ synchronization
But for contagion model:1. Node states are binary2. Asynchrony remains as connectivity increases
![Page 237: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/237.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
80 of 88
Chaotic behavior in coupled systems
Coupled maps are well explored(Kaneko/Kuramoto):
xi,n+1 = f (xi,n) +∑j∈Ni
δi,j f (xj,n)
I Ni = neighborhood of node i
1. Node states are continuous2. Increase δ and neighborhood size |N |
⇒ synchronization
But for contagion model:1. Node states are binary2. Asynchrony remains as connectivity increases
![Page 238: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/238.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
80 of 88
Chaotic behavior in coupled systems
Coupled maps are well explored(Kaneko/Kuramoto):
xi,n+1 = f (xi,n) +∑j∈Ni
δi,j f (xj,n)
I Ni = neighborhood of node i
1. Node states are continuous2. Increase δ and neighborhood size |N |
⇒ synchronization
But for contagion model:1. Node states are binary2. Asynchrony remains as connectivity increases
![Page 239: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/239.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
81 of 88
Bifurcation diagram: Asynchronous updating
0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
α
P(s
| r)/
max
P(s
| r)
![Page 240: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/240.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
82 of 88
References I
[1] S. Bikhchandani, D. Hirshleifer, and I. Welch.A theory of fads, fashion, custom, and culturalchange as informational cascades.J. Polit. Econ., 100:992–1026, 1992.
[2] S. Bikhchandani, D. Hirshleifer, and I. Welch.Learning from the behavior of others: Conformity,fads, and informational cascades.J. Econ. Perspect., 12(3):151–170, 1998. pdf (�)
[3] J. M. Carlson and J. Doyle.Highly optimized tolerance: A mechanism for powerlaws in designed systems.Phys. Rev. E, 60(2):1412–1427, 1999. pdf (�)
![Page 241: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/241.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
83 of 88
References II
[4] J. M. Carlson and J. Doyle.Highly optimized tolerance: Robustness and designin complex systems.Phys. Rev. Lett., 84(11):2529–2532, 2000. pdf (�)
[5] N. A. Christakis and J. H. Fowler.The spread of obesity in a large social network over32 years.New England Journal of Medicine, 357:370–379,2007. pdf (�)
[6] N. A. Christakis and J. H. Fowler.The collective dynamics of smoking in a large socialnetwork.New England Journal of Medicine, 358:2249–2258,2008. pdf (�)
![Page 242: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/242.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
84 of 88
References III
[7] R. B. Cialdini.Influence: Science and Practice.Allyn and Bacon, Boston, MA, 4th edition, 2000.
[8] J. H. Fowler and N. A. Christakis.Dynamic spread of happiness in a large socialnetwork: longitudinal analysis over 20 years in theFramingham Heart Study.BMJ, 337:article #2338, 2008. pdf (�)
[9] M. Gladwell.The Tipping Point.Little, Brown and Company, New York, 2000.
[10] M. Granovetter.Threshold models of collective behavior.Am. J. Sociol., 83(6):1420–1443, 1978. pdf (�)
![Page 243: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/243.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
85 of 88
References IV
[11] M. Granovetter and R. Soong.Threshold models of diversity: Chinese restaurants,residential segregation, and the spiral of silence.Sociological Methodology, 18:69–104, 1988. pdf (�)
[12] M. S. Granovetter and R. Soong.Threshold models of interpersonal effects inconsumer demand.Journal of Economic Behavior & Organization,7:83–99, 1986.Formulates threshold as function of price, andintroduces exogenous supply curve. pdf (�)
[13] E. Katz and P. F. Lazarsfeld.Personal Influence.The Free Press, New York, 1955.
![Page 244: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/244.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
86 of 88
References V
[14] T. Kuran.Now out of never: The element of surprise in theeast european revolution of 1989.World Politics, 44:7–48, 1991. pdf (�)
[15] T. Kuran.Private Truths, Public Lies: The SocialConsequences of Preference Falsification.Harvard University Press, Cambridge, MA, Reprintedition, 1997.
[16] T. C. Schelling.Dynamic models of segregation.J. Math. Sociol., 1:143–186, 1971.
![Page 245: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/245.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
87 of 88
References VI
[17] T. C. Schelling.Hockey helmets, concealed weapons, and daylightsaving: A study of binary choices with externalities.J. Conflict Resolut., 17:381–428, 1973. pdf (�)
[18] T. C. Schelling.Micromotives and Macrobehavior.Norton, New York, 1978.
[19] D. Sornette.Critical Phenomena in Natural Sciences.Springer-Verlag, Berlin, 2nd edition, 2003.
[20] D. J. Watts.A simple model of global cascades on randomnetworks.Proc. Natl. Acad. Sci., 99(9):5766–5771, 2002.pdf (�)
![Page 246: Social Contagion - Principles of Complex Systems …pdodds/teaching/courses/2011-08UVM...CSYS/MATH 300, Fall, 2011 Prof. Peter Dodds Department of Mathematics & Statistics | Center](https://reader035.vdocuments.net/reader035/viewer/2022071301/60a07e97c73ac0525b7d8b2b/html5/thumbnails/246.jpg)
Social Contagion
Social ContagionModelsBackground
Granovetter’s model
Network version
Groups
Chaos
References
88 of 88
References VII
[21] U. Wilensky.Netlogo segregation model.http://ccl.northwestern.edu/netlogo/models/Segregation. Center for ConnectedLearning and Computer-Based Modeling,Northwestern University, Evanston, IL., 1998.