SN- Lecture 11
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The document discusses how network structure impacts behavior, specifically in the context of contagion. It provides an example of a girls' dormitory social network and how a disease might spread through it. Real-life social networks often have properties like heavy-tailed degree distributions, small diameters, and high clustering that influence contagion. These network structures determine the extent to which infections, ideas, or behaviors can diffuse.
SN- Lecture 11
- 2. To understand: Aims Lecture 11 The functioning of contagion in networks Some properties of real life social networks
- 3. How does network structure impact behavior? Networks & Behavior In this lecture we will take the networks as given We are going to see the effect of a network on behavior Simple infections, contagion, diffusion (1 or 0) Choices, decisions - games on networks (strategic interaction ) + + Next Lecture
- 4. However Keep in mind The relation between the outcomes we can get for a given network will affect which links we form There is a co-determination between structure & behavior The macro-micro-macro link For this course, we will look at them separately At the end of the course we will say something more about it
- 5. Example Contagion In a girls dormitory in college, he asked students about their friendship: specifically in the dinning table Jacob Moreno (1960) Two choices (number 1 and number 2 friends you dine with) Who do you dine with? He put all the data into a network
- 6. Example Contagion 1 1 Cora 2 Jean Hellen Robin 2 1 1 Ada 2 1 2 2 Louise Lena Marion 2 1 2 Eva 2 1 Martha 2 1 1 2 1 2 2 Adele Maxine Frances 1 1 2 Anna1 Alice Laura Ella 2 1 Ellen 2 1 Edna 1 2 1 2 Mary Jane Hazel Betty Hilda Ruth Irene 2 1 1 2 2 1 2 1 2 1 2 1 2 1 1 2
- 7. Contagion 1 1 Cora 2 Jean Hellen Robin 2 1 1 Ada 2 1 2 2 Louise Lena Marion 2 1 2 Eva 2 1 Martha 2 1 1 2 1 2 2 Adele Maxine Frances 1 1 2 Anna1 Alice Laura Ella 2 1 Ellen 2 1 Edna 1 2 1 2 Mary Jane Hazel Betty Hilda Ruth Irene 2 1 1 2 2 1 2 1 2 1 2 1 2 1 1 2 Ada is my first choice and Jean my second Direction in the connections
- 8. Contagion 1 1 Cora 2 Jean Hellen Robin 2 1 1 Ada 2 1 2 2 Louise Lena Marion 2 1 2 Eva 2 1 Martha 2 1 1 2 1 2 2 Adele Maxine Frances 1 1 2 Anna1 Alice Laura Ella 2 1 Ellen 2 1 Edna 1 2 1 2 Mary Jane Hazel Betty Hilda Ruth Irene 2 1 1 2 2 1 2 1 2 1 2 1 2 1 1 2 Ada is my first choice and Jean my second Hellen & Robin are my choices Direction in the connections
- 9. Who is the most popular girl? Contagion 1 1 Cora 2 Jean Hellen Robin 2 1 1 Ada 2 1 2 2 Louise Lena Marion 2 1 2 Eva 2 1 Martha 2 1 1 2 1 2 2 Adele Maxine Frances 1 1 2 Anna1 Alice Laura Ella 2 1 Ellen 2 1 Edna 1 2 1 2 Mary Jane Hazel Betty Hilda Ruth Irene 2 1 1 2 2 1 2 1 2 1 2 1 2 1 1 2
- 10. Who is the most popular girl? Contagion 1 1 Cora 2 Jean Hellen Robin 2 1 1 Ada 2 1 2 2 Louise Lena Marion 2 1 2 Eva 2 1 Martha 2 1 1 2 1 2 2 Adele Maxine Frances 1 1 2 Anna1 Alice Laura Ella 2 1 Ellen 2 1 Edna 1 2 1 2 Mary Jane Hazel Betty Hilda Ruth Irene 2 1 1 2 2 1 2 1 2 1 2 1 2 1 1 2
- 11. In directed networks Variants of degree In-degree Number of links from others to me Out-degree Number of links from me to others Reciprocity I choose a person who also chooses me
- 12. Isolated Group Contagion 1 1 Cora 2 Jean Hellen Robin 2 1 1 Ada 2 1 2 2 Louise Lena Marion 2 1 2 Eva 2 1 Martha 2 1 1 2 1 2 2 Adele Maxine Frances 1 1 2 Anna1 Alice Laura Ella 2 1 Ellen 2 1 Edna 1 2 1 2 Mary Jane Hazel Betty Hilda Ruth Irene 2 1 1 2 2 1 2 1 2 1 2 1 2 1 1 2 No reciprocity out degree>in degree
- 13. Size of these isolated groups Important If you are in an isolated group, and a disease outbreaks, it might get stucked in those isolated locations Measures of connectivity & navigation in the network are fundamental for problems of contagion
- 14. Always mutual Contact Cora Jean Hellen Robin Ada Louise Lena Marion Eva Martha Adele Maxine Anna Alice Laura Ella Ellen Edna Mary Jane Hazel Betty Hilda Ruth Irene Frances
- 15. Who are the people in more danger? Contagion Cora Jean Hellen Robin Ada Louise Lena Marion Eva Martha Adele Anna Alice Laura Ella Ellen Edna Mary Jane Hazel Betty Hilda Ruth Irene Frances Maxine I am sick
- 16. & after that? Contagion Cora Jean Hellen Robin Ada Louise Lena Marion Eva Martha Adele Anna Alice Laura Ella Ellen Edna Mary Jane Hazel Betty Hilda Ruth Irene Frances Maxine
- 17. & after that? Contagion Cora Jean Hellen Robin Ada Louise Lena Marion Eva Martha Adele Anna Alice Laura Ella Ellen Edna Mary Jane Hazel Betty Hilda Ruth Irene Frances Maxine
- 18. Contagion The previous were just some questions you can answer using networks More to come up, but first an example
- 20. What’s the extent of diffusion? Other Questions How does it depend on the process as well as the network? Is everyone infected? Are some network architechtures more suitable for contagion?
- 21. Low density - no contagion Some main results Part of the population infected Degree affects who is infected & when Middles density - some probability of contagion High density - sure infection & all infected Network structure matters This is only one side of the problem
- 22. How do they look like? Real life networks It is important to know what kind of networks allow transmission to flow better/worse But, how do real life networks relate to this? Are there universal structural properties? Every network is unique microscopically, but with appropriate definitions, stricking macroscopic commonalities emerge
- 23. Large scale networks Properties Main claim: Typical large scale networks exhibit: Heavy-tailed degree distributions Small diameter High clustering Hubs or connectors Six degrees of separation? Friends of friends are friends
- 24. degree distributions Heavy-tailed Lots of nodes with small degree and few nodes with very high degree Degree Number of nodes
- 25. degree distributions Heavy-tailed Erdös Number Project http://www.oakland.edu/enp/ Paul Erdös 1913-1996 Collaboration network between mathematicians Nodes are mathematicians Link if they coautor a research paper together Paul Erdös is in the center The number of a node is her distance to P.E. P.E. has an Erdös-number = 0 A coauthor of P.E. has Erdös-number = 1 Their coauthors = 2, and so on...
- 26. Network of P.E.’s coauthors Erdös-number
- 27. Degree distributions Details: Size (N) 410,000 authors Size (g) 676,000 links Average (di) 3.36 Erdös number 0 --- 1 person Erdös number 1 --- 504 people Erdös number 2 --- 6593 people Erdös number 3 --- 33605 people Erdös number 4 --- 83642 people Erdös number 5 --- 87760 people Erdös number 6 --- 40014 people Erdös number 7 --- 11591 people Erdös number 8 --- 3146 people Erdös number 9 --- 819 people Erdös number 10 --- 244 people Erdös number 11 --- 68 people Erdös number 12 --- 23 people Erdös number 13 --- 5 people Diameter (N,g) 7.64 Similar network for acting & Kevin Bacon
- 28. Compared to the population size Small Diameter Arguably, every person in the world is at diameter 6 from anyone else http://www.youtube.com/watch?v=HLIyuYwbVnA Think about Milgram’s experiment with the letters Other Networks: Messenger (Lescovec & Horvitz, 2008) Diameter = 6.5; N = 180 millions Facebook (Backstrom et al., 2012) Diameter = 5; N = 721 millions
- 29. Compared to average degree High Clustering How likely two nodes that share a common neighbor are to be neighbors themselves Examples Networks: (Watts, 2003) Movie actor network C.C.=0.79 ; p=0.00027 Neuronal network C.C.=0.28 ; p=0.05
- 30. Checklist Network structure matters for contagion Individual degrees affect who is infected and when Real life networks portray some universal properties Heavy-tailed degree distribution Small diameter compared to the size of the population High clustering compared to the average degree
- 31. Questions?