Social Networks 101

1. Social Networks 101 Prof. Jason Hartline and Prof. Nicole Immorlica

2. Guessing game Experiment: There are three balls in this urn, either or

3. Guessing game Experiment: This is called a blue urn. This is called a yellow urn.

4. Guessing game Experiment: When I call your name, 1. You (and only you) will see a random ball. 2. You must then guess if the urn is a blue urn or a yellow (red) urn, and tell the class your guess. If you guess correctly, you will earn one point.

5. What should the 1st student do? ? ? ? Guess that urn is same color as ball.

6. What should the 2nd student do? 1st guess was blue. ? ? ?

7. What should the 2nd student do? 1st guess was blue. ? ? ? Guess that urn is same color as ball.

8. What should the 3rd student do? 1st and 2nd guesses were blue. ? ? ? Guess that urn is blue no matter what she sees!

9. What should the nth student do? First (n-1) guesses were blue. ? ? ? First 2 students told the truth.

10. If the first two guesses are blue, everyone should guess blue.

11. Staring up at the sky. Choosing a restaurant in a strange town. Self-reinforcing success of best-selling books. Voting for popular candidates.

12. Information cascades When: 1. People make decisions sequentially, 2. and observe actions of earlier people. Information cascade: People abandon own info. in favor of inferences based on others’ actions.

13. Rational Imitation Cascade (not simply peer pressure)

14. Observations • Cascades are easy to start. (every student makes same guess so long as first two students make same guess)

15. Observations 2. Cascades can lead to bad outcomes. (given blue urn, chance of seeing red ball is 1/3, so first two students guess red with prob. (1/3)2 = 1/9)

16. Observations 3. Cascades are fragile. (if a few students show the color of the ball they picked to the entire class, the cascade can be reversed)

17. Conditional probability How likely is it that the urn is yellow given what you’ve seen and heard?

18. Strategy for urns A player should guess yellow if, Pr [ yellow urn | what she saw and heard ] > ½ and blue otherwise.

19. Analysis From setup of experiment, Pr [ ] = Pr [ ] = 1/2

20. Analysis From composition of urns, Pr [ | ] = Pr [ | ] = 2/3

21. Analysis Suppose first student draws : Pr [ | ] x Pr [ ] Pr [ | ] = Pr [ ]

22. Analysis Suppose first student draws : (2/3) Pr [ | ] x (1/2) x Pr [ ] Pr [ | ] = Pr [ ] Pr [ ] = Pr [ | ] x (2/3) x (1/2) Pr [ ] + (1/3) x (1/2) = (1/2) Pr [ | ] x Pr [ ]

23. Analysis Suppose first student draws : then he should guess yellow. Pr [ | ] = (2/3) > (1/2)

24. Analysis Suppose second student draws too: Pr [ | , ] 1st student’s guess 2nd student’s draw

25. Analysis Suppose second student draws too: by a similar analysis, so she also guesses yellow. Pr [ | , ] > (1/2)

26. Analysis Suppose third student draws : Pr [ | , , ] 1st student’s guess 3rd student’s draw 2nd student’s guess

27. Analysis Suppose third student draws : Pr [ | , , ] = (1/3) x (2/3) x (2/3) (1/2) Pr [ , , | ] x Pr [ ] Pr [ , , ]

28. Analysis Pr [ , , ] = (1/3) x (2/3) x (2/3) (1/2) Pr [ , , | ] x Pr [ ] (2/3) x (1/3) x (1/3) (1/2) + Pr [ , , | ] x Pr [ ]

29. Analysis Suppose third student draws : Pr [ | , , ] = (1/3) x (2/3) x (2/3) x (1/2) (1/3) x (2/3) x (2/3) x (1/2) + (2/3) x (1/3) x (1/3) x (1/2) = (2/3) > (1/2)

30. Analysis The best strategy for the third student is to guess yellow no matter what he draws.

31. Other sequences # yellow guesses - # blue guesses Yellow cascade starts. 2 1 0 1 2 3 4 5 6 7 student -1 -2 Blue cascade starts.

32. Q. What is the largest city in the US? A. New York, population 8,175,133

33. Q. What is the 2nd largest city in the US? A. Los Angeles, population 3,729,621

34. Q. What is the 3rd largest city in the US? A. Chicago, population 3,695,598

35. City populations • New York 8,175,133 • Los Angeles 3,792,621 • Chicago 2,695,598 • Houston 2,099,451 • Philadelphia1,526,006 • Phoenix 1,445,632 • San Antonio 1,327,407 • San Diego 1,307,402 • Dallas 1,197,816 • San Jose 945,942

36. City populations • New York 8,175,133 • Los Angeles 3,792,621 • Chicago 2,695,598 230. Berkeley, CA112,580 240. Murfreesboro, TN 108,755 250. Ventura, CA106,433 A few cities with high population Many cities with low population

37. City populations

38. City populations

39. Power Law: The number of cities with population at least k is proportional to k-c for a constant c.

40. Power Law: The number of cities with population > k is proportional to k-c. “fraction of items” “popularity = k”

41. Power Law: Fraction f(k) of items with popularity k is proportional to k-c. f(k) k-c log [f(k)] log [k-c] log [f(k)] -c log [k]

42. A power law is a straight line on a log-log plot.

43. City populations

44. Other examples

45. Why does data exhibit power laws?

46. Previously, … Imitation Cascade

47. Today Imitation Power law

48. Constructing the web • Pages are created in order, named 1, 2, …, N • When created, page j links to a page by • With probability p, picking a page i uniformly at random from 1, …, j-1 • With probability (1-p), pick page i uniformly at random and link to the page that i links too Imitation

49. The rich get richer 2 b) With prob. (1-p), pick page i uniformly at random and link to the page that i links too 3/4 1/4

50. The rich get richer 2 b) With prob. (1-p), pick page i uniformly at random and link to the page that i links too Equivalently, 2 b) With prob. (1-p), pick a page proportional to its in-degree and link to it