Social and Economic Networks Matthew O. Jackson

Transcription

1 Social and Economic Networks Matthew O. Jackson Copyright: Matthew O. Jackson 2016 Please do not post or distribute without permission.

2 Networks and Behavior How does network structure impact behavior? Simple infec9ons, contagion diffusion Opinions, informa9on learning Choices, decisions games on networks

3 Diffusion disease Ideas basic informa9on (know or not know) Buy a product or not (come back to complementari9es later...) contagion

4 Diffusion Ques9ons and Background Bass Model no networks SIR and SIS Models bring in interac9on structure

5 Diffusion Ques9ons and Background Bass Model no networks SIR and SIS Models bring in interac9on structure

6 1.2 1 Frac9on having adopted Coleman, Katz, Menzel (1966) adoption of new drug by doctors more connected earlier adopters Named by 0 others Named by 1,2 others Named by 3+ others Months

7 Diffusion: Coleman, Katz, Menzel (1966) adop9on (prescribing) of new drug by doctors: more connected are earlier adopters Frac:on Adop:ng by: names by 0 others (36) named by 1 or 2 others (56) named by 3+ others (33) 6 months months months months

8 Griliches (1957): Hybrid Corn Diffusion Frac9on Adopted S-Shape, Spatial Pattern... Kentucky Wisonsin Iowa Year

9 Griliches (1957): Hybrid Corn Diffusion S-Shape, Spatial Pattern...

10 Kim et al Honduras: gave out 9ckets for free mul9vitamins, and also for chlorina9on tablets Seeded different villages either by picking injec9on points at random, or by indegree based on survey, or by asking people to name friends and picking randomly from friends.

11 Kim et al No S-Shape

12 Xiong 2015 Rural China: Gave out new crop seedlings First year only 20 households planted them 10 x more profitable than tradi9onal rice, coaon By second year, only way to get seedlings is from someone who has planted them in past Eventually fully diffused

13 Xiong 2015 S-Shape

14 Ques:ons: Extent of diffusion? How does it depend on the par9culars of the process as well as the network? Time paaerns? S-shape? Welfare analysis?

15 Diffusion Ques9ons and Background Bass Model no networks SIR and SIS Models bring in interac9on structure Richer models, dynamics (Leeat)

16 Bass Model A benchmark model with no explicit social structure Two ac9ons/states/behaviors 0 and 1 F(t) frac9on of the popula9on who have adopted ac9on 1 at 9me t

17 Bass Model p rate of spontaneous innova9on/adop9on q rate of imita9on of adop9on df(t)/dt = (p + q F(t))(1-F(t))

18 Solu:on: p rate of spontaneous innova9on/adop9on q rate of imita9on of adop9on df(t)/dt = (p + q F(t))(1-F(t)) F(t) = (1-e -(p+q)t )/ (1+qe -(p+q)t /p)

19 GeTng the S-shape Gives S-shape (if q>p) and tends to 1 in the limit Ini9ally only p maaers, then q takes over Eventually change slows as F(t) approaches 1

20 GeTng the S-shape df(t)/dt = (p + q F(t))(1-F(t)) when F(t)=0, df(t)/dt = p when F(t) nears 1, df(t)/dt = 0 when F(t)=ε, df(t)/dt = ( p + q ε ) (1- ε) to get ini9al convexity: need ( p + q ε ) (1- ε) > p q (1- ε) > p so ini9ally need q > p

21 1 F(t) 0 t

22 Diffusion Ques9ons and Background Bass Model no networks SIR and SIS Models bring in interac9on structure

23 Heterogeneity mavers Can we incorporate network structure in a tractable manner? Simula9ons Simple models of diffusion

24 SIR and SIS Models Bailey (1975) SIR Suscep9ble, Infected, Removed SIS Suscep9ble, Infected, Suscep9ble

25 SIR Model Can infect for some random 9me (e.g., Reed Frost model just one period) Like having some random links Component calcula-ons (under an approxima9on that links independent )

26 Applica:on: Diffusion Idea, disease, computer virus spreads via connec9ons in the network Nodes are linked if one would ``infect the other Will an infec9on take hold? How many nodes/people will it reach?

27 Ques:ons: When do we get diffusion? What is the extent of diffusion? How does it depend on the par9culars of the process as well as the network? Who is likely to be infected earliest?

28 Component Structure Reach of contagion is determined by the component structure Some players or nodes are immune Some links fail to transmit What do components look like of those who are suscep9ble and given links that work

29 Size of the Giant Component: How big is the giant component when there is one? Size of the giant component when 1/n< p < log(n)/n [know that if p << 1/n all isolated, and if log(n)/n <<p then all path connected]

30 Bearman, Moody, and Stovel s High School Romance Data

31 Extent of Diffusion Get nontrivial diffusion if someone in the giant component is infected/adopts Size of the giant component determines likelihood of diffusion and its extent Random network models allow for giant component calcula9ons

32 Calcula:ng the Size of the Giant Component q is frac9on of nodes in largest component look at any node: chance it is in the giant component is q chance that this node is outside of the giant component is the chance that all of its neighbors are outside of the giant component

33 Calcula:ng the Size of the Giant Component Probability that a node is outside of the giant component = 1-q = probability that all of its neighbors are outside = (1-q) d where d is the node s degree

34 Giant Component Size So, probability 1-q that a node is outside of the giant component is 1-q = (1-q) d P(d) Where P(d) is the chance that the node has d neighbors Solve for q

35 solution of q=0 1 (1-q) d P(d) solution 0 1-q 1

36 Erdos-Renyi (1959,1960) G(n,p) Graphs n nodes each link is formed iid with probability p For small p, degree distribu9on has Poisson distribu9on

37 Giant Component Size: Poisson Case Solve 1-q = (1-q) d P(d) when P(d) = [ (n-1) d / d! ] p d e -(n-1)p

38 Giant Component Size: Poisson Case Solve 1-q = (1-q) d P(d) when P(d) = [ (n-1) d / d! ] p d e -(n-1)p so 1-q = e -(n-1)p [(1-q) (n-1)p] d / d!

39 Useful Approxima:ons Defini9on of exponen9al func9on: e x = lim n ( 1+ x/n) n Taylor series approxima9on: e x = 1 + x + x 2 /2! + x 3 /3!... = 0 x d / d! [ f(x) = f(a) + f (a)(x-a)/1! + f (a)(x-a) 2 /2!... ]

40 Giant Component Size: Poisson Case Solve 1-q = (1-q) d P(d) when P(d) = [ (n-1) d / d! ] p d e -(n-1)p so 1-q = e -(n-1)p [(1-q) (n-1)p] d / d! = e -(n-1)p e (n-1)p(1-q) = e -q(n-1)p or - log(1-q) / q = (n-1) p = E[d]

41 Giant Component Size: 1 - log(1-q) / q = E[d] q Series E[d]

42 Poisson p=.01, 50 nodes, E[d]=.5

43 Poisson p=.03, 50 nodes, E[d]= is the threshold for emergence of cycles and a giant component

44 Poisson p=.05, 50 nodes, E[d]=2.5

45 Poisson p=.10, 50 nodes, E[d]=5

46 Giant Component Size and Network Structure So, probability that random node is outside of the giant component is 1-q = (1-q) d P(d) How does this change as we change the degree distribu9on P(d)?

47 Giant Component Size and Network Structure Probability that random node is outside of the giant component is 1-q = (1-q) d P(d) (1-q) d is decreasing and convex in d

48 Giant Component Size and Network Structure Probability that random node is outside of the giant component is 1-q = (1-q) d P(d) (1-q) d is decreasing and convex in d FOSD change in P leads (1-q) d P(d) to decrease,

49 1 (1-q) d P(d) fosd shift lower 1-q, higher q 0 1-q 1

50 Giant Component Size and Network Structure Probability that random node is outside of the giant component is 1-q = (1-q) d P(d) (1-q) d is decreasing and convex in d FOSD change in P leads (1-q) d P(d) to decrease, q must increase; larger giant component

51 Giant Component Size and Network Structure So, probability that random node is outside of the giant component is 1-q = (1-q) d P(d) (1-q) d is decreasing and convex in d FOSD change in P leads rhs to decrease, q must increase; larger giant component MPS change in P leads rhs to increase, q must decrease, smaller giant component

52 1 (1-q) d P(d) higher 1-q, lower q MPS 0 1-q 1

53 Who is infected? Probability of being in the giant component: 1-(1-q) d increasing in d More connected, more likely to be infected (more likely to be infected at any point in 9me...)

54 Study: Christakis-Fowler (2010) Flu spread at Harvard in fall of 2009 Interviewed 319 randomly selected undergrads (``random sample) asked them to name up to 3 friends Iden9fied 425 ``friends

55 Study: Christakis-Fowler (2010) ``Friends should have higher degree (P(d)d/E[d]) Friends should have higher infec9on rate at earlier dates... (e.g., 1-(1-q) d is increasing in d) Es9mate that ``friends infec9on shived 14.7 days earlier than ``random (+/- 3 at 95%) [based on a logis9c func9on regression of infec9on by a given date]

56

57 Naviga:on S9ll following the links of a network Rather than random process of contagion, explicit process of search

58 Ques:ons Milgram experiments: How to find another node so quickly? Not just an issue of the diameter of the network Take advantage of the network structure? Take advantage of who is who?

59 Naviga:on Even more restricted need to follow the network Overlaps become more problema9c with clustering, etc.

60 Network Naviga:on Dodds, Muhamad, Waas (2003) Small Worlds 24,163 chains started s ``send to someone closer to target Target ``John Doe at University X in US, ``. Archival Inspector in Estonia, ``technology consultant in India, ``a policeman in Australia 18 targets in 13 countries 384 reached targets Ini9al par9cipa9on.25, aver first stage.37

61 Social Ties Used:

62 Accoun9ng for Aari9on Aari9on rates at each step Readjus9ng for aari9on n(l) = observed number length L, A L aari9on at L Adjusted n(l) = n(l) / Π L <L (1-A L )

63 How they chose

64 Diffusion Network structure affects diffusion: threshold for infec9on/contagion sharp phase transi9ons extent of infec9on who becomes infected Naviga9on (must?) use social structure

65 Implica:ons For educa9on/immuniza9on Targe9ng nodes for dele9on Endogenizing network? Immuniza9on?