Diffusion

1. Diffusion • Structural Bases of Social Network Diffusion • Dynamic limitations on diffusion • Implications / Applications in the diffusion of Innovations

2. Diffusion • Two factors that affect network diffusion: • Topology • - the shape, or form, of the network • - simple example: one actor cannot pass information to another unless they are either directly or indirectly connected • Time • - the timing of contacts matters • - simple example: an actor cannot pass information he has not yet received.

3. Diffusion Topology features • Connectivity refers to how actors in one part of the network are connected to actors in another part of the network. • Reachability: Is it possible for actor i to reach actor j? This can only be true if there is a chain of contact from one actor to another. • Distance: Given they can be reached, how many steps are they from each other? • Number of paths: How many different paths connect each pair?

4. Network Toplogy Consider the following (much simplified) scenario: • Probability that actor i infects actor j (pij)is a constant over all relations = 0.6 • S & T are connected through the following structure: S T • The probability that S infects T through either path would be: 0.09

5. Why Sexual Networks Matter: Now consider the following (similar?) scenario: S T • Every actor but one has the exact same number of partners • The category-to-category mixing is identical • The distance from S to T is the same (7 steps) • S and T have not changed their behavior • Their partner’s partners have the same behavior • But the probability of an infection moving from S to T is: • = 0.148 • Different outcomes & different potentials for intervention

6. Network Topology: Ego Networks Mixing Matters • The most commonly collected network data are ego-centered. While limited in the structural features, these do provide useful information on broad mixing patterns & relationship timing. • Consider Laumann & Youm’s (1998) treatment of sexual mixing by race and activity level, using data from the NHSLS, to explain the differences in STD rates by race • They find that two factors can largely explain the difference in STD rates: • Intraracially, low activity African Americans are much more likely to have sex with high activity African Americans than are whites • Interracially, sexual networks tend to be contained within race, slowing spread between races

7. Network Topology: Ego Networks • In addition to general category mixing, ego-network data can provide important information on: • Local clustering (if there are relations among ego’s partners. Not usually relevant in heterosexual populations, though very relevant to IDU populations) • Number of partners -- by far the simplest network feature, but also very relevant at the high end • Relationship timing, duration and overlap • By asking about partner’s behavior, you can get some information on the relative risk of each relation. For example, whether a respondents partner has many other partners (though data quality is often at issue).

8. Network Topology: Ego Networks Clustering matters because it re-links people to each other, lowering the efficiency of the transmission network. Clustering also creates pockets where goods can circulate.

9. Network Topology: Partial and Complete Networks Once we move beyond the ego-network, we can start to identify how the pattern of connection changes the disease risk for actors. Two features of the network’s shape are known to be important: Connectivity and Centrality. • Connectivity refers to how actors in one part of the network are connected to actors in another part of the network. • Reachability: Is it possible for actor i to infect actor j? This can only be true if there is an unbroken (and properly time ordered) chain of contact from one actor to another. • Given reachability, three other properties are important: • Distance • Number of paths • Distribution of paths through actors (independence of paths)

10. Reachability example: All romantic contacts reported ongoing in the last 6 months in a moderate sized high school (AddHealth) 2 12 9 Male Female 63 (From Bearman, Moody and Stovel, 2004.)

11. Network Topology: Distance & number of paths • Given that ego can reach alter, distance determines the likelihood of an infection passing from one end of the chain to another. • Diffusion is never certain, so the probability of transmission decreases over distance. • Diffusion increases with each alternative path connecting pairs of people in the network.

12. Probability of Diffusion by distance and number of paths, assume a constant pij of 0.6 1.2 1 10 paths 0.8 5 paths probability 0.6 2 paths 0.4 1 path 0.2 0 2 3 4 5 6 Path distance

13. Probability of Diffusion by distance and number of paths, assume a constant pij of 0.3 0.7 0.6 0.5 0.4 probability 0.3 0.2 0.1 0 2 3 4 5 6 Path distance

14. S T S T Return to our first example: 2 paths 4 paths

15. Reachability in Colorado Springs (Sexual contact only) • High-risk actors over 4 years • 695 people represented • Longest path is 17 steps • Average distance is about 5 steps • Average person is within 3 steps of 75 other people • 137 people connected through 2 independent paths, core of 30 people connected through 4 independent paths (Node size = log of degree)

16. Network Topology: Centrality and Centralization • Centrality refers to (one dimension of) where an actor resides in a sexual network. • Local: compare actors who are at the edge of the network to actors at the center • Global: compare networks that are dominated by a few central actors to those with relative involvement equality

17. Centrality example: Add Health Node size proportional to betweenness centrality Graph is 45% centralized

18. Centrality example: Colorado Springs Node size proportional to betweenness centrality Graph is 27% centralized

19. Network Topology: Effect of Structure

20. Network Topology: Effect of Structure Simulated diffusion curves for the observed network.

21. Network Topology: Effect of Structure The effect of the observed structure can be seen in how diffusion differs from a random network with the same volume

22. Network Topology: Effect of Structure

23. Network Topology: Effect of Structure Mean number of independent paths

24. Network Topology: Effect of Structure Clustering Coefficient

25. Network Topology: Effect of Structure Mean Distance

26. Network Topology: Effect of Structure

27. Network Topology: Effect of Structure

28. Timing Sexual Networks • A focus on contact structure often slights the importance of network dynamics. • Time affects networks in two important ways: • 1) The structure itself goes through phases that are correlated with disease spread • Wasserheit and Aral, 1996. “The dynamic topology of Sexually Transmitted Disease Epidemics” The Journal of Infectious Diseases 74:S201-13 • Rothenberg, et al. 1997 “Using Social Network and Ethnographic Tools to Evaluate Syphilis Transmission” Sexually Transmitted Diseases 25: 154-160 • 2) Relationship timing constrains disease flow • a) by spending more or less time “in-host” • b) by changing the potential direction of disease flow

29. Sexual Relations among A syphilis outbreak Changes in Network Structure Rothenberg et al map the pattern of sexual contact among youth involved in a Syphilis outbreak in Atlanta over a one year period. (Syphilis cases in red) Jan - June, 1995

30. Sexual Relations among A syphilis outbreak July-Dec, 1995

31. Sexual Relations among A syphilis outbreak July-Dec, 1995

32. Data on drug users in Colorado Springs, over 5 years

33. Data on drug users in Colorado Springs, over 5 years

34. Data on drug users in Colorado Springs, over 5 years

35. Data on drug users in Colorado Springs, over 5 years

36. Data on drug users in Colorado Springs, over 5 years

37. What impact does this kind of timing have on diffusion? • The most dramatic effect occurs with the distinction between concurrent and serial relations. • Relations are concurrent whenever an actor has more than one sex partner during the same time interval. Concurrency is dangerous for disease spread because: • a) compared to serially monogamous couples, and STDis not trapped inside a single dyad • b) the std can travel in two directions - through ego - to either of his/her partners at the same time

38. Concurrency and Epidemic Size Morris & Kretzschmar (1995) 1200 800 400 0 0 1 2 3 4 5 6 7 Monogamy Disassortative Random Assortative Population size is 2000, simulation ran over 3 ‘years’

39. Adjusting for other mixing patterns: Variable Constant Concurrent K2 Degree Correlation Bias Coefficient 84.18 357.07 440.38 -557.40 982.31 Each .1 increase in concurrency results in 45 more positive cases Concurrency and disease spread

40. A hypothetical Sexual Contact Network 8 - 9 E C 3 - 7 2 - 5 A B 0 - 1 3 - 5 D F

41. The path graph for a hypothetical contact network E C A B D F

42. Direct Contact Network of 8 people in a ring

43. Implied Contact Network of 8 people in a ring All relations Concurrent

44. Implied Contact Network of 8 people in a ring Mixed Concurrent 2 3 2 1 1 2 2 3

45. Implied Contact Network of 8 people in a ring Serial Monogamy (1) 1 8 2 7 3 6 5 4

46. Implied Contact Network of 8 people in a ring Serial Monogamy (2) 1 8 2 7 3 6 1 4

47. Implied Contact Network of 8 people in a ring Serial Monogamy (3) 1 2 2 1 1 2 1 2

48. Timing Sexual Networks • Network dynamics can have a significant impact on the level of disease flow and each actor’s risk exposure • This work suggests that: • a) Disease outbreaks correlate with ‘phase-shifts’ in the connectivity level • b) Interventions focused on relationship timing, especially concurrency, could have a significant effect on disease spread • c) Measure and models linking network topography to disease flow should account for the timing of romantic relationships

49. Timing Sexual Networks

50. Degree or Connectivity?: Large-scale network model implications: Scale-Free Networks Many large networks are characterized by a highly skewed distribution of the number of partners (degree)