1. Social Cohesion and Connectivity: Diffusion Implications of Relational Structure James Moody The Ohio State University Population Association of America Meetings Minneapolis Minnesota, May 1 – 3, 2003

2. Why do Networks Matter? “To speak of social life is to speak of the association between people – their associating in work and in play, in love and in war, to trade or to worship, to help or to hinder. It is in the social relations men establish that their interests find expression and their desires become realized.” Peter M. Blau, Exchange and Power in Social Life, 1964 "If we ever get to the point of charting a whole city or a whole nation, we would have … a picture of a vast solar system of intangible structures, powerfully influencing conduct, as gravitation does in space. Such an invisible structure underlies society and has its influence in determining the conduct of society as a whole." J.L. Moreno, New York Times, April 13, 1933

3. Why do Networks Matter? The importance of networks is well recognized in demographic work: Behrman, Kohler, and Watkins (2003) Demography 713 – 738 Lusyne, Page and Lievens (2001). Population Studies 281-289 Astone, NM, CA Nathanson, R Schoen, and YJ Kim. (1999) Population and Development Review 1-31 Goldstein (1999) Demography 399-407 Entwisle, Rindfuss. Guilkey,Chamratrithirong; Curran and Sawangdee (1996) Demography 1-11

4. Why do Networks Matter? Social Support Social Influence Diffusion DirectIndirect Data CompanionshipCommunity Peer Pressure / Information Cultural differentiation Receiving / Transmitting Population distribution Local “Ego-network” Global or partial network Mechanism:

5. Why do Networks Matter? Local vision

6. Why do Networks Matter? Global vision

7. Consider the following (much simplified) scenario: Probability that actor i infects actor j (p ij )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: 0.090.09 Why do Networks Matter?

8. Probability of infection over independent paths: The probability that an infectious agent travels from i to j is assumed constant at p ij. The probability that infection passes through multiple links (i to j, and from j to k) is the joint probability of each (link1 and link2 and … link k) = p ij d where d is the path distance. To calculate the probability of infection passing through multiple paths, use the compliment of it not passing through any paths. The probability of not passing through path l is 1-p ij d, and thus the probability of not passing through any path is (1-p ij d ) k, where k is the number of paths Thus, the probability of i infecting j given k independent paths is: Why matter Distance

9. Probability of infection over non-independent paths: - To get the probability that I infects j given that paths intersect at 4, I calculate Using the independent paths formula.formula

10. Now consider the following (similar?) scenario: S T Every actor but one has the exact same number of partners The category-to-category (blue to orange) 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: = 0.1480.148 Different structures create different outcomes Why do Networks Matter?

11. Combining alternative mechanisms with levels of observation, “Why networks matter?” reduces to two classes of related questions: 1)Those dealing with global network structure. The global structure of the network affects how goods can travel throughout the population. The key elements for diffusion are average path distance and connectivity. 2)Those dealing with individual or group position. One’s “risk” for receiving/transmitting a good depends on one’s position in the overall network (“structural embeddedness”) The strength and qualities of direct connections (“direct embeddedness”) Why do Networks Matter?

12. Three Approaches to Network Structure 1. Small World Networks Based on Milgram’s (1967) famous work, the substantive point is that networks are structured such that even when most of our connections are local, any pair of people can be connected by a fairly small number of relational steps.

13. High relative probability that a node’s contacts are connected to each other. Small relative average distance between nodes C=Large, L is Small = SW Graphs Three Approaches to Network Structure 1. Small World Networks

14. In a highly clustered, ordered network, a single random connection will create a shortcut that lowers L dramatically Watts demonstrates that small world properties can occur in graphs with a surprisingly small number of shortcuts Three Approaches to Network Structure 1. Small World Networks

15. Three Approaches to Network Structure 2. Scale-Free Networks Across a large number of substantive settings, Barabási points out that the distribution of network involvement (degree) is highly and characteristically skewed.

16. Many large networks are characterized by a highly skewed distribution of the number of partners (degree) Three Approaches to Network Structure 2. Scale-Free Networks

17. Many large networks are characterized by a highly skewed distribution of the number of partners (degree) Three Approaches to Network Structure 2. Scale-Free Networks

18. Scale-free networks can appear when new nodes enter the network by attaching to already popular nodes (called proportionate mixing in earlier epidemiology models). Scale-free networks are common (WWW, Sexual Networks, Email) Three Approaches to Network Structure 2. Scale-Free Networks

19. Colorado Springs High-Risk (Sexual contact only) Network is power-law distributed, with = -1.3 Three Approaches to Network Structure 2. Scale-Free Networks

20. Hubs make the network fragile to node disruption Three Approaches to Network Structure 2. Scale-Free Networks

21. Hubs make the network fragile to node disruption Three Approaches to Network Structure 2. Scale-Free Networks

22. James Moody and Douglas R. White. “Structural Cohesion and Embeddedness: A hierarchical Conception of Social Groups” American Sociological Review 68:103-127 Three Approaches to Network Structure 3. Structural Cohesion

23. Three Approaches to Network Structure 3. Structural Cohesion An intuitive definition of structural cohesion: A collectivity is structurally cohesive to the extent that the social relations of its members hold it together. Five features: 1.A property describing how a collectivity is united 2.It is a group level property 3.The conception is continuous 4.Rests on observed social relations 5.Is applicable to groups of any size

24. Three Approaches to Network Structure 3. Structural Cohesion The minimum requirement for structural cohesion is that the collection be connected.

25. Three Approaches to Network Structure 3. Structural Cohesion Add relational volume:

26. Three Approaches to Network Structure 3. Structural Cohesion When focused on one node, the system is still fragile. Add relational volume:

27. Three Approaches to Network Structure 3. Structural Cohesion Spreading relations around the structure makes it robust to node removal.

28. Three Approaches to Network Structure 3. Structural Cohesion Formal definition of Structural Cohesion: (a)A group’s structural cohesion is equal to the minimum number of actors who, if removed from the group, would disconnect the group. Equivalently (by Menger’s Theorem): (b)A group’s structural cohesion is equal to the minimum number of independent paths linking each pair of actors in the group.

29. Networks are structurally cohesive if they remain connected even when nodes are removed Node Connectivity 01 23 Three Approaches to Network Structure 3. Structural Cohesion

30. Identified in wide ranging contexts: High School Friendship networks Biotechnology Inter-organizational networks Mexican political networks Kinship networks Structurally cohesive networks are conducive to equality and diffusion, since no node can control the flow of goods through the network. Three Approaches to Network Structure 3. Structural Cohesion

31. 0 0.2 0.4 0.6 0.8 1 1.2 23456 Path distance probability Probability of infection by distance and number of paths, assume a constant p ij of 0.6 10 paths 5 paths 2 paths 1 path Three Approaches to Network Structure 3. Structural Cohesion

32. Three Approaches to Network Structure 3. Structural Cohesion STD diffusion in Colorado Springs Endemic Chlamydia Structure Source: Potterat, Muth, Rothenberg, et. al. 2002. Sex. Trans. Infect 78:152-158

33. Three Approaches to Network Structure 3. Structural Cohesion STD diffusion in Colorado Springs Epidemic Gonorrhea Structure Source: Potterat, Muth, Rothenberg, et. al. 2002. Sex. Trans. Infect 78:152-158 G=410

34. Source: Potterat, Muth, Rothenberg, et. al. 2002. Sex. Trans. Infect 78:152-158 Three Approaches to Network Structure 3. Structural Cohesion STD diffusion in Colorado Springs Epidemic Gonorrhea Structure

35. Structural cohesion gives rise automatically to a clear notion of embeddedness, since cohesive sets nest inside of each other. Three Approaches to Network Structure 3. Structural Cohesion 17 18 19 20 2 22 23 8 11 10 14 12 9 15 16 13 4 1 75 6 3 2

36. Three Approaches to Network Structure 3. Structural Cohesion Structural Embeddedness has proved important for: Adolescent Suicide Adolescent females who are not members of the largest bicomponent are 2 times as likely to contemplate suicide (Bearman and Moody, 2003) Weapon Carrying Adolescents who are not members of the largest bicomponent are 1.37 times more likely to carry weapons to school (Moody, 2003) Adolescent attachment to school Embeddedness is the strongest predictor of attachment to school (Moody & White, 2003), which is a strong predictor of other health outcomes (Resnick, et. al, 1997).

37. Getting Data: Duality of Persons and Groups While global network position matters fundamentally, collecting global network data on (most) social relations is very expensive and time consuming. First priority: develop network sampling and modeling schemes. This work is underway. Identify alternative relations with long-lasting traces Kinship records Public interaction (Frank & Yasumoto, 1998) Identify cohesion through co-membership Brieger’s (1974) work on the duality of persons and groups demonstrated how we can link people (groups) to each other through membership. Data are surprisingly abundant – almost any list can form a basis for co-membership. The resulting group-level network is robust to standard sampling methods.

38. Getting Data: Duality of Persons and Groups 1 2 3 4 5 6 7 8 9 10 A B C D 3 5 1 6 7 8 9 4 2 A B C D Person Group

39. Getting Data: Duality of Persons and Groups Advantages of affiliation networks: Ease of data collection. Data on activities / presence / membership is easy to collect. A simple list of what people do / where they go is all that is needed. Examples include: Formal organizations (clubs, churches, workplaces, etc.) Event attendance (Parties they’ve been at recently, funerals, etc.) Common meeting places (bars they frequent, where they met most recent partner, etc.) Can be time-stamped for greater mixing accuracy Sampling. The resulting data are simply a n-way involvement cross-tabulation. This is a frequency table, which at the group-to-group level, is often quite robust to individual-level sampling, even in the face of heavily skewed involvement levels.

40. Getting Data: Duality of Persons and Groups Disadvantage of affiliation networks: Co-presence does not necessarily imply interaction -The resulting network can be thought of as a likely field of potential interaction, but does not record interaction itself. -This level of potential can be modeled, however, by including a basic ego-network module to then model the association between interaction and co-membership. -In general, we can also make some reasonable assumptions about the relation between interaction and membership based on (a) group size and (b) amount of time spent in the organization.

41. Getting Data: Duality of Persons and Groups -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Same Race SES GPA Both Smoke College Drinking Fight Reciprocity Same Sex Same Clubs Transitivity Intransitivity Same Grade Network Model Coefficients, In school Networks

42. Getting Data: Duality of Persons and Groups The resulting “networks” are cross-tabulations of the number of people that belong to each group: 3 5 1 6 7 8 10 9 4 2 A C B D In general, the minimum node connectivity of the person to person network is going to equal the edge connectivity (valued) of the group to group network. The relative edge connectivity is robust to sampling

43. Structural Implications of group membership There is a strong connection between the literature on “Social Capital” and group membership, which provides a theoretical link between notions of structural cohesion, ideational diffusion, and the duality of groups. Most research on the community building effects of group membership focus on relational volume (c.f. Putnam, 2000). However, to the extent that our interest is in how group membership creates structurally cohesive settings, interaction pattern is more important than volume. Suggestions about the structure of modern life (Pescosolido & Rubin, 2000), suggest that membership patterns should generate loosely coupled group structures.

44. Structural Implications of group membership Structural cohesion increases when membership in various groups are uncorrelated. If membership in group i predicts membership in group j (membership structure is tight), then the resulting groups will be nested. For example, if all Kiwanis members are also Methodists while all Shriners are Catholic If membership in group i is unrelated to membership in group j, then the resulting network will be structurally cohesive, as unconstrained membership links groups across many domains.

45. Paxton (2002) Groups differ in the extent to which members are jointly involved in other groups. We don’t currently have good empirical data on membership tightness, though it should be easy to calculate if collected properly. An untested empirical claim: Membership tightness has declined in the last 100 years.

46. Structural Implications of group membership Two hypothetical examples. A sample of 4000 people, who each visit an average of 1.4 bars.

47. Structural Implications of group membership Two hypothetical examples. Loose membership mixing Tight membership mixing

48. Getting Data: Duality of Persons and Groups What types of “groups” might be of interest to population researchers? Village – to – village networks (Entwisle et al, Demography 1997) If people marry, work, or attend services/festivals across villages, then the village- village links can form a probable contact network.

49. Getting Data: Duality of Persons and Groups What types of “groups” might be of interest to population researchers? Mixing location. If we know where people ‘hook up’ to find partners, we can identify potential STD cores.