1. Epidemic Potential in Human Sexual Networks: Connectivity and The Development of STD Cores James Moody The Ohio State University Institute for Mathematics and its Applications Minneapolis Minnesota, November 17 - 23, 2003

2. Epidemic Potential in Human Sexual Networks: Connectivity and The Development of STD Cores Introduction What features of networks matter? STD Cores Definition Implications for network structure Structural Cohesion Definition Cohesive Blocking: Structure & Position Structural Cohesion = STD cores Three Questions: Implications of Large Scale Net. Models Empirical Evidence for Cohesive Cores Development of Core groups in low-degree networks Future Extensions Extension to dynamic networks

3. Local network involvement The strength and qualities of particular network ties (“direct embeddedness”) Degree, tie strength, condom use, etc One’s position in the overall network (“structural embeddedness”) Centrality, local-network density, transitivity, membership. Global network structure The global structure of the network affects how goods can travel throughout the population. Distance distribution Connectivity structure Among the most challenging tasks for modeling networks is building a robust link from the first to the second. Introduction: Two ways that networks matter:

4. Why do Networks Matter? Local vision

5. Why do Networks Matter? Global vision

6. Why do Networks Matter? Networks are complex & multidimensional, so what aspects of global network structure are we interested in capturing? Substantively, we want to identify aspects of the network that are most important for diffusion of goods through the network. There are a number of options. Simple connectivity is a necessary condition, but consider the complexity within a single connected component, using data from Colorado Springs:

7. 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)

8. Centrality example: Colorado Springs Node size proportional to betweenness centrality Graph is 27% centralized Purely local characteristics are not necessarily correlated with structural embeddedness

9. 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 Why do Networks Matter?

10. STD Cores Infection Paradox in STD spread: The proportion of the total population infected is too low to sustain an epidemic, so why don’t these diseases simply fade away? The answer, proposed generally by a number of researchers*, is that infection is unevenly spread. While infection levels are too low at large to sustain an epidemic, within small (probably local) populations, infection rates are high enough for the disease to remain endemic, and spread from this CORE GROUP to the rest of the population. If this is correct, it suggests that we need to develop network measures of potential STD cores. *John & Curran, 1978; Phillips, Potterat & Rothenberg 1980; Hethcote & Yorke, 1984

11. STD Cores: A potential STD core requires a relational structure that can sustain an infection over long periods. Suggesting a structure that: is robust to disruption. Diseases seem to remain in the face of concerted efforts to destroy them. Individuals enter and leave the network Diseases (often) have short infectious periods magnifies transmission risk A disease that would otherwise dissipate likely gets an epidemiological boost when it enters a core. can accommodate rapid outbreak cycles Gumshoe work on STD outbreaks suggests that small changes in individual behavior can generate rapid changes in disease spread.

12. James Moody and Douglas R. White. 2003. “Structural Cohesion and Embeddedness: A hierarchical Conception of Social Groups” American Sociological Review 68:103-127 Structural Cohesion provides a natural indicator of STD cores. Intuitively, A network 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

13. Structural Cohesion: Definition The minimum requirement for structural cohesion is that the collection be connected.

14. When focused on one node, the system is still vulnerable to targeted attacks Add relational volume: Structural Cohesion: Definition

15. Spreading relations around the structure makes it robust. Structural Cohesion: Definition

16. Two definitions from graph theory: Two paths from i to j in G are node independent if they only have nodes i and j in common. If there is at least one path linking every pair of actors in the graph then it is connected. If there are k node-independent paths connecting every pair, the graph is k-connected and called a k-component. In any component, the path(s) linking two non-adjacent vertices must pass through a subset of other nodes, which if removed, would disconnect them. S, is called an (i,j) cut-set if every path connecting i and j passes through at least one node of S. The node-connectivity, k, of G is the smallest size of any (i,j) cutset in G. Menger’s theorem shows that any graph with node connectivity k is at most k-connected, and any graph that is k-connected has node connectivity k.

17. 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 node independent paths linking each pair of actors in the group. Structural Cohesion: Definition In English:

18. Networks are structurally cohesive if they remain connected even when nodes are removed Node Connectivity 01 23 Structural Cohesion: Definition

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

20. G {7,8,9,10,11 12,13,14,15,16} {1, 2, 3, 4, 5, 6, 7, 17, 18, 19, 20, 21, 22, 23} {7, 8, 11, 14} {1,2,3,4, 5,6,7} {17, 18, 19, 20, 21, 22, 23} A Cohesive Blocking of a network is the enumeration of all connected sets, and their relation to each other. Structural Cohesion: Properties

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22. Connectivity Pairwise Connectivity profile Structural Cohesion: Properties

23. Three requirements for potential STD cores: A structure that: is robust to disruption. Defining characteristic of k-components Allows for a continuous (as opposed to categorical) measure of “coreness” based on the embeddedness levels within the graph. magnifies transmission risk Overlapping k-components act like transmission substations, where high within-component diffusion boosts the likelihood of long-distance transmission from one k-component to other components (lumpy transmission) or to less embedded actors at the fringes (a ‘pump station’). can accommodate rapid outbreak cycles Once disease enters one of these cores, spread is likely robust and rapid. Structural Cohesion = Potential Std Cores?

24. Three Questions: 1) What are the STD Core implications of current large- scale network models? Scale-free models Small-world models 2) How empirically plausible is a structural cohesion model for STD cores? Evidence from STD outbreak investigations Cohesive blocking of the Colorado Springs drug exchange network 3) What is the relationship between local node behavior and the development of structurally cohesive cores? The emergence of core structure in low-degree networks

25. Many large networks are characterized by a highly skewed distribution of the number of partners (degree) Large-scale network model implications: Scale-Free Networks Large Models & STD Cores:

26. Many large networks are characterized by a highly skewed distribution of the number of partners (degree) Large Models & STD Cores: Large-scale network model implications: Scale-Free Networks

27. The scale-free model focuses on the distance- reducing capacity of high-degree nodes: Large Models & STD Cores:

28. Large-scale network model implications: Scale-Free Networks The scale-free model focuses on the distance- reducing capacity of high-degree nodes: Which implies: a thin cohesive blocking structure and a fragile global topography Scale free models work primarily on through distance, as hubs create shortcuts in the graph, not through core-group dynamics. Large Models & STD Cores:

29. Large-scale network model implications: Small-world models 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 Large Models & STD Cores:

30. Large-scale network model implications: Small-world graphs 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 Large Models & STD Cores:

31. Large-scale network model implications: Small-world graphs The ‘cave-man’ version of the SW model suggests a cohesive blocking with everyone embedded at k=2 (the ring), and small sets at k=(n i -1) (the local clusters), for summary blocking that would look something like: T gigi gigi gigi gigi gigi Large Models & STD Cores: Consistent with STD cores

32. Large-scale network model implications: Small-world graphs The lattice version of the SW model suggests a cohesive blocking with everyone embedded at high k, determined by the degree. Since each person is connected to a similar number of overlapping neighbors, determined by distance along the underlying lattice ring, for summary blocking that would look something like: Large Models & STD Cores:

33. Large-scale network model implications: Small-world graphs Thus, while the descriptive logic of the SW model is consistent with STD cores, the empirical measures, particularly the clustering coefficient (transitivity ratio), are insufficient to specify structural cohesion. This will be particularly vexing with heterosexual sex networks, as C is by definition 0. Theoretically, this mismatch follows from the local nature of the transitivity index. Large Models & STD Cores:

34. Almost no evidence of Chlamydia transmission Source: Potterat, Muth, Rothenberg, et. al. 2002. Sex. Trans. Infect 78:152-158 Empirical Evidence 2) Empirical evidence for Structurally Cohesive STD Cores:

35. Epidemic Gonorrhea Structure Source: Potterat, Muth, Rothenberg, et. al. 2002. Sex. Trans. Infect 78:152-158 G=410 Empirical Evidence 2) Empirical evidence for Structurally Cohesive STD Cores:

36. Source: Potterat, Muth, Rothenberg, et. al. 2002. Sex. Trans. Infect 78:152-158 Epidemic Gonorrhea Structure Empirical Evidence 2) Empirical evidence for Structurally Cohesive STD Cores:

37. 3-Component (n=58) Empirical Evidence Project 90, Sex-only network (n=695) 2) Empirical evidence for Structurally Cohesive STD Cores:

38. Empirical Evidence:Project 90, Drug sharing network Connected Bicomponents N=616 Diameter = 13 L = 5.28 Transitivity = 16% Reach 3: 128 Largest BC: 247 K > 4: 318 Max k: 12 2) Empirical evidence for Structurally Cohesive STD Cores:

39. Empirical Evidence:Project 90, Drug sharing network 2) Empirical evidence for Structurally Cohesive STD Cores:

40. Empirical Evidence:Project 90, Drug sharing network 2) Empirical evidence for Structurally Cohesive STD Cores:

41. Empirical Evidence:Project 90, Drug sharing network Multiple 4-components 2) Empirical evidence for Structurally Cohesive STD Cores:

42. 3) Development of STD Cores in Low-degree networks? While much attention has been given to the epidemiological risk of networks with long-tailed degree distributions, how likely are we to see the development of potential STD cores, when everyone in the network has low degree? Low degree networks are particularly important when we consider the short-duration networks, needed for diseases with short infectious windows. Logically bounded: If everyone has degree = 1, then the network will have only isolated dyads. If everyone has degree = 2, then the most expansive network would be a simple cycle. Only when at least some people have 3 ties do we get structures that could resemble empirical data: with distinct communities and cross- group branching.

43. 3) Development of STD Cores in Low-degree networks? Building on recent work on conditional random graphs*, we examine (analytically) the expected size of the largest component for graphs with a given degree distribution, and simulate networks to measure the size of the largest bicomponent. For these simulations, the degree distribution shifts from having a mode of 1 to a mode of 3. We estimate these values on populations of 10,000 nodes, and draw 100 networks for each degree distribution. * Newman, Strogatz, & Watts 2001; Molloy & Reed 1998

44. 3) Development of STD Cores in Low- degree networks?

48. Very small changes in degree generate a quick cascade to large connected components. While not quite as rapid, STD cores follow a similar pattern, emerging rapidly and rising steadily with small changes in the degree distribution. This suggests that, even in the very short run (days or weeks, in some populations) large connected cores can emerge covering the majority of the interacting population, which can sustain disease.

49. Possible Extensions: 1) When viewed dynamically, graphs can have radically different implications for possible diffusion, since infection cannot be passed through relations that have ended. Relationship timing creates one- way streets from a ‘virus-eye-view’ (Moody, 2001). How do we identify potential STD cores in these networks? 2) Extend the model to group overlaps, as in people connected through locations. Building on work such as Martin (2002), we can characterize the probability of belonging to one group as a function of belonging to another (‘tight’ versus ‘loose’ membership spaces). Since edge connectivity of the “location” graph is tied directly to node-connectivity of the “people” graph, and since group membership contours can be sampled, this provides a potential proxy for estimating global characteristics of the network from sample data.

50. References: Barabasi, A.-L. and Albert, R. 1999. “Emergence of Scaling in Random Networks.” Science 286, 509-512. Granovetter, M. S. 1985. "Economic Action and Social Structure: The Problem of Embeddedness." American Journal of Sociology 91:481-510. Hethcote, H. and Yorke, J.A. 1984. Gonorrhea Transmission Dynamics and Control. Springer Verlag, Berlin. Hethcote, H. 2000. “The Mathematics of infectious diseases.” SIAM Review 42, 599-653 Jones, J. and Handcock, M. 2003, “Sexual contacts and epidemic thresholds.” Nature 423, 605-606. Molloy, M. and Reed, B. 1998. “The Size of the Largest Component of a Random Graph on a Fixed Degree Sequence”. Combinatorics, Probability and Computing 7, 295-306. Moody, J. 2000. “The Importance of Relationship Timing for STD Diffusion: Indirect Connectivity and STD Infection Risk.” Social Forces 81, 25-56. Moody, J. and White, D.R. 2003. “Social Cohesion and Embeddedness: A Hierarchical conception of Social Groups.” American Sociological Review 68, 103-127. Newman, M.E.J. 2002. “Spread of Epidemic disease on Networks.” Physical Review E 66 016128. Newman, M.E.J., Strogatz, S.J., and Watts, D.J. 2001. “Random Graphs with arbitrary degree distributions and their applications.” Phys. Rev. E. Phillips, L., Potterat, J., and Rothenberg, R.. 1980. “Focused Interviewing in gonorrhea control.” American Journal of Public Health 70, 705-708. Rothenberg, R.B. et al. 1997. “Using Social Network and Ethnographic Tools to Evaluate Syphilis Transmission.” Sexually Transmitted Diseases 25, 154-160 St. John, R. and Curran, J. 1978. “Epidemiology of gonorrhea”. Sexually Transmitted Diseases 5, 81-82. Watts, Duncan J. 1999. "Networks, Dynamics, and the Small-World Phenomenon." American Journal of Sociology 105:493-527. White, D.R. and Harary, F. 2001. “The Cohesiveness of Blocks in Social Networks: Node Connectivity and Conditional Density.” Sociological Methodology 31, 305-359.

51. 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?

52. 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

53. 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

54. 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?

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