Diffusion Over Dynamic Networks Stanford University May 8, 2007 James Moody Duke University

1. Diffusion Over Dynamic Networks Stanford University May 8, 2007 James Moody Duke University

2. Introduction We live in a connected world: “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

3. Introduction We live in a connected world: "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 These patterns of connection form a social space, that can be seen in multiple contexts:

4. Introduction Source: Linton Freeman “See you in the funny pages” Connections, 23, 2000, 32-42.

5. Introduction High Schools as Networks

8. Introduction • And yet, standard social science analysis methods do not take this space into account. • “For the last thirty years, empirical social research has been dominated by the sample survey. But as usually practiced, …, the survey is a sociological meat grinder, tearing the individual from his social context and guaranteeing that nobody in the study interacts with anyone else in it.” • Allen Barton, 1968 (Quoted in Freeman 2004) • Moreover, the complexity of the relational world makes it impossible to identify social connectivity using only our intuitive understanding. • Social Network Analysis (SNA) provides a set of tools to empirically extend our theoretical intuition of the patterns that construct social structure.

9. Introduction Why do Networks Matter? Local vision

10. Introduction Why do Networks Matter? Local vision

11. Introduction • Why networks matter: • Intuitive: “goods” travel through contacts between actors, which can reflect a power distribution or influence attitudes and behaviors. Our understanding of social life improves if we account for this social space. • Less intuitive: patterns of inter-actor contact can have effects on the spread of “goods” or power dynamics that could not be seen focusing only on individual behavior. • These, ultimately, are often features that rest on the diffusion of some “bit” over the network. We’ll focus today on how that happens.

12. Social Network Data Elements • Social Network data consists of two linked classes of data: • Information on the individuals (aka: actors, nodes, points) • Network nodes are most often people, but can be any other unit capable of being linked to another (schools, countries, organizations, personalities, etc.) • The information about nodes is what we usually collect in standard social science research: demographics, attitudes, behaviors, etc. • Includes the times when the node is active • b) Information on relations among individuals (lines, edges, arcs) • Records a connection between the nodes in the network • Can be valued, directed (arcs), binary or undirected (edges) • One-mode (direct ties between actors) or two-mode (actors share membership in an organization) • Includes the times when the relation is active

13. b d b b d d a c e a a c c e e Social Network Data Elements In general, a relation can be: Binary or Valued Directed or Undirected Directed, binary Undirected, binary b d 1 2 1 3 4 a c e Directed, Valued Undirected, Valued

14. Social Networks & Diffusion “Goods” flow through networks:

15. Social Networks & Diffusion • In addition to* the dyadic probability that one actor passes something to another (pij), two factors affect flow through a network: • Topology • the shape, or form, of the network • - Example: one actor cannot pass information to another unless they are either directly or indirectly connected • Time • - the timing of contact matters • - Example: an actor cannot pass information he has not receive yet *This is a big conditional! – lots of work on how the dyadic transmission rate may differ across populations.

16. Social Networks & Diffusion Three features of the network’s topology are known to be important: Reachability, Distance & Number of Paths (redundancy) • 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? • How efficiently do ties reach new nodes? (How clustered is the network) • Number of paths: How many different paths connect each pair?

17. Social Networks & Diffusion Without full network data, you can’t distinguish actors with limited diffusion potential from those more deeply embedded in a setting. c b a

18. Social Networks & Diffusion Reachability • Given that ego can reach alter, distance determines the likelihood of information passing from one end of the chain to another. • Because flow is rarely certain, the probability of transfer decreases over distance. • However, the probability of transfer increases with each alternative path connecting pairs of people in the network.

19. b f c e d Social Networks & Diffusion Reachability Indirect connections are what make networks systems. One actor can reach another if there is a path in the graph connecting them. a b d a c e f Paths can be directed, leading to a distinction between “strong” and “weak” components

20. Social Networks & Diffusion Reachability • Basic elements in connectivity • A path is a sequence of nodes and edges starting with one node and ending with another, tracing the indirect connection between the two. On a path, you never go backwards or revisit the same node twice. • Example: a  b  cd • A walk is any sequence of nodes and edges, and may go backwards. Example: a  b  c  b c d • A cycle is a path that starts and ends with the same node. Example: a  b  c  a

21. Social Networks & Diffusion Reachability Reachability If you can trace a sequence of relations from one actor to another, then the two are reachable. If there is at least one path connecting every pair of actors in the graph, the graph is connected and is called a component. Intuitively, a component is the set of people who are all connected by a chain of relations.

22. Social Networks & Diffusion Reachability This example contains many components.

23. Social Networks & Diffusion Distance & number of paths Distance is measured by the (weighted) number of relations separating a pair: Actor “a” is: 1 step from 4 2 steps from 5 3 steps from 4 4 steps from 3 5 steps from 1 a

24. Social Networks & Diffusion Distance & number of paths Paths are the different routes one can take. Node-independent paths are particularly important. There are 2 independent paths connecting a and b. b There are many non-independent paths a

25. Social Networks & Diffusion Social Cohesion White, D. R. and F. Harary. 2001. "The Cohesiveness of Blocks in Social Networks: Node Connectivity and Conditional Density." Sociological Methodology 31:305-59. Moody, James and Douglas R. White. 2003. “Structural Cohesion and Embeddedness: A hierarchical Conception of Social Groups” American Sociological Review 68:103-127 White, Douglas R., Jason Owen-Smith, James Moody, & Walter W. Powell (2004) "Networks, Fields, and Organizations: Scale, Topology and Cohesive Embeddings."  Computational and Mathematical Organization Theory. 10:95-117 Moody, James "The Structure of a Social Science Collaboration Network: Disciplinary Cohesion from 1963 to 1999" American Sociological Review. 69:213-238

26. Social Networks & Diffusion Social Cohesion • Networks are structurally cohesive if they remain connected even when nodes are removed. Each of these graphs have the exact same density. 2 3 0 1 Node Connectivity

27. Social Networks & Diffusion Social Cohesion • Formal definition of Structural Cohesion: • 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): • A group’s structural cohesion is equal to the minimum number of node-independent paths linking each pair of actors in the group.

28. Social Networks & Diffusion Social Cohesion Structural cohesion gives rise automatically to a clear notion of embeddedness, since cohesive sets nest inside of each other. 2 3 1 9 10 8 4 11 5 7 12 13 6 14 15 17 16 18 19 20 2 22 23

29. Social Networks & Diffusion Social Cohesion Project 90, Sex-only network (n=695) 3-Component (n=58)

30. Social Networks & Diffusion Social Cohesion Connected Bicomponents IV Drug Sharing Largest BC: 247 k> 4: 318 Max k: 12 Structural Cohesion simultaneously gives us a positional and subgroup analysis.

31. Social Networks & Diffusion Emergence of multiple connectivity by degree distribution Emergent Connectivity in low-degree networks Partner Distribution Component Size/Shape

32. Social Networks & Diffusion Emergence of multiple connectivity by degree distribution Development of STD cores in low-degree networks: rapid transition without stars.

33. 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 Social Networks & Diffusion Distance & number of paths Probability of transfer by distance and number of paths, assume a constant pij of 0.6

34. Social Networks & Diffusion Clustering and diffusion Arcs: 11 Largest component: 12, Clustering: 0 Arcs: 11 Largest component: 8, Clustering: 0.205 Clustering turns network paths back on already identified nodes. This has been well known since at least Rappaport, and is a key feature of the “Biased Network” models in sociology.

35. Social Networks & Diffusion Diffusion features on static graphs

36. Social Networks & Diffusion Example on static graphs

37. Social Networks & Diffusion Example on static graphs Define as a general measure of the “diffusion susceptibility” of a graph as the ratio of the area under the observed curve to the area under the random curve. As this gets smaller than 1.0, you get effectively slower median transmission.

38. Social Networks & Diffusion Example on static graphs

39. Social Networks & Diffusion Example on static graphs

40. Social Networks & Diffusion Centrality • Centrality refers to (one dimension of) location, identifying where an actor resides in a network. • For example, we can compare actors at the edge of the network to actors at the center. • In general, this is a way to formalize intuitive notions about the distinction between insiders and outsiders. • Centrality affects within-network diffusion likelihood – we’ll not talk about this much today.

41. Social Networks & Diffusion Centrality • At the individual level, one dimension of position in the network can be captured through centrality. • Conceptually, centrality is fairly straight forward: we want to identify which nodes are in the ‘center’ of the network. In practice, identifying exactly what we mean by ‘center’ is somewhat complicated, but substantively we often have reason to believe that people at the center are very important. • Three standard centrality measures capture a wide range of “importance” in a network: • Degree • Closeness • Betweenness

42. Social Networks & Diffusion Centrality A common measure of centrality is closeness centrality. An actor is considered important if he/she is relatively close to all other actors. Closeness is based on the inverse of the distance of each actor to every other actor in the network. Closeness Centrality: Normalized Closeness Centrality

43. Social Networks & Diffusion Centrality Closeness Centrality in 4 examples C=0.0 C=1.0 C=0.36 C=0.28

44. Measuring Networks: Flow Time • Two factors that affect network flows: • 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.

45. Measuring Networks: Flow Time Timing in networks • A focus on contact structure has often slighted the importance of network dynamics,though a number of recent pieces are addressing this. • Time affects networks in two important ways: • The structure itself evolves, in ways that will affect the topology an thus flow. • 2) The timing of contact constrains information flow

46. Measuring Networks: Flow Time Drug Relations, Colorado Springs, Year 1 Data on drug users in Colorado Springs, over 5 years

47. Measuring Networks: Flow Time Drug Relations, Colorado Springs, Year 2 Current year in red, past relations in gray

48. Measuring Networks: Flow Time Drug Relations, Colorado Springs, Year 3 Current year in red, past relations in gray

49. Measuring Networks: Flow Time Drug Relations, Colorado Springs, Year 4 Current year in red, past relations in gray

50. Measuring Networks: Flow Time Drug Relations, Colorado Springs, Year 5 Current year in red, past relations in gray