Diffusion Over Dynamic Networks (plus some social network intro since I’m first) NetSci Workshop May 16, 2006 James Mood

1. Diffusion Over Dynamic Networks (plus some social network intro since I’m first) NetSci Workshop May 16, 2006 James Moody This work supported by the Network Modeling Project through the University of Washington: NIH grants DA12831 and HD41877

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.

12. Introduction • Social network analysis is: • a set of relational methods for systematically understanding and identifying connections among actors. SNA is • is motivated by a structural intuition based on ties linking social actors • is grounded in systematic empirical data • draws heavily on graphic imagery • relies on the use of mathematical and/or computational models. (Freeman, 2004) • Social Network Analysis embodies a range of theories relating types of observable social spaces.

13. Introduction • Social Network Basics • Basic data Elements • Basic data structures • Network Analysis Buffet • Networks & Diffusion • Structural constraints on network diffusion • Reachability • Distance • Connectivity • Closeness centrality • Temporal Constraints on network diffusion • Defining dynamic networks • How order constrains flow • Reachability variance w. constant structure • Minimum temporal reachability • New time-dependent network measures • Graph-level measures • Node-level measures • Visualizing Diffusion potential in time-dependent Graphs

14. 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

15. Social Network Data Elements The unit of interest in a network are the combined sets of actors and their relations. We represent actors with points and relations with lines. Actors are referred to variously as: Nodes, vertices or points Relations are referred to variously as: Edges, Arcs, Lines, Ties Example: b d a c e

16. 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

17. Social Network Data Elements • Social network data are substantively divided by the number of modes in the data. • 1-mode data represents edges based on direct contact between actors in the network. All the nodes are of the same type (people, organization, ideas, etc). Examples: • Communication, friendship, giving orders, sending email. • 1-mode data are usually singly reported (each person reports on their friends), but you can use multiple-informant data, which is more common in child development research (Cairns and Cairns).

18. Social Network Data Elements Social network data are substantively divided by the number of modes in the data. 2-mode data represents nodes from two separate classes, where all ties are across classes. Examples: People as members of groups People as authors on papers Words used often by people Events in the life history of people The two modes of the data represent a duality: you can project the data as people connected to people through joint membership in a group, or groups to each other through common membership There may be multiple relations of multiple types connecting nodes in any given substantive setting.

19. Social Network Data Elements Levels of analysis Global-Net Ego-Net Partial-Network

20. Social Network Data Elements We can examine networks across multiple levels: • 1) Ego-network • - Have data on a respondent (ego) and the people they are connected to (alters). Example: 1985 GSS module • - May include estimates of connections among alters • 2) Partial network • - Ego networks plus some amount of tracing to reach contacts of contacts • - Something less than full account of connections among all pairs of actors in the relevant population • - Example: CDC Contact tracing data for STDs

21. Social Network Data Elements We can examine networks across multiple levels: • 3) Complete or “Global” data • - Data on all actors within a particular (relevant) boundary • - Never exactly complete (due to missing data), but boundaries are set • Example: Coauthorship data among all writers in the social sciences, friendships among all students in a classroom • For the most part, I will be discussing issues surrounding global networks.

22. Social Network Data Structures Visualization A good network drawing allows viewers to come away from the image with an almost immediate intuition about the underlying structure of the network being displayed. However, because there are multiple ways to display the same information, and standards for doing so are few, the information content of a network display can be quite variable. Each of these images represents the exact same graph information.

23. Social Network Data Structures Visualization Network visualization helps build intuition, but you have to keep the drawing algorithm in mind. Again, the same graph with two different techniques: Spring embedder layouts Tree-Based layouts (Fair - poor) (good) Most effective for very sparse, regular graphs. Very useful when relations are strongly directed, such as organization charts. Most effective with graphs that have a strong community structure (clustering, etc). Provides a very clear correspondence between social distance and plotted distance Two images of the same network

24. Social Network Data Structures Visualization Another example: Spring embedder layouts Tree-Based layouts (poor) (good) Two layouts of the same network

25. Social Network Data Structures Visualization • Network visualization helps build intuition, but you have to keep the drawing algorithm in mind. • Hierarchy & Tree models • Use optimization routines to add meaning to the vertical dimension of the plot. This makes it possible to easily see who is most central by who is on the top of the figure. These also include some routine for minimizing line-crossing. • Spring Embedder layouts • Work on an analogy to a physical system: ties connecting a pair have ‘springs’ that pull them together. Unconnected nodes have springs that push them apart. The resulting image reflects the balance of these two forces. This usually creates a layout with a close correspondence between physical closeness and network distance. • In the next slides we give examples of successful graph layouts

26. Social Network Data Structures Visualization A spring embedder layout of romantic relations in a single high school. This image “works” because the sparse nature of the graph allows you to easily trace all of the connections without any line crossings.

27. Social Network Data Structures Visualization Using colors to code attributes makes it simpler to compare attributes and relations. This plot compares the effectiveness of two different clustering routines on a school friendship network. Because the spring-embedder model pulls communities close, we would expect cohesive groups to be in the same region of the graph. This is what we see in the RNM solution at the bottom.

28. Social Network Data Structures Visualization

29. Social Network Data Structures

30. Social Network Data Structures

31. Social Network Data Structures Visualization As networks increase in size, the effectiveness of a point-and-line display routines diminishes, because you simply run out of plotting space. You can still get some insight by using the ‘overlap’ that results in from a space-based layout as information. Here we plot a very large and dense network (the standard point-and-line image is in the upper right).

32. Social Network Data Structures Visualization Adding time to social networks is also complicated, as you run out of space to put time in most network figures. One solution is to animate the network. Here we see streaming interaction in a classroom, where the teacher (yellow square) has trouble maintaining order. The SONIA software program (McFarland and Bender-deMoll) will produce these figures.

33. a a b b c c d d e e b d b d a a 1 1 a c e a 1 1 c e b b 1 c c 1 1 1 1 1 1 d d 1 1 e e 1 1 1 1 Social Network Data Structures Data Representations Pictures only take us so far: from pictures to adjacency matrices Undirected, binary Directed, binary

34. a b b a c c b d e d c e e c d a b c d e a 1 1 b 1 c 1 1 1 d 1 1 e 1 1 Social Network Data Structures Data Representations From matrices to lists Arc List Adjacency List a b b a b c c b c d c e d c d e e c e d

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

36. 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

37. 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?

38. 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

39. 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.

40. 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

41. 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

42. 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.

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

44. Social Networks & Diffusion Reachability • In general, components can be directed or undirected. • For a graph with any directed edges, there are two types of components: • Strong components consist of the set(s) of all nodes that are mutually reachable • Weak components consist of the set(s) of all nodes where at least one node can reach the other.

45. 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

46. 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

47. Measuring Networks: Large-Scale Models 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

48. Measuring Networks: Large-Scale Models 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

49. Measuring Networks: Large-Scale Models 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.

50. Measuring Networks: Large-Scale Models 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