Social Network Analysis in Sociolegal Research

1. John Hipp Criminology, Law & Society UC Irvine Social Network Analysis in Sociolegal Research

3. Social Networks Sociogram:A pictorial representation of relationships between actors in a social system • Nodes (vertices, points) represent units of analysis • Edges (arcs, lines, ties) represent relationships among them (Lakon, Godette, & Hipp, 2006)

4. Possible nodes, Possible edges • People • Directed relationships • Ask for advice • Hate/bully • Trust • Love? • Undirected relationships • Friendship • Kinship • Collaborate • Exchange • Binary or valued?

5. Possible nodes, Possible edges • Aggregations of people (organizations, neighborhoods, cities, countries) • Text (paragraphs, sentences) • Court cases • Brain neurons • Residential mobility • Airline flights • Collaboration • Similar concepts • Citations • Timing of firing

6. SociometricSocial NetworkDefined by a specific boundary, information from all actors in a social systemExamples: a classroom of students, all the students in a school, all the members of a church, all organizations in a coalition (Lakon, Godette, & Hipp, 2006)

7. Need to bound the network, and define the types of ties • Must be able to define who is in or not in the network • Sometimes this is straightforward: • Organization • School or classroom • Jury • Sometimes it is not: • Neighborhood • City • Church? • Academic collaborators • Facebook members • Then, which ties are of interest?

8. Overall network structure Network 1 Network 2

9. Migration flow NC counties (race)

10. 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 Translating sociograms to matrices Undirected, binary Directed, binary

11. 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 Translating 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

12. Network Characteristics • Various types of network characteristics • Individual position in network- describe the relative position of network actors • Personal network characteristics -- describes the subgroup one belongs to • Full network characteristics - describes the overall network • Network subgroups - finds subgroups in the larger network

14. Network Measures: individual position

15. Network Characteristics: Isolates • Not tied to anyone in network

16. Network Characteristics: Centrality • Centrality: which nodes are central? • Various strategies: • Degree • Closeness • Betweenness • Information/power

17. Network Characteristics: In-degree Centrality

18. Network Characteristics: Degree Centrality

19. Standardize by the maximum possible ties (N-1)

20. Most central?

21. Network Characteristics: Closeness Centrality • Measures the closeness of an actor to all other actors • Closeness based on geodesic distance: the shortest path between two actors • So compute the average of the inverse distance to all other actors

22. Network Characteristics: Betweeness Centrality • Based on communication flow • Important actors lie on communication paths • Compute the number of geodesic paths between i and k that actor j resides on

23. Network Characteristics: Information Centrality • Information can flow on paths other than the shortest path • This measure uses all paths, and weights them based on their length

24. Network Characteristics: Bonacich Power Centrality • Centrality (prestige) is a function of the prestige of those one is connected to • Actors tied to central actors will have higher prestige/ centrality • A parameter determines the weighting of the global structure

25. Network Characteristics: Bridging

26. Network Characteristics: Bridging • Various measures • Average path length • Lambda Sets • Valente measure • What to do with isolates??? • N-1? • Maximum path length + 1?

27. Network Measures: personal network

28. Personal Network Characteristics: Size • Number of ties: sent, received, both?

29. Personal Network Characteristics: Density

30. Personal Network Chars: Homogeneity • Homogeneity: similarity of social ties based on some characteristic • Homophily is choosing (or maintaining) ties based on similarity • Age, race, education, income, gender • Network range is the extent to which a person’s ties connects them to a diverse set of other actors

31. Network Characteristics: Reciprocated Ties • Directed ties can be in one of three states: • 1) Mutual • 2) Asymmetric • 3) Null • Compute proportion reciprocated ties

32. Personal Network chars:Transitivity • Transitivity (Friends of friends are friends)

33. + + + Social balance/transitivity Balance theory (Heider) is determined based on the product of the edges: “A friend of a friend is a friend” (+)(+)(+) = (+) Balanced “An enemy of my enemy is a friend” - - (-)(+)(-) = (+) Balanced + “An enemy of my enemy is an enemy” - - (-)(-)(-) = (-) Unbalanced - “A Friend of a Friend is an enemy” + + (+)(-)(+) = (-) Unbalanced -

34. Network Characteristics: multiplexity • Based on multiple networks • Suppose studying three types of ties: • Hang out with • Ask advice about personal problems • Have sexual relations • Multiplexity is the degree of overlap between these tie types • A multiplex tie would engage in all three • Personal network: compute average multiplexity of ties

35. Network Measures: full network

36. å X D = - N ( N 1 ) Network Characteristics: Density • The volume of relations in the system is known as density • The proportion of ties out of all possible ties:

37. å X D = N Network Characteristics: Mean degree • The average number of ties per node is known as mean degree:

38. Network Characteristics: Clustering Clustering coefficient: average density of personal networks

39. Network Characteristics: centralization • This captures the degree to which the entire graph is centralized • One approach: the variance of centrality scores (this is the dispersion of centrality)

40. Cohesion/solidarity • Various measures: • Density • Number of ties • Reachability • Ratio of within-group ties to out-group ties

41. Cohesion/solidarity From Moody-White: 1) Reachability is an essential element of relational cohesion. As more paths re-link actors in the group, the ability to ‘hold together’ increases.

42. Network Characteristics: Components • A component is when there is at least one path connecting every pair of actors • If this is the entire graph, the graph is considered connected • Two paths are independent if they only have the two end-nodes in common. • If a graph has two independent paths between every pair, it is a bicomponent. • Similarly for three paths, four, etc. • Cutpoint: a node that , when removed, divides the network

43. Network measure: Hierarchy Linear Hierarchy (all triads transitive) Simple Hierarchy Branched Hierarchy Mixed Hierarchy

44. Network Measures: Finding subgroups

45. Network Characteristics: Finding subgroups • A) Graph theoretical methods: Cliques and extensions of cliques • Cliques, k-cores, k-plexes, k-components • B) Algorithmic methods: search through a network trying to maximize a particular pattern • Adjust assignment of actors to groups until a particular pattern of ties (block diagonal, usually) is identified: • Factions (UCI-NET), NEGOPY (Richards), KliqueFinder (Frank), RNM (Moody), CROWDS (Moody), General Distance & Clustering Methods

46. Network Characteristics: Cliques Clique: a maximal subgraph in which every member of the graph is connected to every other member of the graph. • Properties of cliques: • Density: 1.0 • Everyone connected to n-1 alters • Distance between every pair is 1 • Ratio of within group ties to between group ties is infinite • All triads are transitive • But not real useful: • Too small • Tend to overlap

47. Network Characteristics: k-cores k-cores: Every person connected to at least k other people. Here are two 3-cores. Note that adding a single tie from A to B would make the whole graph a 3-core

48. Network Characteristics: other cliques K-plex: Every member connected to at least n-k other people in the graph (recall in a clique everyone is connected to n-1, so this relaxes that condition. n-clique: Every person is connected by a path of N or less (recall a clique is with distance = 1). N-clan: same as an n-clique, but all paths must be inside the group.

49. Finding subgroups with algorithmic methods • Measure of fit • Typically, this is a function of within group ties to across group ties • Algorithm for maximizing fit • Need to search the network for an optimal fit • Many different available algorithms • Generalized cluster analysis • Sometimes can use the relational distance directly to look for clusters

50. Finding subgroups with algorithmic methods • Cluster analysis creates a distance matrix between each pair of points. • Most commonly this is Euclidean distance (the two dimensions are lat/long) • But could be social dimensions • Can utilize any number of dimensions • For network data, the distance is often the path-distance between pairs