Social Sub-groups

1. Social Sub-groups • Overview • Background: • How do we characterize the social structure of a ‘group’? • Exemplar: Ken Frank and Jeffrey Yasumoto • A discussion of how action is situated within and between social groups. Nice application of a group-detection algorithm on interesting data. • Linton Freeman • UC-Irvine. Long-standing editor of the journal Social Networks. Writes today on the theoretical necessities of a ‘group’. • Rise of StatMath: Modularity fro Newman, Porter, Mucha • Methods: Algorithms & Measures

2. Social Sub-groups Frank & Yasumoto: Action and Structure “...subgroups may define the essential components that contextualize actors’ social ties and relations.” The predominance of subgroups in the literature, “...leaves unanswered how and why rational actors simultaneously sustain their subgroups and the linkages between them.”

3. Social Sub-groups Frank & Yasumoto: Action and Structure They argue that actors seek social capital, defined as the access to resources through social ties, and emphasize two mechanisms: a) Reciprocity Transactions Actors seek to build obligations with others, and thereby gain in the ability to extract resources. b) Enforceable Trust “Social capital is generated by individual members’ disciplined compliance with group expectations.” An indirect, group level effect, that comes through the judicious non-use of negative action. (p.646)

4. Social Sub-groups Frank & Yasumoto: Action and Structure They expect to find evidence of enforceable trust within social subgroups and evidence of reciprocity between such groups. To do so, they must identify primary subgroups within the network. They do so using a density based criterion. Frank’s algorithm iteratively assigns nodes to subgroups until a parameter that maximizes in-group density is reached. Basic model is: logit(Yij)= a + gij Seek to find an assignment of nodes to groups (g) that maximizes fit. This results in a ‘block diagonal’ adjacency matrix, where most of the ties fall along the diagonal.

5. Relations among the French Financial Elite (as drawn by F&Y) Relations within group are weighted heavier than between to generate this picture: Group-weighted MDS

6. Relations among the French Financial Elite (as drawn by F&Y) Treat all edges equal, get a somewhat less clear pattern:

7. # of ties: n A B C D n 2 5 2 2 1 A 5 42 8 1 4 B 2 8 22 3 1 C 2 1 3 8 1 D 1 4 1 1 4 Density: n 0.167 0.139 0.095 0.167 0.111 A 0.139 0.467 0.127 0.028 0.148 B 0.095 0.127 0.629 0.143 0.067 C 0.167 0.028 0.143 0.667 0.111 D 0.111 0.148 0.067 0.111 0.667 Relations among the French Financial Elite: Group to group density table

8. Relations among the French Financial Elite: Given a subgroup structure, how do these groups relate to social capital? Enforceable trust: Look for acts of hostility. A hostile act was any action on the part of one actor that would deprive another actor of access to resources. Note that these were rare. Only 15 overall, likely indicating some level of cohesion in the system as a whole. On the whole, they find that -- net of other focal features and direct ties -- being members of the same sub-group lowers the probability of a negative action between the dyad

9. Relations among the French Financial Elite: They repeat the exercise with positive support. They find that supportive actions are better predicted by friendship (reciprocity) than by subgroup membership. They conclude that this supports the hypothesis that “the potential for enforceable trust within subgroups reduces the relative need to pursue social capital through reciprocity transactions within subgroups.” (p.647) Instead, they find that support occurs between subgroups.

10. Social Sub-groups Lin Freeman: The sociological concept of “Group” Focus on collectivities that are: “Relatively small, informal, and involve close personal ties.” What we would call “Primary Groups” What (network) structure characterizes such a group? Goal: Identify (a) non-overlapping groups that allow one to (b) identify internal group structure.

11. Social Sub-groups Lin Freeman: The sociological concept of “Group” Winship’s Model: 1) Assign people to equivalence classes that are hierarchically nested:

12. Social Sub-groups Lin Freeman: The sociological concept of “Group” Winship’s Model: To assign people to a class, you must first identify the strength of the relation between each pair. Winship’s model says that you define proximity based on interaction such that:

13. Social Sub-groups Lin Freeman: The sociological concept of “Group” Winship’s Model: In words, this means that whatever metric you define, a person is closer to themselves than to anyone else, that the relation be symmetric, and that triads be transitive (which, given the symmetric condition, means that they be complete). You can then identify partitions by scaling the proximity, such that these three conditions are met.

14. A B C D E F G H I J K A .5544443333 B 5 . 544443333 C 55 . 44443333 D 444 . 5553333 E 4445 . 553333 F 44455 . 53333 G 444555 . 3333 H 3333333 . 555 I 33333335 . 55 J 333333355 . 5 K 3333333555 . Social Sub-groups Lin Freeman: The sociological concept of “Group” Winship’s Model:

15. Social Sub-groups Lin Freeman: The sociological concept of “Group” Winship’s Model: total {A-G} {H-K} {A-C} {D-G}

16. Social Sub-groups Lin Freeman: The sociological concept of “Group” Granovetter’s Model: Proceed exactly as in Winship, but treat intransitivity differently when looking at strong or weak ties. If x and y are strongly connected, and y and z are strongly connected, then x and z should be at least weakly connected.

17. Social Sub-groups Lin Freeman: The sociological concept of “Group” Granovetter’s Model: An example of a graph fitting the prohibition against G-intransitive relations.

18. Social Sub-groups The Davis - “Old South” Example

19. Social Sub-groups The Davis - “Old South” Example: Ties > 2

20. Social Sub-groups The Davis - “Old South” Example: Ties > 3

21. Social Sub-groups The Davis - “Old South” Example: Ties > 4 Meets the G-transitivity condition

22. Social Sub-groups The Davis - “Old South” Example: Ties > 5 Stronger than the G-transitivity condition

23. Social Sub-groups Lin Freeman: The sociological concept of “Group” Freeman argues that the G-intransitivity model fits the data best for each of the 7 groups he studies. Substantively, the types of groups this model predicts are very similar to those predicted by the general transitivity model, except re-cast as a valued relation. Empirically, if you want to identify groups based on levels like this, you can use PAJEK and walk through the model in just the same way as we did with “Old South” or you can use UCI-NET (or program it, it’s not hard)

24. Methods: How do we identify primary groups in a network? • A) Classic graph theoretical methods: Cliques and extensions of cliques • Cliques • k-cores • k-plexes • Freeman (1992) Models • K-components (we talked about these already) • B) Algorithmic methods: search through a network trying to maximize for a particular pattern (I.e. like Frank & Yasumoto) • Adjust assignment of actors to groups until a particular pattern of ties (block diagonal, usually) is identified. • Standard models: • - Factions (UCI-NET) • - KliqueFinder (Frank) • RNM/CROWDS/JIGGLE (Moody) • Principle component analysis (PCA) • Flow models (MCL) • Modularity Maximization routines • - General Distance & Clustering Methods

25. Methods: How do we identify primary groups in a network? Graph Theoretical Models. Start with a clique. A clique is defined as a maximal subgraph in which every member of the graph is connected to every other member of the graph. Cliques are collections of nodes where density = 1.0. • 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

26. Methods: How do we identify primary groups in a network? Graph Theoretical Models. In practice, complete cliques are not very useful. They tend to overlap heavily and are limited in their size. Graph theorists have thus relaxed the complete connectivity requirement (with varying degrees of success). See the Moody & White paper on cohesion for a discussion of many of these attempts.

27. Methods: How do we identify primary groups in a network? Graph Theoretical Models. k-cores: Every person connected to at least k other people. Ideally, they would look something like this (here two 3-cores). However, adding a single tie from A to B would make the whole graph a 3-core

28. Methods: How do we identify primary groups in a network? Graph Theoretical Models. Extensions of this idea include: 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. I’ve never had much luck with any of these methods empirically. Real data is usually too messy to work well. You should try them, and gain some intuition for yourself. The place to start is in UCINET.

29. Methods: How do we identify primary groups in a network? Graph Theoretical Models. UCINET will compute all of the best-known graph theoretic treatments for subgroups

30. Methods: How do we identify primary groups in a network? Graph Theoretical Models. Consider running different methods on a known group structure:

31. Methods: How do we identify primary groups in a network? Graph Theoretical Models.

32. Methods: How do we identify primary groups in a network? Graph Theoretical Models. Cliques

33. Methods: How do we identify primary groups in a network? Cliques The only way to get something meaningful from this is to analyze the clique overlap matrix, which is what the “Clique by partion” dataset does, using cluster analysis

34. Methods: How do we identify primary groups in a network? K-Cores (See example, but in this case it works very poorly)

35. Methods: How do we identify primary groups in a network? n-Clique: (Everyone linked by a path of at least length n)

36. Methods: How do we identify primary groups in a network? n-Clique: (Everyone linked by a path of at least length n)

37. Methods: How do we identify primary groups in a network? n-Clan: (Everyone linked by a path of at least length n, but path is INSIDE group)

38. Methods: How do we identify primary groups in a network? K-plex: (each member of a K-plex of size N has N-K ties to other members)

39. Methods: How do we identify primary groups in a network? Strategies for identifying primary groups: Search: 1) Fit Measure: Identify a measure of groupness (usually a function of the number of ties that fall within group compared to the number of ties that fall between group). 2) Algorithm to maximize fit. Once we have the index, we need a clever method for searching through the network to maximize the fit. See: “Jiggle”, “Factions” etc. Destroy: Break apart the network in strategic ways, removing the weakest parts first, what’s left are your primary groups. See “edge betweeness” “MCL” Evade: Don’t look directly, instead find a simpler problem that correlates: Examples: Generalized cluster analysis, Factor Analysis, RM.

40. Methods: How do we identify primary groups in a network? Search: Optimize a partition to fit Segregation Index (Freeman, L. C. 1972. "Segregation in Social Networks." Sociological Methods and Research 6411-30.) Freeman asked how we could identify segregation in a social network. Theoretically, he argues, if a given attribute (group label) does not matter for social relations, then relations should be distributed randomly with respect to the attribute. Thus, the difference between the number of cross-group ties expected by chance and the number observed measures segregation.

41. Methods: How do we identify primary groups in a network? Search: Optimize a partition to fit Consider the (hypothetical) network below. There are two attributes in this network: people with Blue eyes and Brown eyes and people who are square or not (they must be hip).

42. Blue Brown Blue 6 17 Brown 17 16 Hip Square Hip 20 3 Square 3 30 Methods: How do we identify primary groups in a network? Search: Optimize a partition to fit Segregation Index Mixing Matrix: Seg = -0.25 Seg = 0.78

43. Methods: How do we identify primary groups in a network? Search: Optimize a partition to fit Segregation Index One problem with the segregation index is that it is not ‘margin free.’ That is, if you were to change the distribution of the category of interest (say race) by a constant but not the core association between race and friendship choice, you can get a different segregation level. One antidote to this problem is to use odds ratios. In this case, and odds ratio tells us the relative likelihood that two people in the same category will choose each other as friends.

44. Methods: How do we identify primary groups in a network? Search: Optimize a partition to fit The second problem is that the Segregation index has no clear maximum – if every node is assigned to a single group the value can be higher than if everyone is assigned to the “right” group. This means you can’t just keep adjusting nodes until you see a best fit, but instead have to look for changes in fit. The modularity score solves this problem by re-organizing the expectation in a way that forces the value to 0 if everyone is in a single group.

45. Methods: How do we identify primary groups in a network? Search: Optimize a partition to fit We can also measure the extent that ties fall within clusters with the modularity score: Where: s indexes clusters in the network ls is the number of lines in cluster s ds is the sum of the degrees of s L is the total number of lines M has the advantage of going to 0 if there is only 1 group, which means maximizing the score is sensible

46. Methods: How do we identify primary groups in a network? Search: Optimize a partition to fit Modularity Scores Comparison to Segregation Index – comparing values for known solutions Modularity Score Plotted against Segregation Index for various nets

47. Methods: How do we identify primary groups in a network? Search: Optimize a partition to fit Number of groups  In-group Density 

48. Methods: How do we identify primary groups in a network? Search: Optimize a partition to fit • Factions in UCI-NET • Multiple options for the exact factor maximized. I recommend either the density or the correlation function, and I would calculate the distance in each case. • Frank’s KliqueFinder (the AJS paper we just read) • Moody’s crowds / Jiggle has elements of this • Generalized blockmodel in PAJEK • iGraph (R) has a couple that see this sort

49. Methods: How do we identify primary groups in a network? Search: Optimize a partition to fit Factions in UCI-NET

50. Factions in UCI-NET