Connectivity & Cohesion

1. Connectivity & Cohesion • Overview • Background: • Small World = Connected • What distinguishes simple connection from cohesion? • Moody & White • Argument • Measure • Bearman, Faris & Moody • Argument • Method • Methods: • Identify components and bicomponents

2. Connectivity & Cohesion Background: 1) Durkheim: What is social solidarity? 2) Simmel: Dyad and Triad 3) Small world: What does it mean to be connected? 4) Can we move beyond small-group ideas of cohesion?

3. Connectivity & Cohesion What are the essential elements of solidarity? 1) Ideological: Common Consciousness 2) Relational: Structural Cohesion • Groups that are ‘held together’ well • Groups should have ‘connectedness’ • cohesion = a “field of forces” that keep people in the group • “resistance of the group to disruptive forces” • “sticking together”

4. Connectivity & Cohesion Analytically, most of these definitions & operationalizations of cohesion do not distinguish the social fact of cohesion from the psychological or behavior outcomes resulting from cohesion. Def. 1: “A collectivity is cohesive to the extent that the social relations of its members hold it together.” What network pattern embodies all the elements of this intuitive definition?

5. Connectivity & Cohesion • This definition contains 5 essential elements: • Focuses on what holds the group together • Expressed as a group level property • The conception is continuous • Rests on observable social relations • Applies to groups of any size

6. Connectivity & Cohesion 1) Actors must be connected: a collection of isolates is not cohesive. Minimally cohesive: a single path connects everyone Not cohesive

7. Connectivity & Cohesion 1) Reachability is an essential element of relational cohesion. As more paths re-link actors in the group, the ability to ‘hold together’ increases. The important feature is not the density of relations, but the pattern. Cohesion increases as # of paths connecting people increases

8. D = . 25 D = . 25 Connectivity & Cohesion Consider the minimally cohesive group: Moving a line keeps density constant, but changes reachability.

9. Removal of 1 person destroys the group. D = . 39 D = . 25 Connectivity & Cohesion What if density increases, but through a single person?

10. Connectivity & Cohesion Cohesion increases as the number of independent paths in the network increases. Ties through a single person are minimally cohesive. D = . 39 More cohesive D = . 39 Minimal cohesion

11. Connectivity & Cohesion Substantive differences between networks connected through a single actor and those connected through many. Minimally Cohesive Strongly Cohesive Power is centralized Power is decentralized Information is concentrated Information is distributed Expect actor inequality Actor equality Vulnerable to unilateral action Robust to unilateral action Segmented structure Even structure Def 2. “A group is structurally cohesive to the extent that multiple independent relational paths among all pairs of members hold it together.”

12. Connectivity & Cohesion

13. Connectivity & Cohesion Formalize the argument: If there is a path between every node in a graph, the graph is connected, and called a component. In every component, the paths linking actors i and j must pass through a set of nodes, S, that if removed would disconnect the graph. The number of nodes in the smallest S is equal to the number of independent paths connecting i and j.

14. 2 1 3 4 5 7 6 8 Connectivity & Cohesion Components and cut-sets: Every path from 1 to 8 must go through 4. S(1,8) = 4, and N(1,8)=1. That is, the graph is a component.

15. 2 1 3 4 5 7 6 8 Connectivity & Cohesion In this graph, there are multiple paths connecting nodes 1 and 8. Components and cut-sets: 1 But only 2 of them are independent. 5 2 4 3 8 1 1 6 6 2 5 4 7 7 3 8 6 5 8 8 N(1,8) = 2. 7 8 8

16. Connectivity & Cohesion The relation between cut-set size and number of paths leads to the two versions of our final definition: Def 3a “A group’s structural cohesion is equal to the minimum number of actors who, if removed from the group, would disconnect the group.” Def 3b “A group’s structural cohesion is equal to the minimum number of independent paths linking each pair of actors in the group.” These two definitions are equivalent.

17. Connectivity & Cohesion Some graph theoretic properties of k-components 1) Every member of a k-components must have at least k-ties. If a person has less than k ties, then there would be fewer than k paths connecting them to the rest of the network. 2) A graph where every person has k-ties is not necessarily a k-component. That is, (1) does not work in reverse. Structures can have high degree, but low connectivity. 3) Two k-components can only overlap by k-1 members. If the k-components overlap by more than k-1 members, then there would be at least k paths connecting the two components, and they would be a single k-component. 4) A clique is n-1 connected. 5) k-components can be nested, such that a k+l component is contained within a k-component.

18. Connectivity & Cohesion Nested connectivity sets: An operationalization of embeddedness. 2 3 1 9 10 8 4 11 5 7 12 13 6 14 15 17 16 18 19 20 21 22 23

19. Connectivity & Cohesion Nested connectivity sets: An operationalization of embeddedness. “Embeddedness” refers to the fact that economic action and outcomes, like all social action and outcomes, are affected by actors’ dyadic (pairwise) relations and by the structure of the overall network of relations. As a shorthand, I will refer to these as the relational and the structural aspects of embeddedness. The structural aspect is especially crucial to keep in mind because it is easy to slip into “dyadic atomization,” a type of reductionism. (Granovetter 1992:33, italics in original)

20. Connectivity & Cohesion Nested connectivity sets: An operationalization of embeddedness. 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}

21. Connectivity & Cohesion Empirical Examples: a) Embeddedness and School Attachment b) Political similarity among Large American Firms

22. Connectivity & Cohesion School Attachment

23. Connectivity & Cohesion Business Political Action

24. Connectivity & Cohesion • Theoretical Implications: • Resource and Risk Flow • Structural cohesion increases the probability of diffusion in a network, particularly if flow depends on individual behavior (as opposed to edge capacity).

25. Probability of infection by distance and number of paths, assume a constant pij of 0.6 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

26. Connectivity & Cohesion • Theoretical Implications: • Community & Class Formation • Community is conceptualized as a structurally cohesive group, and class reproduction is generated by information/resource flow within that group. • Power • Structurally cohesive groups are fundamentally more equal than are groups dominated by relations through a single person, since nobody can monopolize resource flow.

27. Connectivity & Cohesion Blocking the Future: Uses bicomponents to identify historical cases. Argument: The Danto Problem: Sociologically, the future can always change the meaning of a past event, as new information changes the significance of a past event. Examples: 1) The battle of Wounded Knee 2) If we were to discover Clinton was from Mars 3) Battles over the meaning of historical monuments & events (such as Pearl Harbor, or dropping the bomb on Hiroshima, etc.) Not an issue just of data: An “Ideal Chronicler” would have the same problem. The problem of doing history is identifying a case: telling a convincing story, that is robust to changes in our knowledge and our understanding of relations among past events.

28. Connectivity & Cohesion Blocking the Future: Uses bicomponents to identify historical cases. Basic argument: The meaning of an event is conditioned by its position in a sequence of interrelated events. If we can capture the structure of interrelation among events, we can identify the unique features that define an historical case. We propose that multiple connectivity (here bicomponents) linking narratives provide just such a way of casing historical events.

29. Connectivity & Cohesion Blocking the Future An example: Sewell’s account of “Inventing Revolution at the Bastille”. France is nearly bankrupt Dispute over National Assembly Set of ‘crises’ Food problems

30. Connectivity & Cohesion Blocking the Future The problem with these kinds of narratives, is that small changes in facts or understanding of events changes the entire flow of the narrative. Strong theories (i.e. parsimonious) generate weak structures. In contrast, we propose connecting multiple “histories” and based on the resulting pattern, induce historical cases.

31. Connectivity & Cohesion Blocking the Future The empirical setting: A small village in northern china (Liu Ling), reporting on events surrounding the communist revolution. The data: Life stories of 14 people in the village.

32. Connectivity & Cohesion Blocking the Future Kinship structure of the storytellers. Different positions in the village yield different insights into their life stories.

33. Connectivity & Cohesion Blocking the Future An example of a villager life story

34. Connectivity & Cohesion Blocking the Future Traditional summary of events in Liu Ling (condensed)

35. Connectivity & Cohesion Blocking the Future Combining all individual stories:

36. Blocking the Future Of the nearly 2000 total events, about 1500 are linked in a single component

37. Blocking the Future Of the nearly 1500 total events, about 500 are linked in a single bicomponent. This is our candidate for a ‘case’.

38. Blocking the Future Same figure, with dark cases being representatives of the events in the summary history of the village.

39. Blocking the Future Case Resilience to Perturbation Adding Edges at Random Subtracting Edges at Random 1.0 Adjusted Rand Statistic 0.9 0.8 1 2 3 5 7 10 1 2 3 5 7 10 Number of Edges Changed Number of Edges Changed