Connectivity

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
Connectivity Cohesion
Consider the minimally cohesive group
Moving a line keeps density constant, but changes
reachability.
9
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
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
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
3
4
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 groups structural cohesion is equal
to the minimum number of actors who, if removed
from the group, would disconnect the group. Def
3b A groups 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 kl
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
7
5
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
199233, 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 Sewells 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