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Ivan Chase: Structure from process David Krackhardt:

- Overview
- Background:
- Hierarchy surrounds us: what is it?
- Micro foundations of social stratification

- Action --> Structure, not attributes

- Deliberate Structure w. in organizations
- Measures for the extent of hierarchy

Examples of Hierarchical Systems

Linear Hierarchy

(all triads transitive)

Simple

Hierarchy

Branched

Hierarchy

Mixed

Hierarchy

Chase’s Question: Where does hierarchy come from?

Hierarchy surrounds us, in natural (animal and human) and controlled (laboratory, organizations) settings. How do we account for it?

- Most previous research focuses on the static structure of hierarchy
- Often consider the attributes of actors: strength, race, gender, education, size, etc.

Chase’s Question: Where does hierarchy come from?

- The “Correlational Model”
- Individual’s position in the hierarchy is due to their attributes (physical, social, etc.)
- Mathematically, for the correlational model to be true, the correspondence between attributes and rank in the hierarchy would have to be extremely high (Pearson correlation of > .9). (See Chase, 1974 for details)

Chase’s Question: Where does hierarchy come from?

- The “Pairwise interaction” model
- Pairwise differences in each dyad account for position in the hierarchy.
- “...it is assumed that each member of a group has a pairwise contest with each other member, that the winner of a contest dominates the loser in the group hierarchy, and that an individual has a particular probability of success in each contest.”
- Model implies that there be one individual with a .95 probability of beating every other individual, another with a .95 probability of beating everyone but the most dominant, and so forth down the line.
- The required conditions simply do not hold. As such, this explanation for where the hierarchy comes from cannot hold.

B

C

D

E

Chase’s Question: Where does hierarchy come from?

Chase focuses on the simple mathematical fact: Every linear hierarchy must contain all transitive triads. That is, the triad census for the network must have only 3 T triads.

Number of

Type triads

----------------------

1 - 3

-----------------------

2 - 012 0

3 - 102 0

4 - 021D 0

5 - 021U 0

6 - 021C 0

7 - 111D 0

8 - 111U 0

9 - 030T 10

10 - 030C 0

11 - 201 0

12 - 120D 0

13 - 120U 0

14 - 120C 0

15 - 210 0

16 - 300 0

---------------------------

Sum (2 - 16): 10

What process could generate all 030T triads?

A

B

C

B

C

Transitive (030T) triad

Intransitive (030C) triad

Chase’s Question: Where does hierarchy come from?

The elements: Dominance relations must by asymmetric, thus, the set of possible triads is limited.

Why Chase Finds Linear Hierarchy:

Triad transitions (w/ Random Expectations) for Dominance Relations.

P( 3 C) =

.5*.5=.25

p=.5

030C

p=.5

021C

p=.5

p=1.

P( 3 T)=

(.5*.5 + .25*1

+.25*1) = .75

p=.25

003

012

030T

p=1

021D

p=.25

p=1

021U

That ensure a transitive hierarchy

The “Double Attack” Strategy:

The first attacker quickly attacks the bystander. This means we arrive at 21D, and any action on the part of the other two chickens will lead to a transitive triad.

003

012

030T

021D

The “Double Receive” Strategy:

The first attacker dominates B, and then the bystander quickly dominates B as well, leading to 21U, and any dominance between the first and second attacker will lead to a transitive triple.

003

012

030T

021U

030C

Dominance Strategies

That may not lead to a transitive hierarchy

“Attack the Attacker”

The bystander attacks the first attacker. This could lead to a cyclic triad, and thus thwart hierarchy.

021C

003

012

030T

021C

“Pass on the attack”

The one who is attacked, attacks the bystander. Again, this could lead to a cycle, and thus thwart hierarchy.

003

012

030T

24 Chase Chicken Triads

( 0 stay)

1

( 0 stay)

030C

1

021C

2

(1 stays)

1

23

(6 Fully Transitive)

17

012

003

(17 stay)

1

030 T

021D

4

4

Most Common Path

Domination Reversal

021U

New Domination

( 0 stay)

Graph Theoretic Dimensions of Informal Organizations

Moving beyond dominance relations in animals, what can SNA tell us about dominance in organizations?

Krackhardt argues that an ‘Outree” is the archetype of hierarchy.

- Krackhardt focuses on 4 dimensions:
- 1) Connectedness
- 2) Digraph hierarchic
- 3) digraph efficiency
- 4) least upper bound

(what are the allowed triad types for an out-tree?)

Graph Theoretic Dimensions of Informal Organizations

Connectedness: The digraph is connected if the underlying graph is a component. We can measure the extent of connectedness through reachability.

Where V is the number of pairs that are not reachable, and N is the number of people in the network.

1 2 3 4 5

1 0 1 2 1 0

2 1 0 1 2 0

3 2 1 0 3 0

4 1 2 3 0 0

5 0 0 0 0 0

Graph:

1 2 3 4 5

1 0 1 0 1 0

2 1 0 1 0 0

3 0 1 0 0 0

4 1 0 0 0 0

5 0 0 0 0 0

Digraph:

1 2 3 4 5

1 0 1 0 1 0

2 0 0 1 0 0

3 0 0 0 0 0

4 0 0 0 0 0

5 0 0 0 0 0

1

4

2

5

3

Graph Theoretic Dimensions of Informal Organizations

How to calculate Connectedness:

V = # of zeros in the upper diagonal of Reach:

V = 4.

C = 1 - [4/((5*4)/2)] = 1 - 4/1 = .6

1 2 3 4 5

1 0 1 1 1 0

2 1 0 1 1 0

3 1 1 0 1 0

4 1 1 1 0 0

5 0 0 0 0 0

Reach:

1 2 3 4 5

1 0 1 2 1 0

2 1 0 1 2 0

3 2 1 0 3 0

4 1 2 3 0 0

5 0 0 0 0 0

1

4

2

5

3

Graph Theoretic Dimensions of Informal Organizations

How to calculate Connectedness:

This is equivalent to the density of the reachability matrix.

D = SR/(N(N-1))

= 12 /(5*4)

= .6

Graph Theoretic Dimensions of Informal Organizations

Graph Hierarchy: The extent to which people are asymmetrically reachable.

Where V is the number of symmetrically reachable pairs in the

network. Max(V) is the number of pairs where i can reach j or j can reach i.

4

2

5

3

Graph Theoretic Dimensions of Informal Organizations

Graph Hierarchy: An example

Dreachable

1 2 3 4 5

1 0 1 2 1 0

2 0 0 1 0 0

3 0 1 0 0 0

4 0 0 0 0 0

5 0 0 0 0 0

Digraph:

1 2 3 4 5

1 0 1 0 1 0

2 0 0 1 0 0

3 0 1 0 0 0

4 0 0 0 0 0

5 0 0 0 0 0

Dreach

1 2 3 4 5

1 0 1 2 1 0

2 0 0 1 0 0

3 0 1 0 0 0

4 0 0 0 0 0

5 0 0 0 0 0

V = 1

Max(V) = 4

H = 1/4 = .25

Graph Theoretic Dimensions of Informal Organizations

Graph Efficiency: The extent to which there are extra lines in the graph, given the number of components.

Where v is the number of excess links and max(v) is the maximum possible number of excess links

4

2

6

5

3

7

Graph Theoretic Dimensions of Informal Organizations

Graph Efficiency:

The minimum number of lines in a connected component is N-1 (assuming symmetry, only use the upper half of the adjacency matrix).

In this example, the first component contains 4 nodes and thus the minimum required lines is 3. There are 4 lines, and thus V1= 4-3 = 1.

The second component contains 3 nodes and thus minimum connectivity is = 2, there are 3 so V2 = 1. Summed over all components V=2.

The maximum number of lines would occur if every node was connected to every other, and equals N(N-1)/2. For the first component Max(V1) = (6-3)=3. For the second, Max(V2) = (3-2)=1, so Max(V) = 4.

Efficiency = (1- 2/4 ) = .5

1

2

Graph Theoretic Dimensions of Informal Organizations

Graph Efficiency:

Steps to calculate efficiency:

a) identify all components in the graph

b) for each component (i) do:

i) calculate Vi

= S(Gi)/2 - Ni-1;

ii) calculate Max(Vi)

= Ni(Ni-1) - (Ni-1)

c) V = S(Vi), Max(V)= S(Max(Vi)

d) efficiency = 1 = V/Max(V)

Substantively, this must be a function of the average density of the components in the graph.

Graph Theoretic Dimensions of Informal Organizations

Least Upper Boundedness: This condition looks at how many ‘roots’ there are in the tree. The LUB for any pair of actors is the closest person who can reach both of them. In a formal hierarchy, every pair should have at least one LUB.

E

In this case, E is the LUB for (A,D), B is the LUB for (F,G), H is the LUB for (D,C), etc.

H

B

G

C

F

A

D

Graph Theoretic Dimensions of Informal Organizations

Least Upper Boundedness: You get a violation of LUB if two people in the organization do not have an (eventual) common boss.

Here, persons 4 and 7 do not have an LUB.

1 2 3 4 5 6 7 8 9

1 1 1 1 2 2 2

2 1 1 1

3 1 1

4 1

5 1

6 1 1 1 2

7 1 1

8 1

9 1

Reachable matrix

1 2 3 4 5 6 7 8 9

1 1 1 1 1 1 1

2 1 1 1

3 1 1

4 1

5 1

6 1 1 1 1

7 1 1

8 1

9 1

Graph Theoretic Dimensions of Informal Organizations

Least Upper Boundedness: Calculate LUB by looking at reachability.

(Note that I set the diagonal = 1)

A violation occurs whenever a pair is not reachable by at least one common node. We can get common reachability through matrix multiplication

1 2 3 4 5 6 7 8 9

1 1 1 1 1 1 1

2 1 1 1

3 1 1

4 1

5 1

6 1 1 1 1

7 1 1

8 1

9 1

Reachable Trans

1 2 3 4 5 6 7 8 9

1 1

2 1 1

3 1 1

4 1 1 1

5 1 1 1

6 1

7 1 1

8 1 1

9 1 1 1 1 1

Graph Theoretic Dimensions of Informal Organizations

Least Upper Boundedness: Calculate LUB by looking at reachability.

Common Reach

1 2 3 4 5 6 7 8 9

1 1 1 1 1 1 1

2 1 2 1 2 2 1

3 1 1 2 1 1 2

4 1 2 1 3 2 1

5 1 2 1 2 3 1

6 1 1 1 1

7 1 2 1 2

8 1 1 2 1

9 1 1 2 1 1 1 2 1 5

X

=

(R by S)

(S by R)

(R by R)

Any place with a zero indicates a pair that does not have a LUB.

R`*R = CR

Graph Theoretic Dimensions of Informal Organizations

Least Upper Boundedness: Calculate LUB by looking at reachability.

Where V = number of pairs that have no LUB, summed over all components, and:

Other characteristics of Hierarchy:

- DAG: Directed, Acyclic, Graph
- Graph that:
- contains no cycles
- at least one node has in-degree

- Graph that:
- Rank Cluster
- Graph in which some number of nodes are mutually reachable, but asymmetrically reachable between groups.

- Tree
- A DAG with only one root

- Centralization
- We’ll return to this when we get to centralization

Another method: Approximation based on permutation

One characteristic of a hierarchy is that most of the ties fall on the upper triangle of the adjacency matrix. Thus, one way to get an order is by juggling the rows and columns until most of the ties are in the upper triangle.

1 1 1 1 1 1 1 1 1

1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8

1 1

2 1

3

4 1 1

5 1

6 1 1

7

8 1 1 1

9 1 1 1

1 1

11 1

12 1 1

13 1 1 1

14 1 1 1

15

16

17 1

18 1

14 1 1 1

9 1 1 1

1 1

2 1

11 1

5 1

1 1

8 1 1 1

18 1

17 1

4 1 1

6 1 1

12 1 1

3

7

15

16

Another method: Approximation based on permutation

Re-ordered matrix

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