Hierarchy

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Hierarchy

• Overview
• Background
• Hierarchy surrounds us what is it?
• Micro foundations of social stratification
• Ivan Chase Structure from process
• Action --gt Structure, not attributes
• David Krackhardt
• Deliberate Structure w. in organizations
• Measures for the extent of hierarchy

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Examples of Hierarchical Systems
Linear Hierarchy (all triads transitive)
Simple Hierarchy
Branched Hierarchy
Mixed Hierarchy
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Examples of Similar Non-Hierarchical Systems
Line Graph
Acyclic Cycle
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Chases 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.

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Chases Question Where does hierarchy come from?

• The Correlational Model
• Individuals 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 gt .9). (See Chase,
  1974 for details)

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Chases 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.

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Chases 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)
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What process could generate all 030T triads?
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Chases Question Where does hierarchy come from?
The elements Dominance relations must by
asymmetric, thus, the set of possible triads is
limited.
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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
p1.
P( 3 T) (.5.5 .251 .251) .75
p.25
003
012
030T
p1
021D
p.25
p1
021U
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Dominance Strategies 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
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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
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The evidence 24 Chase Chicken Triads
( 0 stay)
1
( 0 stay)
030C
1
021C

2
(1 stays)
1
23
(6 Fully Transitive)
17
003
012
(17 stay)
1
030 T
021D
4
4
Most Common Path
Domination Reversal
021U
New Domination
( 0 stay)
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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?)
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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.
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Graph Theoretic Dimensions of Informal
Organizations
How to calculate Connectedness
V of zeros in the upper diagonal of Reach
V 4.
C 1 - 4/((54)/2) 1 - 4/1 .6
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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 /(54)
.6
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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.
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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
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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
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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 V2. 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
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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.
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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
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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.
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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
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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.
RR CR
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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
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Other characteristics of Hierarchy

• DAG Directed, Acyclic, Graph
• Graph that
• contains no cycles
• at least one node has in-degree
• 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
• Well return to this when we get to centralization

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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
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Another method Approximation based on permutation
Re-ordered matrix