Chapter 8 Embedding Chapter 8 Embedding
Introduction
 Density
 Reciprocity
 Transitivity
 Clustering
 Group-external and group-internal ties
 Krackhardt's graph theoretical dimensions of hierarchy
Unit of analysis
Micro
 Actor:






People

Subgroups
Organizations
Collectives/aggregat
es
 Communities
 Nation-states
Macro/aggregates

3

 Binary ties-present, or absent
 Directed relations
 有无关系？是否是双向关系？(reciprocal)
 还是单向(asymmetrical)？
Density
The more actors are connected to one another, the more dense the network
will be.
Binary data: the number of present ties/the number of all possible ties
Undirected network: n(n-1)/2 = 2n-1 possible pairs of actors.
L
Δ = n ( n  1) / 2
Directed network: n(n-1)*2/2 = 2n-2possible lines.
ΔD =
L
n ( n  1)
Valued data: the average strength of ties across all possible ties
Density
1
3
5
1
1
3
1
5
1
1
Density=4/(3*2)
Reciprocity
 Directed data, four possible dyadic relations
• Actor: reciprocity pair (AB)/ all possible pairs (AB, BC, AC)
 the number of pairs with a reciprocity tie/the number of pairs
with any tie. (AB)/(AB, BC)
 Relations: reciprocal tie(AB, BA)/all possible pairs (AB, BA,
BC, CB, AC, CA)
• Arc method:
 The number of reciprocal ties/ total number of actual ties
(AB,BA)/(AB, BA, BC)
Reciprocity
 Un-directed data, four possible types of triadic relations
 Directed-data, 16 possible types of triadic relations
Tu = the number of
isomorphism class u
9
Cg3
T=(T003,
T012, ...,T300)‟
=(003, 012, 102, 021D,
021U, 021C, 111D,
111U, 030T, 030C,
201, 120D, 120U,
120C, 210, 300)
Transtivity
Clustering
 “6-degrees” phenomenon, How close actors are together
 “Clique-like” local neighborhood, the tendency towards
dense local neighborhoods.
 Clustering: the density in neighborhood
Weight,
possible
number of
pairs
Group-external and group-internal ties
 Measure of group based on comparing the numbers of
ties within groups and between groups.
 E-I Index
 This value can range from 1 to -1, but for a given
network density and group sizes its range may be
restricted and so it can be rescaled. The index is also
calculated for each group and for each individual actor.
=36/(14+50)
=42/(66+24)
Krackhardt's graph theoretical
dimensions of hierarchy
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?)
Connectedness: The digraph is connected if the underlying graph is a component.
We can measure the extent of connectedness through reachability.


V
Connectedn ess  1  

 N ( N  1) / 2 
Where V is the number of pairs that are not reachable, and N is the
number of people in the network.
How to calculate Connectedness:
1
4
2
3
5
1
2
3
4
5
Digraph:
1 2 3 4 5
0 1 0 1 0
0 0 1 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
1
2
3
4
5
Graph:
1 2 3 4
0 1 0 1
1 0 1 0
0 1 0 0
1 0 0 0
0 0 0 0
5
0
0
0
0
0
1
2
3
4
5
Reach:
1 2 3 4
0 1 2 1
1 0 1 2
2 1 0 3
1 2 3 0
0 0 0 0
V = # of zeros in the upper diagonal of Reach:
V = 4.
C = 1 - [4/((5*4)/2)] = 1 - 4/1 = .6
5
0
0
0
0
0
How to calculate Connectedness:
1
4
2
3
5
1
2
3
4
5
Reach:
1 2 3 4
0 1 2 1
1 0 1 2
2 1 0 3
1 2 3 0
0 0 0 0
This is equivalent to the density of the
reachability matrix.
5
0
0
0
0
0
Reachable:
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
D = SR/(N(N-1))
= 12 /(5*4)
= .6
Graph Hierarchy: The extent to which people are asymmetrically reachable.
Graph Hierarchy


V
 1 

 max( V ) 
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.
Graph Hierarchy: An example
1
4
2
3
5
1
2
3
4
5
Digraph:
1 2 3 4 5
0 1 0 1 0
0 0 1 0 0
0 1 0 0 0
0 0 0 0 0
0 0 0 0 0
1
2
3
4
5
1
0
0
0
0
0
V=1
Max(V) = 4
H = 1/4 = .25
Dreach
2 3 4
1 2 1
0 1 0
1 0 0
0 0 0
0 0 0
5
0
0
0
0
0
1
2
3
4
5
Dreachable
1 2 3 4 5
0 1 2 1 0
0 0 1 0 0
0 1 0 0 0
0 0 0 0 0
0 0 0 0 0
Graph Efficiency: The extent to which there are extra lines in the graph, given the
number of components.


V
Graph effi ciency  1  

max(
V
)


Where v is the number of excess links and max(v)
is the maximum possible number of excess links
The minimum number of lines in a connected
component is N-1 (assuming symmetry, only use
the upper half of the adjacency matrix).
Graph Efficiency:
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.
1
4
2
1
6
3
5
7
2
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
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.
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
H
B
F
A
G
C
D
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.
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.
Least Upper Boundedness: Calculate LUB by looking at reachability.
1
2
3
4
5
6
7
8
9
Distance matrix
1 2 3 4 5 6 7 8 9
1 1 1 2 2
2
1
1 1
1
1
1
1
1 1 1 2
1
1
1
1
1
2
3
4
5
6
7
8
9
Reachable matrix
1 2 3 4 5 6 7 8 9
1 1 1 1 1
1
1
1 1
1
1
1
1
1 1 1 1
1
1
1
1
(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
Graph Theoretic Dimensions of Informal Organizations
Least Upper Boundedness: Calculate LUB by looking at reachability.
1
2
3
4
5
6
7
8
9
Reachable Trans
1 2 3 4 5 6 7 8 9
1
1 1
1
1
1 1
1
1 1
1
1
1 1
1
1
1
1
1 1
1
X
1
2
3
4
5
6
7
8
9
Reachable matrix
1 2 3 4 5 6 7 8 9
1 1 1 1 1
1
1
1 1
1
1
1
1
1 1 1 1
1
1
1
1
(R by S)
(S by R)
R`*R = CR
=
1
2
3
4
5
6
7
8
9
1
1
1
1
1
1
0
0
0
1
Common Reach
2 3 4 5 6 7 8
1 1 1 1 0 0 0
2 1 2 2 0 0 0
1 2 1 1 0 0 0
2 1 3 2 0 0 0
2 1 2 3 0 0 0
0 0 0 0 1 1 1
0 0 0 0 1 2 1
0 0 0 0 1 1 2
1 2 1 1 1 2 1
9
1
1
2
1
1
1
2
1
5
(R by R)
Any place with a zero
indicates a pair that
does not have a LUB.
Least Upper Boundedness: Calculate LUB by looking at reachability.


V
LUB  1  

max(
V
)


Where V = number of pairs that have no LUB, summed over all components, and:
Max (V n ) 
( N n  1)( N n  2 )
2
Bearman, Peter S., James Moody, and Katherine Stovel. &quot;Chains
of Affection: The Structure of Adolescent Romantic and Sexual
Networks1.&quot; American Journal of Sociology 110.1 (2004): 44-91.