Social Balance & Transitivity. Overview Background: Basic Balance Theory Extensions to directed graphs Basic Elements: Affect P  O  X Triads and Triplets Among Actors Among actors and Objects Theoretical Implications: Micro foundations of macro structure
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Heider’s work on cognition of social situations, which can be boiled down to the relations among three ‘actors’:
Object
X
P
O
Other
Person
Heider was interested in the correspondence of P and O, given their beliefs about X
x
x
x
x
x
x
x

+

+
+

+

+

+

+

+

o
o
o
o
o
o
o
o
p
p
p
p
p
p
p
p
+
+



+

+
Social Balance & Transitivity
Each dyad (PO, PX, OX) can take on one of two values: + or 
8 POX triples:
Two Relations:
Like:
+
Dislike

x
x
x
x
x
x
x
x


+

+
+

+
+
+


+
+


+


+

+
+

o
o
o
o
o
o
o
o
p
p
p
p
p
p
p
p

+

+
+

+


+

+
The 8 triples can be reduced if we ignore the distinction between POX:

+
+

+

+



+
+
We determine balance based on the product of the edges:
“A friend of a friend is a friend”
(+)(+)(+) = (+)
Balanced
“An enemy of my enemy is a friend”
()(+)() = ()
Balanced
“An enemy of my enemy is an enemy”
()()() = ()
Unbalanced
“A Friend of a Friend is an enemy”
(+)()(+) = ()
Unbalanced
+
+

+
+
+

+
+



Heider argued that unbalanced triads would be unstable: They should transform toward balance
Become Friends
Become Enemies
Become Enemies
IF such a balancing process were active throughout the graph, all intransitive triads would be eliminated from the network. This would result in one of two possible graphs (Balance Theorem):
Complete Clique
Balanced Opposition
Friends with
Enemies with
Empirically, we often find that graphs break up into more than two groups. What does this imply for balance theory?
It turns out, that if you allow all negative triads, you can get a graph with many clusters. That is, instead of treating ()()() as an forbidden triad, treat it as allowed. This implies that the micro rule is different: negative ties among enemies are not as motivating as positive ties.
Empirically, we also rarely have symetric relations (at least on affect) thus we need to identify balance in undireced relations. Directed dyads can be in one of three states:
1) Mutual
2) Asymmetric
3) Null
Every triad is composed of 3 dyads, and we can identify triads based on the number of each type, called the MAN label system
i
j
j
i
k
k
Balance in directed relations
Actors seek out transitive relations, and avoid intransitive relations. A triple is transitive
If:
&
then:
Once we admit directed relations, we need to decompose triads into their constituent triples.
Ordered Triples:
a
b
c;
a
c
Transitive
b
a
c
b;
a
b
Vacuous
a
c;
b
c
b
Vacuous
a
c
b
c
a;
b
a
Intransitive
120C
a
b;
c
b
c
Intransitive
c
b
a;
c
a
Vacuous
(0)
(1)
(2)
(3)
(4)
(5)
(6)
003
012
102
111D
201
210
300
021D
111U
120D
Intransitive
Transitive
021U
030T
120U
Mixed
021C
030C
120C
An Example of the triad census
Type Number of triads

1  003 21

2  012 26
3  102 11
4  021D 1
5  021U 5
6  021C 3
7  111D 2
8  111U 5
9  030T 3
10  030C 1
11  201 1
12  120D 1
13  120U 1
14  120C 1
15  210 1
16  300 1

Sum (2  16): 63
As with undirected graphs, you can use the type of triads allowed to characterize the total graph. But now the potential patterns are much more diverse
1) All triads are 030T:
A perfect linear hierarchy.
1
1
1
1
Cluster Structure, allows triads: {003, 300, 102}
N*
Eugene Johnsen (1985, 1986) specifies a number of structures that result from various triad configurations
M
M
N*
N*
N*
N*
M
M
A*
A*
A*
A*
A*
A*
A*
A*
M
N*
M
M
N*
M
M
PRC{300,102, 003, 120D, 120U, 030T, 021D, 021U} Ranked Cluster:
1
0
0
0
0
1
1
0
0
0
1
0
1
0
0
1
1
1
1
0
1
1
1
0
1
And many more...
Substantively, specifying a set of triads defines a behavioral mechanism, and we can use the distribution of triads in a network to test whether the hypothesized mechanism is active.
We do this by (1) counting the number of each triad type in a given network and (2) comparing it to the expected number, given some random distribution of ties in the network.
See Wasserman and Faust, Chapter 14 for computation details, and the SPAN manual for SAS code that will generate these distributions, if you so choose.
BA
CL
RC
R2C
TR
HC
39+
p1
p2
p3
p4
Triad:
003
012
102
021D
021U
021C
111D
111U
030T
030C
201
120D
120U
120C
210
300
Triad MicroModels:
BA: Ballance (Cartwright and Harary, ‘56) CL: Clustering Model (Davis. ‘67)
RC: Ranked Cluster (Davis & Leinhardt, ‘72) R2C: Ranked 2Clusters (Johnsen, ‘85)
TR: Transitivity (Davis and Leinhardt, ‘71) HC: Hierarchical Cliques (Johnsen, ‘85)
39+: Model that fits D&L’s 742 mats N :3972 p1p4: Johnsen, 1986. Process Agreement
Models.
Structural Indices based on the distribution of triads
The observed distribution of triads can be fit to the hypothesized structures using weighting vectors for each type of triad.
Where:
l = 16 element weighting vector for the triad types
T = the observed triad census
mT= the expected value of T
ST = the variancecovariance matrix for T
102
111D
201
210
300
021D
111U
120D
021U
030T
120U
021C
030C
120C
Triad Census Distributions
Standardized Difference from Expected
Data from Add Health
400
300
200
tvalue
100
0
100
For the Add Health data, the observed distribution of the tau statistic for various models was:
Indicating that a rankedcluster model fits the best.
So far, we’ve focused on the graph ‘at equilibrium.’ That is, we have hypothesized structures once people have made all the choices they are going to make. What we have not done, is really look closely at the implication of changing relations.
That is, we might say that triad 030C should not occur, but what would a change in this triad imply from the standpoint of the actor making a relational change?
Transition to a Vacuous Triple
030C
120C
102
Transition to a Transitive Triple
Transition to an Intransitive Triple
111U
021C
201
012
300
111D
003
210
021D
120U
030T
021U
120D
030C
120C
102
111U
201
021C
003
111D
012
210
300
021D
120U
030T
021U
120D
Observed triad transition patterns, from Hallinan’s data.