Social%20Balance%20

1
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
• Implications for networks dynamics

2
Social Balance Transitivity
Heiders 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
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Social Balance Transitivity
Each dyad (PO, PX, OX) can take on one of two
values or -
8 POX triples
Two Relations
Like

Dislike
-
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Social Balance Transitivity
The 8 triples can be reduced if we ignore the
distinction between POX
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Social Balance Transitivity
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
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Social Balance Transitivity
Heider argued that unbalanced triads would be
unstable They should transform toward balance
Become Friends
Become Enemies
Become Enemies
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Social Balance Transitivity
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
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Social Balance Transitivity
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.
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Social Balance Transitivity
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
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Social Balance Transitivity
Balance in directed relations Actors seek
out transitive relations, and avoid intransitive
relations. A triple is transitive

If
then

• A property of triples within triads
• Assumes directed relations
• The saliency of a triad may differ for each
  actor, depending on their position within the
  triad.

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Social Balance Transitivity
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
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Network Sub-Structure Triads
(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
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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
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Social Balance Transitivity
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.
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Social Balance Transitivity
Triads allowed 300, 102
N
M
M
1
0
1
0
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Social Balance Transitivity
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
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Social Balance Transitivity
PRC300,102, 003, 120D, 120U, 030T, 021D, 021U
Ranked Cluster
1
0
0
0
0
1
1
0
0
0
1
1
0
0
0
1
1
1
1
0
1
1
1
1
0
And many more...
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Social Balance Transitivity
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.
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Social Balance Transitivity
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Social Balance Transitivity
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
variance-covariance matrix for T
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Triad Census Distributions
Standardized Difference from Expected Data from
Add Health
400
300
200
t-value
100
0
-100
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Social Balance Transitivity
For the Add Health data, the observed
distribution of the tau statistic for various
models was
Indicating that a ranked-cluster model fits the
best.
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Social Balance Transitivity
So far, weve 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?
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Social Balance Transitivity
Transition to a Vacuous Triple
030C
120C
102
Transition to a Transitive Triple
Transition to an Intransitive Triple
111U
021C
201
012
111D
300
003
210
021D
120U
030T
021U
120D
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Social Balance Transitivity
Observed triad transition patterns, from
Hallinans data.