social balance & transitivity

Social Balance & TransitivityOverview

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

Social Balance & Transitivity

Heider’s work on cognition of social situations, which can be boiled down to the relations among three ‘actors’:

P O

X

Person Other

Object

Heider was interested in the correspondence of P and O, given their beliefs about X

+

-

Like:

Dislike

Two Relations:


Each dyad (PO, PX, OX) can take on one of two values: + or -

8 POX triples:

++

+p o

x--

+p o

x

-+

-p o

x

+-

-p o

x

++

-p o

x

-+

+p o

x

+-

+p o

x

--

-p o

x

Social Balance & TransitivityThe 8 triples can be reduced if we ignore the distinction between POX:

++

+p o

x--

+p o

x

-+

-p o

x

+-

-p o

x

++

-p o

x

-+

+p o

x

+-

+p o

x

--

-p o

x

++

+

--

+

++

-

--

-

Social Balance & TransitivityWe determine balance based on the product of the edges:

++

+

--

+

++

-

--

-

(+)(+)(+) = (+)

(-)(+)(-) = (-)

(-)(-)(-) = (-)

(+)(-)(+) = (-)

Balanced

Balanced

Unbalanced

Unbalanced

“A friend of a friend is a friend”

“An enemy of my enemy is a friend”

“An enemy of my enemy is an enemy”

“A Friend of a Friend is an enemy”


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):

Friends withEnemies with

Balanced Opposition Complete Clique


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) Mutual2) Asymmetric3) 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


Balance in directed relations

Actors seek out transitive relations, and avoid intransitive relations. A triple is transitive

• 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.

i j & j k

i k

If:

then:

120Ca

b

c

Ordered Triples:

a b c; Transitivea ca c b; Vacuousa bb a Vacuousc; b cb c a; Intransitiveb ac a b; Intransitivec bc b a; Vacuousc a


Once we admit directed relations, we need to decompose triads into their constituent triples.

Network Sub-Structure: Triads

003

(0)

012

(1)

102

021D

021U

021C

(2)

111D

111U

030T

030C

(3)

201

120D

120U

120C

(4)

210

(5)

300

(6)

Intransitive

Transitive

Mixed

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.


Triads allowed: {300, 102}

M M

N*

110

0


Cluster Structure, allows triads: {003, 300, 102}

M MN*

M MN*

N* N*N*

Eugene Johnsen (1985, 1986) specifies a number of structures that result from various triad configurations

11

11

PRC{300,102, 003, 120D, 120U, 030T, 021D, 021U} Ranked Cluster:

M MN*

M MN*

M

A*A*

A*A*

A*A*

A*A*


11

11

1

1111

011

11 0

0

00 0 0 0

0 00 0

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.

Social Balance & TransitivityTriad:

003

012

102

021D

021U

021C

111D

111U

030T

030C

201

120D

120U

120C

210

300

BA

Triad Micro-Models:BA: Ballance (Cartwright and Harary, ‘56) CL: Clustering Model (Davis. ‘67)RC: Ranked Cluster (Davis & Leinhardt, ‘72) R2C: Ranked 2-Clusters (Johnsen, ‘85)TR: Transitivity (Davis and Leinhardt, ‘71) HC: Hierarchical Cliques (Johnsen, ‘85)39+: Model that fits D&L’s 742 mats N :39-72 p1-p4: Johnsen, 1986. Process Agreement Models.

CL RC R2C TR HC 39+ p1 p2 p3 p4

Social Balance & TransitivityStructural 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.

llμlTl

T

T

)()(l

Where:l = 16 element weighting vector for the triad typesT = the observed triad census T= the expected value of TT = the variance-covariance matrix for T

-100

0

100

200

300

400

t-val

ue

Triad Census DistributionsStandardized Difference from Expected

Data from Add Health

012 102 021D 021U 021C 111D 111U 030T 030C 201 120D 120U 120C 210 300


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.


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?


003

102

021D

021U

030C

111D

111U

030T

201

120D

120U

120C

210 300012

021C

Transition to a Vacuous TripleTransition to a Transitive TripleTransition to an Intransitive Triple

003

102

021D

030T

201

120U

120C

210 300012

021C

021U

111D

111U

030C

120D


Observed triad transition patterns, from Hallinan’s data.

social balance & transitivity

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