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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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slide1

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
slide2

Social Balance & Transitivity

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

slide3

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

-

slide4

Social Balance & Transitivity

x

x

x

x

x

x

x

x

-

-

+

-

+

+

-

+

+

+

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-

+

+

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+

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-

+

-

+

+

-

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:

slide5

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

slide6

Social Balance & Transitivity

+

+

-

+

+

+

-

+

+

-

-

-

Heider argued that unbalanced triads would be unstable: They should transform toward balance

Become Friends

Become Enemies

Become Enemies

slide7

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

slide8

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.

slide9

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

slide10

Social Balance & Transitivity

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:

  • 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.
slide11

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

slide12

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

slide13

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

slide14

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.

slide15

Social Balance & Transitivity

Triads allowed: {300, 102}

N*

M

M

1

0

0

1

slide16

Social Balance & Transitivity

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

slide17

Social Balance & Transitivity

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...

slide18

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.

slide19

Social Balance & Transitivity

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 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.

slide20

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

slide21

012

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

t-value

100

0

-100

slide22

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.

slide23

Social Balance & Transitivity

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?

slide24

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

300

111D

003

210

021D

120U

030T

021U

120D

slide25

Social Balance & Transitivity

030C

120C

102

111U

201

021C

003

111D

012

210

300

021D

120U

030T

021U

120D

Observed triad transition patterns, from Hallinan’s data.

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