Social Balance & Transitivity
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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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Social Balance & Transitivity

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Social balance transitivity

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


  • Social balance transitivity

    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


    Social balance transitivity

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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

    -


    Social balance transitivity

    Social Balance & Transitivity

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


    Social balance transitivity

    Social Balance & Transitivity

    -

    +

    +

    -

    +

    -

    +

    -

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    +

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    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


    Social balance transitivity

    Social Balance & Transitivity

    +

    +

    -

    +

    +

    +

    -

    +

    +

    -

    -

    -

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

    Become Friends

    Become Enemies

    Become Enemies


    Social balance transitivity

    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


    Social balance transitivity

    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.


    Social balance transitivity

    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


    Social balance transitivity

    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.


    Social balance transitivity

    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


    Social balance transitivity

    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


    Social balance transitivity

    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


    Social balance transitivity

    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.


    Social balance transitivity

    Social Balance & Transitivity

    Triads allowed: {300, 102}

    N*

    M

    M

    1

    0

    0

    1


    Social balance transitivity

    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


    Social balance transitivity

    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...


    Social balance transitivity

    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.


    Social balance transitivity

    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.


    Social balance transitivity

    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


    Social balance transitivity

    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


    Social balance transitivity

    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.


    Social balance transitivity

    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?


    Social balance transitivity

    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


    Social balance transitivity

    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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