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

Geographic history of an email petition.
Petition to save NPR, Jan ‘01

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

Each dyad (PO, PX, OX) can take on one of two values: + or - Two Relations: 8 POX triples: + p o x x x x Like: - - - - + + + o p o p o p + - - x x Dislike + - + - - p o p o - + x x - + - + p o p o - +

The 8 triples can be reduced if we ignore the distinction between POX: x x x x - + + - - + + - p o p o p o p o + + - - x x + - + - p o p o - + x x - + - + p o p o - + - - + + - - + + + + - -

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 + - - (-)(-)(-) = (-) Unbalanced “An enemy of my enemy is an enemy” - “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 symmetric relations (at least on affect) thus we need to identify balance in undirected 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

Balance in directed relations Actors seek out transitive relations, and avoid intransitive relations. A triple is transitive If: i j & j k then: i k 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.

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 b a c; b c a c Vacuous b c a; b a 120C Intransitive c a b; c b Intransitive c b a; c a Vacuous

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

An Example of the triad census
Type Number of triads D U C D U T C D U C Sum (2 - 16):

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} N* M M 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* 1 N* M M

PRC{300,102, 003, 120D, 120U, 030T, 021D, 021U} Ranked Cluster: M N* A* 1 1 1 1 1 1 1 1 1 1 1 1 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.

Triad: 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 : p1-p4: Johnsen, Process Agreement Models. CL RC R2C TR HC 39+ p1 p2 p3 p4

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

Standardized Difference from Expected
Triad Census Distributions Standardized Difference from Expected Data from Add Health 400 300 200 t-value 100 -100 012 102 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.

Testing Theories of Friendship
Standardized Coefficients from an Exponential Random Graph Model 0.8 0.6 0.4 0.2 -0.2 -0.4 -0.6 SES GPA Fight College Transitivity Same Sex Drinking Intransitivity Same Grade Same Clubs Same Race Both Smoke Reciprocity

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 210 300 012 021C vacuous transition Increases # transitive Vacuous triad Intransitive triad Transitive triad Decreases # intransitive Decreases # transitive Increases # intransitive

Random Walk TRIAD 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 100 200 300

Transitive = .5, Intran & Pos = 0. TRIAD 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ITTER 100 200 300

TRIAD 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ITTER 100 200 300

Strong Negative (INT=-2) TRIAD 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ITTER 100 200 300

Observed triad transition patterns, from Hallinan’s data. 003 102 021D 030T 201 120U 120C 210 300 012 021C 021U 111D 111U 030C 120D

Strong Transitivity Simulation Density Transitivity

Strong Intransitivity Simulation

Strong Intransitivity Simulation Ideal-typical results

Moderate values on both Simulation

A Brief History of Balance Through Time Newcomb, PI layout

A Brief History of Balance Through Time

Social Balance & Transitivity

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