Social Balance Transitivity Overview Background Basic Balance Theory

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

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

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

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

Social Balance & Transitivity We determine balance based on the product of the edges: + + (+)(+)(+) = (+) Balanced “A friend of a friend is a friend” - (-)(+)(-) = (-) Balanced “An enemy of my enemy is a friend” - (-)(-)(-) = (-) Unbalanced “An enemy of my enemy is an enemy” + (+)(-)(+) = (-) + + + - Unbalanced “A Friend of a Friend is an enemy”

Social Balance & Transitivity Heider argued that unbalanced triads would be unstable: They should transform toward balance + + Become Friends - Become Enemies + + + + - - -

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 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 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

Social Balance & Transitivity 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.

Social Balance & Transitivity Once we admit directed relations, we need to decompose triads into their constituent triples. Ordered Triples: b a c 120 C a a b b c c b c a b c; b; c; a; b; a; a a b b c c c b c a b a Transitive Vacuous Intransitive Vacuous

Network Sub-Structure: Triads (0) (1) (2) (3) (4) (5) 003 012 102 111 D 201 210 021 D 111 U 120 D (6) 300 Intransitive Transitive 021 U 030 T 120 U 021 C 030 C 120 C Mixed

An Example of the triad census Type Number of triads -------------------1 - 003 21 -------------------2 - 012 26 3 - 102 11 4 - 021 D 1 5 - 021 U 5 6 - 021 C 3 7 - 111 D 2 8 - 111 U 5 9 - 030 T 3 10 - 030 C 1 11 - 201 1 12 - 120 D 1 13 - 120 U 1 14 - 120 C 1 15 - 210 1 16 - 300 1 -------------------Sum (2 - 16): 63

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 030 T: A perfect linear hierarchy.

Social Balance & Transitivity Triads allowed: {300, 102} N* M M 1 0 0 1

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* 1 M 1 1 M

Social Balance & Transitivity PRC{300, 102, 003, 120 D, 120 U, 030 T, 021 D, 021 U} Ranked Cluster: M A* A* N* M M A* A* 1 1 1 0 0 1 1 1 0 0 0 0 0 1 A* A* N* M M And many more. . .

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 Triad: BA CL RC R 2 C TR HC 39+ p 1 p 2 p 3 003 012 102 021 D 021 U 021 C 111 D 111 U 030 T 030 C 201 120 D 120 U 120 C 210 300 Triad Micro-Models: BA: Ballance (Cartwright and Harary, ‘ 56) CL: Clustering Model (Davis. ‘ 67) RC: Ranked Cluster (Davis & Leinhardt, ‘ 72) R 2 C: 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 p 1 -p 4: Johnsen, 1986. Process Agreement Models. p 4

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 m. T= the expected value of T ST = the variance-covariance matrix for T

Triad Census Distributions Standardized Difference from Expected Data from Add Health 400 t-value 300 200 100 0 -100 012 102 021 D 021 U 021 C 111 D 111 U 030 T 030 C 201 120 D 120 U 120 C 210 300

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.

Testing Theories of Friendship Standardized Coefficients from an Exponential Random Graph Model 0. 8 0. 6 0. 4 0. 2 0 -0. 2 -0. 4 -0. 6 t gh Fi ng ki rin e ok e ac R Sm e x bs lu Se C de ra G ge le ol D C th Bo S PA G SE m e e e m m m Sa Sa ty vi iti ns ity v iti s an ity oc r ip ec tra In Tr R

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 030 C 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 102 030 C 120 C 111 U 021 C 201 003 012 111 D 021 D 210 300 120 U vacuous transition Increases # transitive 030 T 021 U Decreases # intransitive 120 D Decreases # transitive Increases # intransitive Vacuous triad Intransitive triad Transitive triad

Social Balance & Transitivity

Social Balance & Transitivity Random Walk TRIAD 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 100 200 300

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

Social Balance & Transitivity Trans=1. 0 TRIAD 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 100 200 ITTER 300

Social Balance & Transitivity Strong Negative (INT=-2) TRIAD 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 100 200 ITTER 300

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