Unit 4: Dyads and Triads, Reciprocity and Transitivity

ICPSR University of Michigan, Ann Arbor Summer 2015 Instructor: Ann McCranie

•  There has been much theorizing over the years about structural properties (mutuality, reciprocity, balance theory, transitivity) which are manifested at the structural level of dyads and triads.

•  Simmel argued that the triad is the fundamental unit of social analysis.

•  Patterns of relations that appear in these small constituent parts of the network can result in larger structural patterns that influence the whole network.

Subgraphs, Dyads, and Triads Much of social network analysis involves the study of smaller pieces of a network, particularly those that arise from using graph theoretic ideas to split up a graph.

•  Subgraphs •  A graph Gs is a subgraph of G if the set of nodes Gs, and the set of

lines in Gs is a subset of the lines in the graph G. •  There are a variety of kinds of subgraphs:

•  subgraph •  node-generated subgraph •  line-generated subgraph

•  Dyads •  A dyad, representing a pair of actors and the possible relational ties

between them, is a (node-generated) subgraph consisting of a pair of nodes and the possible line between the nodes.

•  Triads •  A triad is a subgraph consisting of three nodes and the possible lines

among them.

Subgraphs-Dyads

Three dyadic isomorphism classes for directed graphs:

– null dyads have no arcs – asymmetric dyad has an arc between the

two nodes going in one direction or the other, but not both.

– mutual dyads have two arcs between the nodes, one going in one direction, and the other going in the opposite direction.

Reciprocity

•  At a network level, one way of thinking about the cohesion of a network. •  How strong is the tendency to return a tie

in this network? As with other measures, there are different ways to consider this concept: at the level of the dyad or at the level of the arc.

Reciprocity – dyad based •  Dyad based reciprocity (most commonly

reported) is the number of reciprocated dyads divided by the number of adjacent dyads.

Reciprocity – Arc-based •  Less commonly reported, this is the

number of reciprocated arcs divided by the total number of arcs.

You can also consider the reciprocity of each actor. •  For instance, the dyad-based reciprocity of

each actor is the number of mutual dyads they are in divided by the number of other nodes to which they are adjacent.

Triadic Analysis One level of analysis (other types: dyadic, individual, group, subgroup) Assuming data are directed, directional, and dichotomous (with one relation) Historical reasons, mathematically completed

Triads: Historical Perspective •  Heider (1958) Theory of balance

•  Focused on individual’s perception of social cognitive processes which gave rise to a triad P-O-X Person-Other Individual-Object

•  Festinger (1954) and “Cognitive Dissonance”

•  Structural balance extended by Cartwright & Harary focus on a set of individuals instead of just one individual

Triadic Analysis •  Takes into account all the different

combinations of three individuals and examines the interactions between the three individuals

A D

C B

A-B-C A-B-D B-D-C A-D-C

Triadic Analysis

Describes directed interactions between three individuals

Total of 16 different triads (Wasserman & Faust, 1994)

Each triad is represented by 3 numbers and a letter (if present)

Triadic Analysis

Describes directed interactions between three individuals

Total of 16 different triads (Wasserman & Faust, 1994)

Each triad is represented by 3 numbers and a letter (if present)

•  1st=Number of mutual dyads

Triadic Analysis

Describes directed interactions between three individuals

Total of 16 different triads (Wasserman & Faust, 1994)

Each triad is represented by 3 numbers and a letter (if present)

•  1st=Number of mutual dyads •  2nd=Number of asymmetric dyads

Triadic Analysis

Holland & Leinhardt (1970) Davis & Leinhardt (1972)

Describes directed interactions between three individuals

Total of 16 different triads (Wasserman & Faust, 1994)

Each triad is represented by 3 numbers and a letter (if present)

•  1st=Number of mutual dyads •  2nd=Number of asymmetric dyads •  3rd=Number of null dyads

Triadic Analysis

•  Letter (if present after the triad represents a state) •  “D” Down •  “U” Up •  “T” Transitive •  “C” Cyclic

•  Number of triads that are present (g choose 3) where g=number of nodes

16 isomorphism classes for triads

Example

A-B-C=030T

A-B-D=111U

B-D-C=111D

A-D-C=012

A D

C B

030T

111D

012

111U

Example

A D

C B

A-B-C=030T

A-B-D=111U

B-D-C=111D

A-D-C=012

030T

111U

111D

012

Example

A D

C B

A-B-C=030T

A-B-D=111U

B-D-C=111D

A-D-C=012

030T

111D

012

111U

Example

A D

C B

A-B-C=030T

A-B-D=111U

B-D-C=111D

A-D-C=012

030T

111D

012

111U

Example

A D

C B

A-B-C=030T

A-B-D=111U

B-D-C=111D

A-D-C=012

030T

111D

012

111U

•  In order to consider transitivity, we have to consider the order of choices.

•  If x chooses y and y chooses z, does x choose z?

•  If x->y and y->z, we have a non-vacuous triad. •  If x -> z, then we have a transitive triplet. •  If x does not choose z, then it is

intransitive.

Each one of these triads can be broken down into 6 ordered triples – order matters

x

y z

See Wasserman and Faust, pg. 572 for a full list of the transitive and intransitive triples included in each of the 16 isomorphism classes.

Transitivity: Why does it matter? •  At a network level, it tells you something

about the “clustering” of the network. •  At an individual level, it tells you about

the degree to which an actor exists in a tightly bound group, or if they have connections outside their own group.

On voting behavior, from Connected (2010) by Christakis and Fowler, p 184

Number of non-vacuous transitive ordered triples: 3 (1 from a 120C, 2 from a 120U)

Number of triples of all kinds: 60 (10 triads x 6 ordered triplets in each) Number of triples in which i-->j and j-->k: 8 Number of triangles with at least 2 legs: 18 Number of triangles with 3 legs: 3 Percentage of all ordered triples: 5.00% (3/60) Transitivity: % of ordered triples in which i-->j and j-->k that are

transitive: 37.50% (3/8) Transitivity: % of triangles with at least 2 legs that have

3 legs: 16.67% (3/18)

Transitivity Triad Census for dataset ErdosRenyi10 ErdosReny --------- 1 003 0.000 2 012 2.000 3 102 2.000 4 021D 1.000 5 021U 0.000 6 021C 1.000 7 111D 1.000 8 111U 1.000 9 030T 0.000 10 030C 0.000 11 201 0.000 12 120D 0.000 13 120U 1.000 14 120C 1.000 15 210 0.000 16 300 0.000

“Legs” in a ordered triple •  Number of triangles with at least 2 legs:

•  This UCINET output refers to the number of ordered triples in which this happens at least twice: A chooses B, B chooses C, A chooses C.

•  This is different than the number of ordered triples in which A->B AND B->C (a non-vacuous triple that can either be transitive or intransitive.)

•  Think of the “2 leg” as a loosening of the definition of transitivity – you will find more potentially transitive triples this way. •  However, for the triple to actually BE transitive, then the

following must still hold: A->B, B->C, AND A->C.

Want to prove it to yourself?

There is a worksheet on the website under the slides for Day 4 that lists all 60 ordered triples in this example network, and how many “legs” they have. The above is an excerpt of all triples that have at least two legs (18), but you can also see the 8 that are non-vacuous (meaning they could be transitive or intransitive, because A->B and B->C). Also, you can see the 3 transitive ordered triplets.

More on triads and intransitivity

•  Granovetter (1973, “Strength of Weak Ties”) argues that some triadic arrangements are uncommon in social relations (like the “forbidden triad” of 201), but that means that the ties that CAN be activated here, “weak ties” are rare, important and powerful – for instance, for getting a job.

If you’re really into it…

•  A puzzle concerning triads in social networks: Graph constraints and the triad census, K. Faust Social Networks 32 221--233 (2010)