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 IndividualObject
•  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
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
Holland & Leinhardt (1970)
Davis & Leinhardt (1972)
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
030T
A-B-D=111U
111U
B-D-C=111D
A
D
C
B
111D
A-D-C=012
012
Example
A-B-C=030T
030T
A-B-D=111U
111U
B-D-C=111D
A
D
C
B
111D
A-D-C=012
012
Example
A-B-C=030T
030T
A-B-D=111U
111U
B-D-C=111D
A
D
C
B
111D
A-D-C=012
012
Example
A-B-C=030T
030T
A-B-D=111U
111U
B-D-C=111D
A
D
C
B
111D
A-D-C=012
012
Example
A-B-C=030T
030T
A-B-D=111U
111U
B-D-C=111D
A
D
C
B
111D
A-D-C=012
012
•  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 nonvacuous 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
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
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)
“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)