Triad Trajectories and Mental
Health Transitions in Adolescents
A Working Paper
G. Robin Gauthier and James Moody
Department of Sociology
Duke University
Durham, North Carolina
Thanks to Jeffery Smith, Duke Sociology
Outline
• Past Findings and Literatures
– Social Network and Mental Health literature
Triads
• Balance Theory (Heider)
• Davis and Leinhardt, 1972; Hallinan, 1976; Bearman
and Moody, 2004
Add Health data
Triad transition model
Social Networks and Mental Health
Network Characteristics
•Density
•Size
•Multiplexy
•Strength of tie
Social Support (Pearlin,1999)
•Chronic stressors
•Acute stressors
•Social Support; instrumental
and emotional
•Coping strategies
•Mastery
Triads
• Advantages of studying triads
– Characterize macro structure from the distribution
of triads (Tau Statistic, Holland and Leinhardt)
– smallest unit with a group structure independent
of the individuals within it (Simmel)
• competition, alliances, or mediation
• Hierarchy (Mayhew, Chase)
– The triad can be used to study both local and
global structure
• Granovetter, 1973
Balance theory:
A Friend of a friend...
Heider
Transitivity in Directed Graphs
• Holland and Leinhardt (1971)
– MAN labelling system
– Tau Statistic
Ranked Cluster Model
– Davis and Leinhardt (1972)
Triples within triads: Strength of
(In)transitivity
• Any triad can be decomposed into its 6 triples
• Constituent triples, like triads can be characterized as
vacuous, transitive or intransitive
• The order of these triples matter for the level
of in(transitivity) experienced by each node
Triads are characterized by more or
less intransitivity, rather than
intransitivity or transitivity
Triad type 120c and constituent triples
b
a-b-c; a-c
a-c-b; a-b
b-a-c; b-c
b-c-a; b-a
c-a-b; c-b
c-b-a; c-a
a
Transitive
Intransitive
c
Transitive
Vacuous
Vacuous
Intransitive
Intransitive
Vacuous
021C 111D 111U 030T 030C 201 120D 120U 120C 210 300
0
0
0
1
0
0
2
2
1
3 6
1
1
1
0
3
2
0
0
2
1 0
Reproduced from Wasserman and Faust (1994), pp. 572
Triad Transitivity Levels
003
030T
021C
012
120D
111D
102
120U
111U
021D
021U
300
030C
201
Legend
Vacuous
Transitive
120C
210
Intransitive
Mixed
Research: Triad trajectories,
intransitivity and mental health
Sorenson and Hallinan (1976)
Model of dynamic triadic change using a
Markov chain approach
Bearman and Moody (2004)
Adolescent girls in intransitive relationships
are more likely to experience suicidal thoughts
Transition Model
•
•
•
•
•
•
•
Triads can be characterized by trajectories
Depressive symptoms change over time
People are embedded in multiple triad trajectories
A given triad can have more or less transitivity
There is a change in intransitivity associated with each unique triad path
Individuals within a triad can experience different levels of transitivity
Individuals can experience different levels of intransitivity in different
triads
• Research Question: how do the paths by which triads become more or less
transitive matter (or not) in determining how depressive symptoms
change over time-given that individuals are embedded in multiple
complicated trajectories simultaneously.
Add Health data
• Nationally representative longitudinal study of adolescents in
grades 7-12 during the 1994-’95 school year
• 80 high schools and two of their feeders, yielding in total 132
schools
• One in-school survey, four detailed in-home follow-ups, the
most recent in 2008
• Longitudinal network data has been collected on 12 of the
schools, this paper uses 9 of these
– In two of these, nominations were restricted to one male
and one female friend per student
– Two schools are (provisionally) dropped from the analysis
due to their size
Working Measures
Depression transition measure
Wave I to II
4 question CES-D subset
Wave II to III
19 question CES-D subset
Explanatory Variable
Trajectory level summary score of intransitivity
Controls
Grade, gender, race, school, parental SES, gpa, in-degree,
out-degree, social isolation, attachment to school
Empirical Distribution of Triad Types
Wave 2
Wave 1
T. type
Count
Pr()
Pr Non-null
T. type
Count
Pr()
1857060
0.8965
Pr Non-null
003
1670440
0.8064
003
012
318138
0.1536
012
157338
0.0760
102
6246
0.0030
102
1907
0.0009
021d 8073
0.0039
021d
2116
0.0013
021u 7663
0.0037
021u
2806
0.00102
021c
0.0052
0.1780
021c
2806
0.0014
0.0557
111d 783
0.0004
0.0129
111d
115
0.00006
0.0023
111u 793
0.0004
0.0130
111u
170
0.00008
0.0033
030t
885
0.0004
0.0145
030t
163
0.00008
0.0004
030c
172
0.00008
0.0028
030c
24
0.00001
0.00008
201
60
0.00003
0.0009
201
4
0.000002
0.00008
120d 82
0.00004
0.0013
120d
4
0.000002
0.0002
120u 56
0.00002
0.0009
120u
8
0.000004
0.00004
120c
86
0.00004
0.0141
120c
6
0.000003
0.0001
210
51
0.0002
0.0008
210
2
0.0000001 0.00004
300
47092
0.0227
0.7731
300
47112
0.02274
10846
Legend
Null
Transitive
Intransitive
Mixed
0.9345
One-Step Trajectory Model
102
030C
120C
111U
021C
201
003
111
D
012
021D
210
300
120U
Vacuous triad
030T
021U
Transitive triad
120D
Intransitive triad
Mixed triad
Decreases # intransitive
Increases # intransitive
Choosing a Model
Absolute distance matrix
-the transitions are made up of the one-step
transitions which minimize the length of the
path
Observed probability matrix
-transitions are determined by weighting the
probability of transition by the observed
probabilities of the adjacent triad type
Assumptions
• Triads transition one edge at a time
• At each transition point, triads move into the
most empirically probable adjacent type
• An edge once cut or added is not re-cut or readded at the next transition point, but may be
at subsequent transition points
Divergent Paths: An Example
102
111D
201
012
300
210
021C
120C
030C
Vacuous triad
Transitive triad
Vacuous triad
Mixed triad
Distance weight
Observed
Probability Weight
Both models
Directions for Analysis
• Model transition dependence
– Moody and Smith, forthcoming
•
•
•
•
Block models
Ties within and across blocks-role structure
Clique analysis
Group affiliations
– How do the effects of in(transitivity) differ by context