Dynamic Social Balance
James Moody
Ohio State University
Freeman Award Presentation
Sunbelt Social Network Conference, February 2005
My formal collaboration network
Dynamic Social Balance
Outline
Dynamic Social Balance
1. Introduction
2. Adolescent Friendship Structure
a. Hierarchy
b. Network Change
3. Theory
a. Traditional Social Balance Models
b. A Local-change Balance Model
4. Observed Results
5. Simulation Results
6. An Addendum on Asymmetry…
Dynamic Social Balance
Introduction
A Gallery of Friendship Networks
588 adolescents from a
poor, urban, southern
school.
(Source: Add Health)
Dynamic Social Balance
Introduction
A Gallery of Friendship Networks
1744 adolescents from a
lower-middle class, urban,
school in the West.
(Source: Add Health)
Dynamic Social Balance
Introduction
A Gallery of Friendship Networks
413 adolescents from a
Upper-class, urban, school
in the Midwest.
(Source: Add Health)
Dynamic Social Balance
Introduction
A Gallery of Friendship Networks
776 adolescents from a
working-class, all-white,
suburban, school in the
Midwest.
(Source: Add Health)
Dynamic Social Balance
Introduction
A Gallery of Friendship Networks
678 adolescents from a
working-class, all-white,
rural, school in the
Midwest.
Across all of these settings
(and many more) we can
literally see the differences
imposed by classic ‘Blau
space’ features of youth
communities. Race,
grades, SES etc. often
shape the gross topography
of school friendship
networks.
(Source: Add Health)
Dynamic Social Balance
Introduction
Distribution of Popularity
Community type
Size
By size and city type
Dynamic Social Balance
Introduction
While we revel in the diversity of social settings, a primary motivation
for social theory is to explain common features across settings and
account for social differentiation endogenously.
•Cartwright, Harary, Davis, Leinhardt, Johnsen: Clustering &
hierarchy in social networks
•Chase: The development of dominance
•Gould: Peer influence embellishments on quality stratification
•Mark: Social Differentiation from first principles
•McFarland: Development of ritualized structure in dynamic
networks
Adolescent friendship networks vary on myriad surface features, but do
these networks have a common structural form and if so how can we
explain it?
Dynamic Social Balance
Adolescent Friendship Networks: Data
Data
•I use the National Longitudinal Survey of Adolescent Health (Add Health). This
is a nationally representative survey of adolescents in school (7th through 12
grade), with (approximately) complete network data in 129 schools, including
data over time for a smaller subset of schools.
•Each students named up to 5 best male and 5 best female friends.
•Nominations outside of the school were allowed, but not matched.
•These data are available through the Carolina Population Center:
Methods
•Features of the global network structure are identified through triad distribution
methods and block models
•Specific hypotheses about social balance are tested with exponential random
graph models*
•Dynamic implications for these models are derived from simulation studies
grounded in the observed data.
*pseudo-likelihood logit approximations for ERGM
Dynamic Social Balance
Adolescent Friendship Networks: Triad distributions
Endogenous Building Blocks: A periodic table of social elements:
(0)
(1)
(2)
(3)
(4)
(5)
(6)
003
012
102
111D
201
210
300
021D
111U
120D
021U
030T
120U
021C
030C
120C
Dynamic Social Balance
Adolescent Friendship Networks: Triad distributions
The distribution of triads found in any network determines its final
structure. For example, if all triads are 030T, then the network
must be a perfect linear hierarchy.
Triads Observed:
030T
Dynamic Social Balance
Adolescent Friendship Networks: Triad distributions
Triads Observed:
102
300
N*
M
M
Dynamic Social Balance
Adolescent Friendship Networks: Triad distributions
Triads Observed:
102
300
003
N*
M
M
N*
N*
N*
N*
M
M
“Cluster”
Dynamic Social Balance
Adolescent Friendship Networks: Triad distributions
Triads Observed:
003
012
021D
021U
030T
120D
120U
300
M
A*
A*
A*
A*
N*
Eugene Johnsen (1985,
1986) specifies a number of
structures that result from
various triad
configurations:
M
M
A*
A*
A*
A*
N*
M
M
“Ranked Cluster”
Dynamic Social Balance
Adolescent Friendship Networks: Triad distributions
The observed distribution of triads can be fit to the hypothesized structures
using weighting vectors for each type of triad, and formulas for the
conditional expectation of the triad counts.
 (l ) 
(lT  lμ T )
l T l
Where:
l = 16 element weighting vector for the triad types
T = the observed triad census
mT= the expected value of T
T = the variance-covariance matrix for T
Dynamic Social Balance
Adolescent Friendship Networks: Triad distributions
For the 129 Add Health school networks, the observed distribution of
the tau statistic for various models is:
Suggesting that the “ranked-cluster” models beat random chance in all schools.
Dynamic Social Balance
Adolescent Friendship
Networks: Block Models
Regular equivalence can be
identified by disaggregating
triad distributions into the
positions that nodes occupy
within triads (Hummell and
Sodeur 1990; Burt 1990).
This creates a set of triad
position profiles that you can
then cluster over to identify
equivalent classes
003
021C_S
030T_S
120U_E
012_S
021C_B
030T_B
120U_S
012_E
021C_E
030T_E
120C_S
012_I
111D_S
030C
120C_B
102_D
111D_B
201_S
102_I
111D_E
201_B
210_S
021D_S
111U_S
120D_S
210_B
021D_E
111U_B
120D_E
021U_S
111U_E
021U_E
120C_E
210_E
300
Dynamic Social Balance
Adolescent Friendship Networks: Block Models
If we use block models instead, over all 129 networks, we find a similar clear
hierarchy in each school, differing only in the number of levels that might
form a ‘semi-periphery’ position in the network.
Over half of the networks had one of these 6 image networks
Dynamic Social Balance
Adolescent Friendship Networks: Relational stability
While the structure appears constant, relations are fluid:
Percent of T2 relations that were also T1 relations
Time 2
T1
T2
Time 3
Dynamic Social Balance
Adolescent Friendship Networks: Position stability
An individual’s position in the status hierarchy is also not stable:
Jefferson
Sunshine
Dynamic Social Balance
Adolescent Friendship Networks
These results suggest that:
• All of the school networks have a rank-strata structure
• The structure remains constant even though nearly half of all
relationships are new
•People’s position in the popularity distribution is fluid
What social process will explain a stable macro-structure in the face of
dynamic relations?
Dynamic Social Balance
Traditional models for directed graphs
Classic balance theory offers a set of simple local rules for relational change:
•A friend of a friend is a friend
•My enemy’s enemy is my friend.
Extended to directed relations, balance is typically operationalized as
transitivity:
If:
i
j
&
j
k
then:
i
k
Actors seek out transitive relations and avoid intransitive relations
Dynamic Social Balance
Traditional models for directed graphs
Classic balance theory offers a set of simple local rules for relational change:
•A friend of a friend is a friend
•My enemy’s enemy is my friend.
(0)
(1)
(2)
(3)
(4)
(5)
(6)
003
012
102
111D
201
210
300
021D
111U
120D
Intransitive
Transitive
021U
030T
120U
021C
030C
120C
Mixed
Dynamic Social Balance
Traditional models for directed graphs
Support for the classic balance model is strong, based on an over-representation of
transitive triads in observed networks:
•Davis (1970) finds support in 742 different networks, which was further
specified by Johnsen (1985)
•Hallinan’s work on schoolchildren (1974)
•Numerous studies of the Newcomb Data (Dorien et al 1996, for example)
•…and the extent of order holds even net of clustering imposed through focal
activity (Feld, 1980).
But two troubling points remain:
•Equilibrium models suggest networks should crystallize into stable structures.
•Observed networks always contain intransitive patterns (i.e. t210) much more
frequently than expected by chance.
•My goal is to specify a systematic balance model that can account for both of these
points.
Dynamic Social Balance
New models for directed graphs
Two crucial insights help inform a modified approach to social balance:
•Triples instead of triads. Operationalizing balance theory as
transitivity allows us to simplify the behavioral assumptions (cf.
Hummel and Soduer (1987, 1990.
•Structural implications differ depending on your position in the
network.
•Carley and Krackhardt (1996) show this clearly at the dyad
level, and we would expect similar effects at the triple level.
•Examine relational change directly. Instead of assuming that
intransitive relations resolve in equilibrium, we need to ask the microimplications of moving from one structural state to another.
•This allows us distinguish transitivity seeking from intransitivity
avoidance.
Dynamic Social Balance
New models for directed graphs
102
030C
120C
111U
021C
201
003
012
111D
021D
300
210
120U
vacuous transition
Increases # transitive
Decreases # intransitive
030T
Decreases # transitive
Increases # intransitive
021U
120D
Vacuous triad
Intransitive triad
Transitive triad
(some transitions will both increase transitivity & decrease intransitivity – the effects are independent – they are colored here for net balance)
Dynamic Social Balance
New models for directed graphs: Triad Transitions
Observed triad transition patterns, from Sorensen and Hallinan (1976)
030C
120C
102
111U
201
021C
111D
003
012
210
021D
021U
120U
030T
120D
300
Dynamic Social Balance
Triad-Transition models on observed data
The triad transition model can be tested on observed graphs within
the ERGM (p*) framework by specifying the triad-transition counts
weighted by the number of transitive and intransitive triples that
would be created in each transition.
Here I use the pseudo-likelihood approximations based on a dyadic
logit model (Wasserman and Pattison, et al).
The model includes additional parameters for dyadic properties,
individual expansiveness and attractiveness, out-of-school ties, and
reciprocity.
I estimate this model on the Add Health networks, creating a
distribution of parameter scores across all networks.
Dynamic Social Balance
Triad-Transition models on observed data
ERGM Coefficient Distributions*
0.8
Endogenous
Focal Orgs.
Dyadic Similarity/Distance.
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
*Coefficients based on pseudo-likelihood approximations, here standardized so they fit well on the page…
Dynamic Social Balance
Triad-Transition Simulations
Based on these results, I simulate networks dynamically:
•Tie probabilities are based on separate parameters for seeking
transitivity and avoiding intransitivity, using the triad change
counts.
•Adds a parameter to limit the marginal returns to forming new
relations, effectively dampening (but not hard-coding) outdegree.
•Reciprocity & dyad similarity parameters are held constant
across all simulations.
•As iterations pass, actors adjust their ties based on the resulting
model probabilities, allowing the graph to evolve in response to
others’ changes.
Dynamic Social Balance
Triad-Transition Simulations
Final Graph Transitivity
R2 = 0.82
Dynamic Social Balance
Triad-Transition Simulations
Structural Stability
Correlation of network structure at tfinal with t-5%
R2 = 0.54
Dynamic Social Balance
Triad-Transition Simulations
Total Graph Transitivity
At moderate transitivity/intransitivity
A single simulation run, showing the wide swings in graph transitivity. Similar trends evident in reciprocity,
though the number of arcs and general shape (variance/skew) of the popularity distribution does not
fluctuate much.
Dynamic Social Balance
A dyadic extension: Gould’s asymmetry avoidance rule
Under moderate parameter values, these simulations meet the empirical
requirements:
•
Systematic balance-based action can create a dynamic equilibrium.
•
Graphs evaluated at any of the later points in the simulation have
high rank-cluster tau values.
•
We observe more t210 triads than we would expect by chance.
The key features of this model are:
•
balance is treated as a parameter that scales from weak to strong.
•
the focus for actor behavior is the emotional return to relational
change, not the total elimination of particular triads.
•
transitivity seeking has different implications than intransitivity
avoidance.
•
But the simulations are somewhat sensitive to parameter changes.
•
Some runs suggest that once the network passes a particular
structural threshold, a ‘lock-in’ process takes hold and graphs do
not change much.
Dynamic Social Balance
A dyadic extension: Gould’s asymmetry avoidance rule
•What effect of asymmetry? Consider these triads:
030T
120D
120U
The current model rewards reciprocation, but does not penalize
asymmetry, so these triads are stable for 2 of the 3 actors.
Gould suggests that people will not maintain a relation if it is not
reciprocated, and that’s also exactly what we see in the Add Health data.
Adding a parameter that says actors avoid long-term asymmetry will make
these three triads temporarily attractive, but unstable in the long run.
(run network balance simulation now)
Dynamic Social Balance
Conclusion
•Specific:
•A social balance model that takes seriously the process of avoiding
intransitive settings or seeking transitive ones fits the patterns found
in Add Health
•Transitivity seeking creates more stability than intransitivity
avoidance
•Endogenous balance is only a part of the model:
•Dyadic attributes and focal organization set the constraints in
real graphs.
•The most robust dynamic models are those that include a dyad
level cost for repeated asymmetry.
Dynamic Social Balance
Conclusion
•General:
•There are many other local dynamic processes models to specify.
For example:
• Actors seeking to maximize structural holes are effectively
seeking the t201 triad, but how do they get there? Do different
routes to t201 imply different actor motivations?
•Actors move on multiple relations simultaneously, implying a
network with compound edges, and thus a more complicated
(but finite and specifiable) triad census.
•Micro structures of more than 3 nodes (4-cycles, etc).
•If we can specify the things that actors do with respect to their
relations, we can build these models in any context.