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. Introduction Adolescent Friendship Structure Hierarchy
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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
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
Size
Community type
By size and city type
Dynamic Social Balance
Introduction
Dynamic Social Balance
Adolescent Friendship Networks: 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
030T
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:
102
Dynamic Social Balance
Adolescent Friendship Networks: Triad distributions
Triads Observed:
300
N*
M
M
003
N*
300
M
M
102
N*
N*
N*
N*
M
M
Dynamic Social Balance
Adolescent Friendship Networks: Triad distributions
Triads Observed:
“Cluster”
003
012
300
030T
021D
A*
A*
A*
A*
120D
A*
A*
A*
021U
A*
120U
M
N*
M
M
N*
M
M
Dynamic Social Balance
Adolescent Friendship Networks: Triad distributions
Triads Observed:
Eugene Johnsen (1985, 1986) specifies a number of structures that result from various triad configurations:
“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.
Where:
l = 16 element weighting vector for the triad types
T = the observed triad census
mT= the expected value of T
ST = 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.
120D_S
003
021C_S
120D_E
021C_B
012_S
021C_E
120U_S
012_E
120U_E
111D_S
012_I
120C_S
111D_B
102_D
120C_B
111D_E
102_I
030T_S
120C_E
111U_S
030T_B
021D_S
210_S
111U_B
030T_E
021D_E
210_B
111U_E
021U_S
210_E
030C
021U_E
201_S
300
201_B
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
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 3
Time 2
T1
T2
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
i
j
j
i
k
k
Dynamic Social Balance
Traditional models for directed graphs
If:
&
then:
Actors seek out transitive relations and avoid intransitive relations
Intransitive
Transitive
Mixed
Dynamic Social Balance
Traditional models for directed graphs
(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
Traditional models for directed graphs
Dynamic Social Balance
New models for directed graphs
vacuous transition
Increases # transitive
Decreases # intransitive
Decreases # transitive
Increases # intransitive
Vacuous triad
Intransitive triad
Transitive triad
Dynamic Social Balance
New models for directed graphs
030C
120C
102
111U
021C
201
012
300
111D
003
210
021D
120U
030T
021U
120D
(some transitions will both increase transitivity & decrease intransitivity – the effects are independent – they are colored here for net balance)
030C
111U
111D
021U
120D
Dynamic Social Balance
New models for directed graphs: Triad Transitions
Observed triad transition patterns, from Sorensen and Hallinan (1976)
120C
102
201
021C
003
012
210
300
021D
120U
030T
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.
Transitivity
Intransitivity
Reciprocity
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
GPA
SES
Fight
College
Drinking
Same Sex
Same Race
Both Smoke
Same Clubs
Same Grade
*Coefficients based on pseudo-likelihood approximations, here standardized so they fit well on the page…
Dynamic Social Balance
Triad-Transition Simulations
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
030T
120D
120U
Dynamic Social Balance
A dyadic extension: Gould’s asymmetry avoidance rule
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
Dynamic Social Balance
Conclusion