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Dynamic Social Balance

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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

- Introduction
- Adolescent Friendship Structure
- Hierarchy
- Network Change

- Theory
- Traditional Social Balance Models
- A Local-change Balance Model

- Observed Results
- Simulation Results
- 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

Size

Community type

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

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:

- 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.
- Methods
- Features of the global network structure are identified through triad distributionmethods 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

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

- 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

i

j

j

i

k

k

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:

&

then:

Actors seek out transitive relations and avoid intransitive relations

Intransitive

Transitive

Mixed

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

021U

030T

120U

021C

030C

120C

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).

- 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.

- Structural implications differ depending on your position in the network.
- Examine relational change directly. Instead of assuming that intransitive relations resolve in equilibrium, we need to ask the micro-implications of moving from one structural state to another.
- This allows us distinguish transitivity seeking from intransitivity avoidance.

- Triples instead of triads. Operationalizing balance theory as transitivity allows us to simplify the behavioral assumptions (cf. Hummel and Soduer (1987, 1990.

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

- 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) out-degree.
- 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.

- Butthe 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.

030T

120D

120U

Dynamic Social Balance

A dyadic extension: Gould’s asymmetry avoidance rule

- What effect of asymmetry? Consider these triads:

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.

- There are many other local dynamic processes models to specify. For example: