Network Dynamics: Coevolution of Social Network · 2010-04-23 · V. Forward Model Selection Three Main Steps •1) network dynamics are only considered without taking the behavior

Network Dynamics: Coevolution of Social Network

Modeling the Co-evolution of Networks and Behavior

Jeongyoon Lee

March, 26, 2010

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• Snijders, Steglich, and Schweinberger. (2005).

Modeling the Co-evolution of Networks and Behavior.

In Longitudinal models in the behavioral and related

sciences, edited by K. van Montfort, H. Oud and A.

Satorra: Lawrence Erlbaum

• Burk, Kerr, and Stattin. (2008). The co-evolution of

early adolescent friendship networks, school

involvement and delinquent behaviors. Review

française de sociologie. 49(3): 499-522

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Overview

I. The Joint Dynamics of Networks and

Behavior

II. Notation and Data Structure

III. Modeling of Network and Behavior

Coevolution

IV. Method of Moments Estimation(MoM)

V. Model Selection

VI. Example: Burk, Kerr, and Stattin. (2008)

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I. The Joint Dynamics of Networks and Behaviors


• Change of the patterns of relations

◦ trust, social support, communication, even web links and co-authorship ties

◦ E.g., Burk et al. (2008) Friendship network

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Friendship network at Time1

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Friendship network at Time 4

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Friendship network at Time 5

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• Social changes are affected by:

◦ Social networks: expressed by tie variables

◦ Behavioral variables: expressed by actors’ characteristics

◦ Mutual interdependence


• Social network dynamics depending on actors’ characteristics:

◦ Patterns of homophily:

Preference for similarity in friendship selection (McPherson et al. 2001)

◦ Patterns of exchange (Strategic partner/ alliance)

• Actors’ characteristics depending on social network

◦ Patterns of assimilation

Diffusion of innovations in a professional community (Valente 1995)

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• “The change of network structure is often referred to as selection (Lazarsfeld and Merton 1954); the change of individual characteristics of social actors depends on the characteristics of others to whom they are tied which are called as influence (Friedkin 1998).”

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Behavior (tn) Behavior(tn+1)

Network (tn ) Network (tn+1)

Persistence (?)

Persistence (?)

Source: Steglich et al. Analyzing coevolution of social networks and behavioral dimenison with SIENA, SIENA workshop, 2007

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II. Notation and Data Structure

II. Notations and Data Structure

• Network: relations ◦ X(t) : relation

• Actors’ attributes: behavior or actor characteristics ◦ assumed to be ordered discrete, each having a finite interval of integer

values as its range

◦ Zhi : the value of actor i on the hth attribute

• Time (t) dependence is indicated “X = X(t)” and “Zh = Zh(t)

• Covariates◦ dyadic covariate (i.e., depending on a pair of actors) : wh = (whij) 1≤i,j≤n

◦ individual covariate (i.e., actor-dependent) : vh = (vh1, …, vhn)

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III. Modeling

III. Modeling

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Behavior (tn) Behavior(tn+1)

Network (tn ) Network (tn+1)

Persistence (?)

Persistence (?)


III. Modeling

• The process of network-behavioral coevolution

• Network change

◦ Considering the possible network-behavioral configurations (states)

◦ e.g, Burk et al. (2008) # of Actor=445 students = n

Dichotomous tie =2n(n-1)

Actor characteristics = delinquency scores divided into 5 categories = k = kn

(5445 actor configurations) * (2445*444 tie configurations)

(kn actor configurations) * (2n(n-1) tie configurations)

kn * 2n(n-1) states

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III. Modeling

• The process of network-behavioral coevolution

• Behavioral change◦ an emergent group level result of the network actors’ individual decisions

◦ These decisions are modeled as being the results of myopic optimization by each actor of an objective function

• Modeling of big two domains of decisions and two sub-decisions:◦ Decisions about network neighbors

When can actor i make a decision? (rate function)

Which decision does actor i make? (objective function)

◦ Decisions about own behavior When can actor i make a decision? (rate function)

Which decision does actor i make? (objective function)

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III. Modeling Four Assumptions

1) Observations at the discrete time options t1 < t2 < … < tM are the outcomes of an underlying process Y(t)=(X(t), Z1(t), …,ZH(t)) that is a Markov process with continuous time parameter t.

2) At any given moment t, all actors act conditionally independently of each other, given the current state Y(t) of the process.

3) The changes which an actor applies at time t to his/her network ties (thus, the change in Xi ) and the changes made about his/her behavioral characteristics (change in Zhi) are all conditionally independent of each other.

4) When an actor makes a change in either the vector of outgoing tie variable Xij (j =1, …, n; j≠i) or in the behavior vector (Z1i, …, ZHi), not more than one variable Xij or Zhi can be changes at one instant, and in the value of Zhi only increases or decreases by one unit are permitted.

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III. Modeling

• How does the model look like?

◦ State space Pair (X, Z)(t) contains adjacency matrix x and vector(s) of

behavioral variables z at times point t

◦ Stochastic process Coevolution is modeled by specifying transition probabilities

between such states (x,z)(t1) and (x,z)(t2)

◦ Continuous time model Evolution can be modeled in smaller units (‘micro steps’)

Observed changes are interpreted as resulting from a sequence of micro steps

Source: Steglich et al. Analyzing coevolution of social networks and behavioral dimension with SIENA, SIENA workshop, 2007

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III. Modeling

• Micro steps◦ Network micro steps:

(x,z)(t1) and (x,z)(t2) differ in one tie variable Xij only.

◦ Behavioral micro stpes:

(x,z)(t1) and (x,z)(t2) differ in one behavioral score variable Zj only.

• Actor-driven model◦ Micro steps are modeled as outcomes of an actor’s decisions

◦ These decisions are conditionally independent, given the current state of the process


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III. Modeling

1. Rate functions

• When can actor i make a decision?

2. Objective functions

• Which decision does actor i make?

Evaluation function

Endowment function

Random residuals

3. Model components

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III. Modeling1. Rate functions

• When can actor i make a decision? (λ)

• When changing either a tie variable Xij or a behavioral variable Zhi, these are randomly determined and follow Poisson processes, the waiting times being modeled by exponential distributions with parameters given by so-called rate functions λ.

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III. Modeling1. Rate functions

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• For each actor i,

◦ one rate function for the network (denoted λi[X])

◦ one rate function for each behavioral dimension (denoted λi

[Zh]).

III. Modeling2. Objective functions

• Which decision does actor i make?

• Objective functions = which changes are made; assuming that actors i myopically optimize an objective function over the set of possible micro steps they can make.

◦ Evaluation function (f)

◦ Endowment function (g)

◦ Random residual (e)

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• For network decisions taken by actor I, starting from the current state Y(t) and optimizing the new state y under the constrains defined by the type of micro step, the objective function optimized is,

• For behavioral decisions,

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• Evaluation function (f)

◦ measuring the satisfaction of actor i with a given network-behavioral configuration

evaluation of the network:

evaluation of behavioral variables:

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• Endowment function (g)

◦ measuring a component of the satisfaction with a given network-behavioral configuration that will be lost when the value of a variable Xij or Zhi is changed, but which was obtained without ‘cost’ when this value was obtained

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• Network endowment effects: assessing systematic difference b/t the creation and the dissolution of ties that cannot be captured by the evaluation function.

• Endowment function for network changes from y0 to y:

,

where

= the endowment value of the tie =1

I{A}= indicator function of the condition A

If the condition is satisfied, the value is equal to 1

If the condition is not satisfied, the value is equal to 0

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• Behavioral endowment effects: assessing similar asymmetries b/t moving upwards on a behavioral dimension and moving downwards

• Endowment function for Behavioral variables is written, for the change from y0 to y:

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• Random residuals◦ capturing noise i.e., unexplained influences

◦ assumed to be independent and to follow a type-I extreme value distribution

• Network decisions the resulting choice probabilities are

• Behavioral decisions the formula is

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III. ModelingEvaluation & Endowment functions

• The difference b/t evaluation function and endowment function: ◦ evaluation function (f) depends only on the new state y

◦ however, endowment function (g) depends both on the hypothetical new state y and the current sate Y(t) that is the immediate precursor of y.

◦ evaluation function (f) considers increases of the tie and behavior variables

◦ however, endowment function (g) considers the satisfaction lost when decreasing values of tie variables (Xij) and behavior variable (Zhi)

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III. Modeling3. Model components

Evaluation function Endowment function

Network 1. Outdegree effect2. Reciprocity effect3. Transitivity effect 4. Number of geodesic distance two effect, or indirect relations effect5. Attribute-related similarity6. Main effect of a dyadic covariate w

1. Outdegree effect2. Reciprocity effect3. Transitivity effect 4. Number of geodesic distance two effect, or indirect relations effect5. Attribute-related similarity6. Main effect of a dyadic covariate w

Behavioral 1. Tendency2. Attribute-related similarity3. Dependence on other

behaviors h’

1. Tendency2. Attribute-related similarity3. Dependence on other

behaviors h’

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III. Modeling

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Ne

two

rkEvo

lutio

n

Rate functions “When can actor i make a

decision?”

Objective functions“Which decision does actor i

make?”

Evaluation function (f)

1. Outdegree effect2. Reciprocity effect3. Transitivity effect 4. Number of geodesic distance two effect, or indirect relations effect5. Attribute-related similarity6. Main effect of a dyadic covariate w

Endowment Function (g)

Random

Be

havio

ralEvo

lutio

n

Rate functions

Objective functions

Evaluation (f) 1. Tendency2. Attribute-related similarity3. Dependence on other

behaviors h’Endowment (g)

Random

III. ModelingTransition intensities

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The matrix of transition intensities has the following elements, where y=(x,z) is the current and y^ the next outcome

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• For a general statistical model with data Y and parameter θ , the MoM estimator based on the statistic u(Y) is defined as the parameter value of θ^ for which the expected and observed values of u(Y) are the same,

◦ Eθ^ (u(Y))= u(y)

• The components of θ in this model are the following parameters: ρ, α, β, γ

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• The components of θ

◦ ρ, α, β, γ

1) ρm : constant factors in the rate functions

2) α : the other parameter in the rate functions

3) β : weights in the evaluation functions

4) γ : weights in the endowment functions

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V. Forward Model SelectionThree Main Steps


• 1) network dynamics are only considered without taking the behavior into account.

◦ testing whether more complex network structuring (e.g., transitivity) increases the fit of the model

◦ Under the continuous-time Markov model assumption: Processes shaping the dyads are independent (DI)

◦ Testing Ho : independent dyad processes Including triadic dependencies (e.g., the number of transitive

triplets)

Rejection of Ho : DI assumption is indefensible

Suggesting use of Neyman-Rao score tests

◦ Argument for continuing the analysis with actor-driven models

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• 2) If dyad do not follow independent processes, the main dependencies between dyad processes should be captured by actor-driven models for the network dynamics along with simple specifications of the behavior dynamics.

Ho : Independence of the network and behavior dynamics

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• 3) statistical modeling by actor-driven models for joint network and behavioral dynamics

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Examples

• Burk, Kerr, and Stattin. (2008). The co-evolution of early adolescent friendship networks, school involvement and delinquent behaviors. Review française de sociologie. 49(3): 499-522

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Research Question

1) How dynamic are early adolescent friendship networks?

2) What are the most prominent features of network dynamics?

3) What are the relative contributions of friends’ behaviors on early adolescents’ school involvement and delinquency?

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Research Methods

Methods

• Model of network-behavioral dynamics with a five-year longitudinal sample of an entire 4th

grade cohort of Swedish youth from a small community

• Identifying the structural characteristics of these networks and behavioral tendencies of the adolescent participants

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MethodsModel of network and behavioral dynamics

• After seeing Network dynamics and Behavioral dynamics ,

• Using forward selection three steps procedure,

◦ 1) testing whether more complex network structuring (i.e., transitivity) increases the fit of the model

◦ 2) testing whether network evolution is independent of behavioral evolution

◦ 3) building a model including all parameters of interest

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Methodsforward selection three steps procedure: Step 1

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Research ResultsResearch Question 2 & 3

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Result 2-1Q2. What are the most prominent features of network dynamics?

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Results 3Q3. What are the relative contributions of friends’ behaviors on early

adolescents’ school involvement and delinquency?

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Research Implications

Research Implications

• If homophilic selection effects are found to be more important, this suggests a focus on preventing the establishment of antisocial relationships;

• Whereas, if influence is deemed to be the relatively more important mechanism, this suggests a focus on disrupting relationships that have already formed (intervention).

• This study suggests that effects for both processes related to delinquency, with selection demonstrating slightly stronger effects.

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Questions

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Documents

Network Dynamics: Coevolution of Social Network · 2010-04-23 · V. Forward Model Selection Three Main Steps •1) network dynamics are only considered without taking the behavior