Introduction to Introduction to Statistical Models for Statistical Models for longitudinal network longitudinal network

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Stochastic actor-based Stochastic actor-based modelsmodels

Kayo Fujimoto, Ph.D.Kayo Fujimoto, Ph.D.

Stochastic actor-based modelStochastic actor-based model(Snijders 2001, 2005)(Snijders 2001, 2005)

Actor-orientedActor-oriented modelingmodeling– Methodological individualismMethodological individualism

Modeled as a consequence of actors: Modeled as a consequence of actors: – Making new choicesMaking new choices– Withdrawing existing choiceWithdrawing existing choice– Functions Functions that actors try to maximizethat actors try to maximize

Continuous-time Markov chain Continuous-time Markov chain modelsmodels– Simulation modelsSimulation models

Dependent variableDependent variable

Changing relation network Changing relation network – Number of changed tiesNumber of changed ties between between

consecutive observationsconsecutive observations

Independent Variables Independent Variables

Change in the network (DV) Change in the network (DV) is is modeled as the stochastic result of modeled as the stochastic result of network effectsnetwork effects (such as reciprocity, (such as reciprocity, transitivity, etc.) and transitivity, etc.) and covariatescovariates

Model assumptions Model assumptions

Full knowledge of the Full knowledge of the present present networknetwork

All actors All actors control their outgoing control their outgoing relationsrelations– A specific actor i has the opportunity to A specific actor i has the opportunity to

change their relations change their relations one at a timeone at a time at at stochastic moment tstochastic moment t at a at a rate rate ρρmm

Model specification: changes of Model specification: changes of single relationssingle relations

Three types of effectsThree types of effects

Rate function effectsRate function effects– Models the Models the speedspeed by which the DV by which the DV

changeschanges Objective function effectsObjective function effects

– Models the actorsModels the actors’’ satisfactionsatisfaction with with their their local network configurationlocal network configuration

Endowment function effectsEndowment function effects– Model the Model the loss of satisfactionloss of satisfaction incurred incurred

when existing network ties are dissolvedwhen existing network ties are dissolved

Objective function effectObjective function effect

Determines probabilistically the Determines probabilistically the tie tie changeschanges made by the actors made by the actors

Defined as a Defined as a function of the function of the networknetwork – regarded from the regarded from the perspective of the perspective of the

focus actorfocus actor Depends on Depends on parametersparameters

– estimated from the dataestimated from the data

Objective function Objective function

Network evaluation function for actor iNetwork evaluation function for actor i– Degree of satisfaction for each actor i in Degree of satisfaction for each actor i in

relation xrelation x

( )i k ikkf x s x

k are parameters

iks x areeffects

Structural effects Structural effects (examples)(examples)

Outdegree effect (density effect)Outdegree effect (density effect)

Reciprocity effectReciprocity effect

Triad effect (transitivity, cycle, balance etc.)Triad effect (transitivity, cycle, balance etc.)

1( )i i ijjs x x x

2 ( )i ij jijs x x x

3 ,( )i ij ih jhj h

s x x x x

Examples Examples ––transitive triplets transitive triplets effecteffect

Covariate effects (V)Covariate effects (V)

Covariate-ego effect (sender effect)Covariate-ego effect (sender effect)– Whether actors with higher V values tend to Whether actors with higher V values tend to

nominate more friends and hence have a higher nominate more friends and hence have a higher outdegree outdegree

Covariate-alter effect (receiver effect)Covariate-alter effect (receiver effect)– Whether actors with higher V values tend to be Whether actors with higher V values tend to be

nominated by more others and hence have nominated by more others and hence have higher indegreeshigher indegrees

Covariate-similarity effect (homophily)Covariate-similarity effect (homophily)– Whether ties tend to occur more often between Whether ties tend to occur more often between

actors with similar values o n Vactors with similar values o n V

Covariate effects Covariate effects (examples)(examples)

Covariate-ego effect (covariate-related Covariate-ego effect (covariate-related activity)activity)

Covariate-alter effect (covariate-related Covariate-alter effect (covariate-related popularity)popularity)

Same covariate effect (homophily)Same covariate effect (homophily)

4 ( )i i is x v x

5 ( )i ij jjs x x v

6 ( )i ij i jjs x x I v v

Objective function Objective function

Actor i chooses alter j that Actor i chooses alter j that maximizemaximize the value of her objective function the value of her objective function fi(x)fi(x)

Plus Plus random elementrandom element (Gumbel (Gumbel distdist’’n)n)– The part of the actorThe part of the actor’’s preference that is s preference that is

not represented by the systematic not represented by the systematic component of fi(x) component of fi(x)

Model Parameters Model Parameters

Estimated from observed dataEstimated from observed data Stochastic simulation models Stochastic simulation models

– MCMC algorithm MCMC algorithm – Approximate the solution of the Method Approximate the solution of the Method

of Momentof Moment

Estimation in SIENAEstimation in SIENA

Choose Choose statistics statistics Obtain Obtain parametersparameters such that the such that the

expected valuesexpected values of the statistics of the statistics are equal to the are equal to the observed valuesobserved values– Expected valuesExpected values are approximated as are approximated as

the averages over a lot of simulated the averages over a lot of simulated networknetwork

– Observed valuesObserved values are calculated from are calculated from the dataset (the dataset (target valuestarget values))

Estimation in SIENAEstimation in SIENA Iterative stochastic simulation algorithmIterative stochastic simulation algorithm In phase 1In phase 1: the sensitivity of the statistics to the : the sensitivity of the statistics to the

parameters is determinedparameters is determined In phase 2In phase 2: : provisional parameter valuesprovisional parameter values are are

updatedupdated– Simulate a networkSimulate a network based on provisional parameter based on provisional parameter

valuesvalues– Compute Compute thethe deviations deviations between these between these simulated simulated

statisticsstatistics and and target valuestarget values – Update parameter valuesUpdate parameter values

In phase 3In phase 3: the final results of phase 2 is used : the final results of phase 2 is used and checked if the and checked if the average statisticaverage statistic of many of many simulated networks are close to the simulated networks are close to the targeted targeted valuesvalues – t statisticst statistics for deviations from targets for deviations from targets

Longitudinal network dynamic Longitudinal network dynamic modelsmodels

Actor-oriented modelsActor-oriented models of Snijders of Snijders and colleaguesand colleagues– Assumption: network change driven by Assumption: network change driven by

actorsactors seeking to optimize particular seeking to optimize particular structural positionsstructural positions

Longitudinal versions of ERGMLongitudinal versions of ERGM (tie-based version of the model)(tie-based version of the model)– Assumption: network change driven by Assumption: network change driven by

change in change in tie variablestie variables (particular (particular social neighborhood of other ties)social neighborhood of other ties)

ReferencesReferences

Snijders, T.A.B.(2001). The statistical Snijders, T.A.B.(2001). The statistical evaluation of social network evaluation of social network dynamics, Sociological Methodologydynamics, Sociological Methodology

Snijders, T.A.B.(2005). Models for Snijders, T.A.B.(2005). Models for longitudinal network data, chapter 11 longitudinal network data, chapter 11 in Carrington, P., Scott, J, Wasserman in Carrington, P., Scott, J, Wasserman S (eds), models and methods in S (eds), models and methods in social network analysis. New York: social network analysis. New York: Cambridge University Press.Cambridge University Press.