Hierarchical Network Models Motivation Social Networks in Education - PowerPoint PPT Presentation

Hierarchical Network Models

Motivation Social Networks in Education Independent network replications ◮ Teacher networks within schools (Frank et al., 2004; Moolenaar et al., 2010; Spillane et al., 2012; Weinbaum et al., 2008) ◮ Student networks within classes/schools (Gest and Rodkin, 2011; Harris et al., 2008)

Motivation Social Networks in Education Independent network replications ◮ Teacher networks within schools (Frank et al., 2004; Moolenaar et al., 2010; Spillane et al., 2012; Weinbaum et al., 2008) ◮ Student networks within classes/schools (Gest and Rodkin, 2011; Harris et al., 2008) How can we accommodate multiple networks using social network models? ◮ Modeling multiple networks simultaneously ◮ Estimating treatment effects

Hierarchical Network Models (Sweet et al., 2013) POPULATION Θ 1 Θ K Θ 2 Θ K-1 . . . … … … …

Hierarchical Network Models K � P ( Y | X , Θ) = P ( Y k | X k = ( X 1 k , . . . , X P k ) , Θ k = ( θ 1 k , . . . , θ Q k )) k =1 (Θ 1 , . . . , Θ K ) ∼ F (Θ 1 , . . . , Θ K | W 1 , . . . , W K , ψ ) , ◮ P ( Y k | X k , Θ k ) is a model for a single network Y k with covariates X k and parameters Θ k ◮ W k can model a variety of dependence assumptions across networks ◮ ψ may specify additional hierarchical structure on Θ

Examples ◮ Multilevel Social Relations Model (Snijders and Kenny, 1999) ◮ Extends the Social Relations Model (Kenny and La Voie, 1984) ◮ Multilevel p 2 Model (Zijlstra et al., 2006) ◮ Extends the p 2 Model (van Duijn et al., 2004) ◮ Hierarchical Mixed Membership Stochastic Blockmodel (Sweet et al., 2014) ◮ Extends the Mixed Membership Stochastic Blockmodel (Airoldi et al., 2008), related to Stochastic Blockmodels (Snijders and Nowicki, 1997) ◮ Hierarchical Latent Space Model (HLSM Sweet et al., 2013) ◮ Extends the Latent Space Model (Hoff et al., 2002)

Hierarchical Network Models K � P ( Y | X , Θ) = SBM for Y k k =1 (Θ 1 , . . . , Θ K ) ∼ F (Θ 1 , . . . , Θ K | W 1 , . . . , W K , ψ ) , ◮ P ( Y k | X k , Θ k ) is a model for a single network Y k with covariates X k and parameters Θ k ◮ W k can model a variety of dependence assumptions across networks ◮ ψ may specify additional hierarchical structure on Θ

Hierarchical Network Models K � P ( Y | X , Θ) = LSM for Y k k =1 (Θ 1 , . . . , Θ K ) ∼ F (Θ 1 , . . . , Θ K | W 1 , . . . , W K , ψ ) , ◮ P ( Y k | X k , Θ k ) is a model for a single network Y k with covariates X k and parameters Θ k ◮ W k can model a variety of dependence assumptions across networks ◮ ψ may specify additional hierarchical structure on Θ

Hierarchical Latent Space Model (HLSM) K � P ( Y | X , Θ) = P ( Y k | β k , X k , Z k ) k =1 ( β 1 , . . . , β K ) ∼ F ( µ, σ 2 ) , ◮ µ and σ 2 specify additional hierarchical structure on β

Hierarchical Latent Space Model (HLSM) n k K � � P ( Y | X , Θ) = P ( Y ijk | β k , X ijk , Z ik , Z jk ) k =1 i � = j ( β 1 , . . . , β K ) ∼ N ( µ, σ 2 ) , ◮ µ and σ 2 specify additional hierarchical structure on β

Hierarchical Latent Space Model (HLSM) 1 2 3 4 5 6 6 6 6 6 4 4 4 4 4 2 2 2 2 2 0 0 0 0 0 -2 -2 -2 -2 -2 -4 -4 -4 -4 -4 -6 -6 -6 -6 -6 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 logit P [ Y ijk = 1] = β 0 k + β 1 k X 1 ijk + · · · + β pk X pijk − | Z ik − Z jk | ◮ Y ijk = 1 indicates a tie from i to j in network k ◮ X ijk is a node-, tie-, or network- level covariate ◮ Z ik is the latent space position for actor i in network k

HLSM for Pitts & Spillane Data Model 1: β ’s are fixed logit P [ Y ijk = 1] = β 0 + β 1 X 1 ijk + β 2 X 2 ijk + β 3 X ijk − | Z ik − Z jk | ◮ β ’s are fixed across networks X ijk ’s are all indicator covariates ◮ X 1 ijk – similar teaching experience ◮ X 2 ijk – similar belief in innovative instruction ◮ X 3 ijk – teaching the same grade

HLSM for Pitts & Spillane Data Model 1: β ’s are fixed logit P [ Y ijk = 1] = β 0 + β 1 X 1 ijk + β 2 X 2 ijk + β 3 X ijk − | Z ik − Z jk | ◮ β ’s are fixed across networks Model 2: β ’s vary logit P [ Y ijk = 1] = β 0 k + β 1 k X 1 ijk + β 2 k X 2 ijk + β 3 k X ijk − | Z ik − Z jk | ◮ β ’s vary across networks but are sampled from a common population, e.g. β 1 k ∼ N ( µ, σ 2 )

Fitting the HLSM logit P [ Y ijk = 1] = β 0 k + β 1 k X 1 ijk − | Z ik − Z jk | �� 0 � a � �� 0 Z ik ∼ MVN , , i = 1 , . . . , n k 0 0 a β 0 k ∼ N ( µ 0 , σ 2 0 ) , k = 1 , . . . , K β 1 k ∼ N ( µ 1 , σ 2 1 ) , k = 1 , . . . , K µ 0 ∼ N ( b 1 , b 2 ) µ 1 ∼ N ( c 1 , c 2 ) σ 2 0 ∼ Inv − Gamma ( d 1 , d 2 ) σ 2 1 ∼ Inv − Gamma ( e 1 , e 2 )

Model Fit: Effect of Teaching the Same Grade Separate Network Models * * * 15 15 10 10 5 5 0 0 -5 -5 -10 -10 * * 2 4 6 8 10 12 14 HLSM 15 10 5 0 -5 -10 2 4 6 8 10 12 14 School

Results: Effect of Teaching the Same Grade Fitting the Model logit P [ Y ijk = 1] = β 0 k + · · · + β 3 k X 3 ijk − | Z ik − Z jk | β 3 k ∼ N ( µ 3 , σ 2 3 ) , k = 1 , . . . , K 2 µ 3 σ 3 0.8 0.8 0.6 0.4 0.4 0.2 0.0 0.0 -1 0 1 2 3 2 4 6 8

Summary ◮ Introduction to SNA ◮ Network representations, visualizations, and statistics ◮ Generative network models ◮ R code includes network statistics, plotting, and dyadic independence models ◮ CIDnetworks ◮ Incorporates generative model components (LSM, SBM, covariates) ◮ Accommodates binary or continuous data ◮ R code includes network plots, simulating networks, fitting a variety of models, and viewing results ◮ Future work includes incorporating HNMs ◮ What methods, summaries, models, plots, etc would you want in this package? ◮ HNMs ◮ Models the naturally occurring independent networks found in education ◮ Can estimate individual–, tie–, and network– level covariate effects ◮ R code includes network plots, fitting a HLSM, and plotting results ◮ Other models exist – e.g. HMMSBM ◮ What data or projects would benefit from these models? What other types of HNMs would you like to have?

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