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Dyadic models and exponential random graph models Modeling network ties Marijtje van Duijn University of Groningen [email protected] Marijtje van Duijn, Dyadic models and exponential random graph models , May 10, 2010 slide 1/28 Outline 1 Introduction 2 The p1 model 3 The p2 model 4 Application 5 The exponential random graph model (ERGM) Marijtje van Duijn, Dyadic models and exponential random graph models , May 10, 2010 slide 2/28 Social network analysis Aims understanding the network structure by visualization description (statistical) modeling Marijtje van Duijn, Dyadic models and exponential random graph models , May 10, 2010 slide 3/28 Social network analysis Aims understanding the network structure by visualization description (statistical) modeling Challenges with statistical modeling complex dependence structure non-normal distributions Marijtje van Duijn, Dyadic models and exponential random graph models , May 10, 2010 slide 3/28 Social network data Require at least tie or structural variable between two actors i and j: Yij set of actors Marijtje van Duijn, Dyadic models and exponential random graph models , May 10, 2010 slide 4/28 Social network data Require at least tie or structural variable between two actors i and j: Yij set of actors Can be extended with more than one tie variable actor attributes (composition variables) (Xi ) Marijtje van Duijn, Dyadic models and exponential random graph models , May 10, 2010 slide 4/28 Social network data Require at least tie or structural variable between two actors i and j: Yij set of actors Can be extended with more than one tie variable actor attributes (composition variables) (Xi ) Social relational system Marijtje van Duijn, Dyadic models and exponential random graph models , May 10, 2010 slide 4/28 Types of networks one-mode: one set of actors (complete network); usually well-defined (population). two-mode: two sets of actors or one set of actors and one set of events ego-centered or personal networks; usually a sample. Types of ties directed/undirected dichotomous/valued Today: one-mode directed dichotomous network data Marijtje van Duijn, Dyadic models and exponential random graph models , May 10, 2010 slide 5/28 Types of networks one-mode: one set of actors (complete network); usually well-defined (population). two-mode: two sets of actors or one set of actors and one set of events ego-centered or personal networks; usually a sample. Types of ties directed/undirected dichotomous/valued Today: one-mode directed dichotomous network data Special cases: multiple complete networks longitudinal complete networks Marijtje van Duijn, Dyadic models and exponential random graph models , May 10, 2010 slide 5/28 Goal of social network analysis Describing or finding structure, by description by network statistics comparing actors by covariates identifying subgroups (categorizing actors) modeling or explaining (dyadic) relationships (possibly over time) Marijtje van Duijn, Dyadic models and exponential random graph models , May 10, 2010 slide 6/28 Social network analysis answers research questions like 1 How can we describe and summarize properties of the network(s) and/or actors? 2 How can we compare or categorize actors and their (endogenous or exogenous) characteristics? Marijtje van Duijn, Dyadic models and exponential random graph models , May 10, 2010 slide 7/28 Social network analysis answers research questions like 1 How can we describe and summarize properties of the network(s) and/or actors? 2 How can we compare or categorize actors and their (endogenous or exogenous) characteristics? 3 How can we describe and model the association between the ties within one network, and between networks? 4 How can we describe and model the association between the network ties and (endogenous or exogenous) actor characteristics? Marijtje van Duijn, Dyadic models and exponential random graph models , May 10, 2010 slide 7/28 Social network analysis answers research questions like 1 How can we describe and summarize properties of the network(s) and/or actors? 2 How can we compare or categorize actors and their (endogenous or exogenous) characteristics? 3 How can we describe and model the association between the ties within one network, and between networks? 4 How can we describe and model the association between the network ties and (endogenous or exogenous) actor characteristics? 5 How do networks develop over time and how do they influence each other? Marijtje van Duijn, Dyadic models and exponential random graph models , May 10, 2010 slide 7/28 Software for social network analysis StOCNET UCINET/Netdraw NetMiner Ora ... R-packages sna, network, latentnet, statnet Pajek Which software is to be preferred? Choice of software depends on type of statistical analysis... Choice of statistical analysis depends on research question... Marijtje van Duijn, Dyadic models and exponential random graph models , May 10, 2010 slide 8/28 Major issues Statistical issues modeling of dependency between arcs estimation of the (consequentially complex) models Between p1 /p2 and p∗ /ERGM modeling four possible dyadic outcomes modeling complete network (using dyadic and more complex properties) Marijtje van Duijn, Dyadic models and exponential random graph models , May 10, 2010 slide 9/28 Four dyadic outcomes Null (0,0) Two asymmetric (0,1), (1,0) Mutual dyads (1,1) Marijtje van Duijn, Dyadic models and exponential random graph models , May 10, 2010 slide 10/28 Outline 1 Introduction 2 The p1 model 3 The p2 model 4 Application 5 The exponential random graph model (ERGM) Marijtje van Duijn, Dyadic models and exponential random graph models , May 10, 2010 slide 11/28 Model for dyadic outcomes Basic formula: sort of (polytomous) logistic regression i →j j →i reciprocity P{Xij = x1 , Xji = x2 } = exp{x1 (µ + αi + βj } ∗ exp{x2 (µ + αj + βi )} ∗ exp{x1 x2 ρ}/kij x1 , x2 = 0, 1 kij : normalizing constant µ: density ρ: reciprocity αi : outgoingness (sender effect) βj : popularity (receiver effect) Marijtje van Duijn, Dyadic models and exponential random graph models , May 10, 2010 slide 12/28 Outline 1 Introduction 2 The p1 model 3 The p2 model 4 Application 5 The exponential random graph model (ERGM) Marijtje van Duijn, Dyadic models and exponential random graph models , May 10, 2010 slide 13/28 The p2 model p1 plus covariates: more regression equations Further modeling of outgoingness (sender) and popularity parameters (receiver) with actor-dependent explanatory variables Wi for the sender; Wj for the receiver. αi = Wi γ1 + Ai , i = 1...n βj = Wj γ2 + Bj , j = 1 . . . n accounting for dependence of ties to and from the same actor: var(Ai ) = σA2 , var(Bi ) = σB2 , cov(Ai , Bi ) = σAB for all i cov(Ai , Aj ) =cov(Bi , Bj ) =cov(Ai , Bj ) = 0 for i 6= j. Marijtje van Duijn, Dyadic models and exponential random graph models , May 10, 2010 slide 14/28 Further modeling of density and reciprocity with dyadic explanatory variables Zij1 and Zij2 (possibly derived from actor attributes W ). µij = µ + Zij1 δ1 , ρij = ρ + Zij2 δ2 , Zij2 = Zji2 Summary of p2 model parameters the basic parameters µ and ρ regression coefficients γ, δ variance components: variances σA2 and σB2 and the covariance σAB of the actor effects. Marijtje van Duijn, Dyadic models and exponential random graph models , May 10, 2010 slide 15/28 Interpretation of p2 parameters Actor-related effects: pos/neg sender effect in/decreases outgoing tie probability pos/neg receiver effect in/decreases ingoing tie probability Marijtje van Duijn, Dyadic models and exponential random graph models , May 10, 2010 slide 16/28 Interpretation of p2 parameters Actor-related effects: pos/neg sender effect in/decreases outgoing tie probability pos/neg receiver effect in/decreases ingoing tie probability Dyad-related effects: pos/neg density effect in/decreases any dyadic tie probability pos/neg density effect in/decreases mutual tie probability Note: in addition to density effect - like an interaction effect Dyadic attributes derived from actor attributes: absolute difference – the same from all perspectives of the dyad difference - positive/negative from sender/receiver perspectives therefore not suitable for reciprocity effect Marijtje van Duijn, Dyadic models and exponential random graph models , May 10, 2010 slide 16/28 Some connections with other models in the area of generalized linear models p2 is a bivariate logistic regression model p2 is a generalized linear mixed model p2 is a cross-nested multilevel model p2 is the dichotomous counterpart of the Social Relations Model (Snijders & Kenny, 1996) p2 belongs to the exponential family but is not an ERGM (Therefore) estimation with MCMC Marijtje van Duijn, Dyadic models and exponential random graph models , May 10, 2010 slide 17/28 p1 and p2 in StOCNET Marijtje van Duijn, Dyadic models and exponential random graph models , May 10, 2010 slide 18/28 Outline 1 Introduction 2 The p1 model 3 The p2 model 4 Application 5 The exponential random graph model (ERGM) Marijtje van Duijn, Dyadic models and exponential random graph models , May 10, 2010 slide 19/28 Application to EIES data Short description of data 32 researchers interacting via email 4 research areas actor information on citations (status) dyadic information on intensity of communication, hierarchy and distance (based on citations) Marijtje van Duijn, Dyadic models and exponential random graph models , May 10, 2010 slide 20/28 p1 and p2 in StOCNET Marijtje van Duijn, Dyadic models and exponential random graph models , May 10, 2010 slide 21/28 Research questions For instance Can we predict friendship between researchers if we know how many email interactions they have? Do researchers on a conference prefer to get acquainted with colleagues from the same research area or do they rather interact with colleagues with a high citation index? Could it be that email contact, homophily and scientific status are all important to explain friendship or acquaintanceship, and if so, which of these effects is strongest? How strong are these effects when controlled for earlier friendship? Marijtje van Duijn, Dyadic models and exponential random graph models , May 10, 2010 slide 22/28 Results Density Model 0 −3.01 (0.27) Model 1 −4.64 (0.51) 0.420 (0.054) −0.230 (0.054) 0.486 (0.20) Model 2 −4.07 (0.56) 0.421 (0.065) −0.368 (0.085) 0.531 (0.20) 3.82 (0.45) 3.32 (0.48) 2.51 (0.59) 0.358 (0.16) Communication Distance Same field Network time 1 Reciprocity Distance Marijtje van Duijn, Dyadic models and exponential random graph models , May 10, 2010 Model 3 −5.15 (0.77) 0.469 (0.076) −0.346 (0.125) 0.298 (0.38) 6.31 (0.64) 1.18 (0.73) 0.443 (0.28) slide 23/28 Results-cont. Sender status Receiver status Sender variance Receiver variance Sender/Receiver covariance Deviance (approx.) 1.06 (0.40) 1.46 (0.52) −0.829 (0.40) 663.7 Marijtje van Duijn, Dyadic models and exponential random graph models , May 10, 2010 0.169 (0.12) 0.117 (0.080) 3.31 (1.18) 0.746 (0.34) −1.15 (0.51) 537.6 0.160 (0.14) 0.0965 (0.088) 3.47 (1.33) 0.751 (0.33) −1.18 (0.52) 532.8 0.114 (0.14) 0.0533 (0.099) 2.88 (1.32) 0.754 (0.45) −0.963 (0.55) 265.8 slide 24/28 Outline 1 Introduction 2 The p1 model 3 The p2 model 4 Application 5 The exponential random graph model (ERGM) Marijtje van Duijn, Dyadic models and exponential random graph models , May 10, 2010 slide 25/28 Exponential random graph models - p∗ Family of probability functions for complete network Pθ {Y = y} = exp θ0 u(y) − ψ(θ) Y (di)graph u = u(y) vector of sufficient statistics ψ(θ) norming constant Important features Modeling of complete network Choice of sufficient statistics Typically more than dyadic dependence Marijtje van Duijn, Dyadic models and exponential random graph models , May 10, 2010 slide 26/28 p1 is an exponential random graph model sufficient statistics are number of ties y++ number of mutuals P i<j yij yji in-degrees yi+ out-degrees y+j with some restrictions on the parameters Marijtje van Duijn, Dyadic models and exponential random graph models , May 10, 2010 slide 27/28 Examples of typical sufficient statistics dyad counts (mutuals, in- and out-2-stars, two-paths) triad counts (digraph of three nodes, 6 types) k -stars (several types, weighted versions) Extension with covariates possible Computational details left out Marijtje van Duijn, Dyadic models and exponential random graph models , May 10, 2010 slide 28/28