Download Dyadic models and exponential random graph models

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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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p1 and p2 in StOCNET
Marijtje van Duijn, Dyadic models and exponential random graph models , May 10, 2010
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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
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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
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p1 and p2 in StOCNET
Marijtje van Duijn, Dyadic models and exponential random graph models , May 10, 2010
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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
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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)
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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
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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
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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
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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
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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
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