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Exponential-family random network modelsfor social networks
Mark S. Handcock
Ian E. Fellows
Department of StatisticsUniversity of California - Los Angeles
Supported by NIH NIDA Grant DA012831, NICHD Grant HD041877, NSFaward MMS-0851555 and the DoD ONR MURI award N00014-08-1-1015.
SAMSI Computational Methods in Social Sciences (CMSS) Workshop,August 18-22 2013
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Statistical Models for Social Networks
NotationA social network is defined as a set of n social “actors”, a socialrelationship between each pair of actors, and a set of variables on thoseactors/pairs.
Yij
=
(1 relationship from actor i to actor j
0 otherwise
call Y ⌘ [Yij
]n⇥n
a graph
a N = n(n � 1) binary array
X be n ⇥ q matrix of actor variates
call (Y ,X ) a network
The basic problem of stochastic modeling is to specify a distributionfor X ,Y i.e., P(Y = y ,X = x)
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Statistical Models for Social Networks
NotationA social network is defined as a set of n social “actors”, a socialrelationship between each pair of actors, and a set of variables on thoseactors/pairs.
Yij
=
(1 relationship from actor i to actor j
0 otherwise
call Y ⌘ [Yij
]n⇥n
a graph
a N = n(n � 1) binary array
X be n ⇥ q matrix of actor variates
call (Y ,X ) a network
The basic problem of stochastic modeling is to specify a distributionfor X ,Y i.e., P(Y = y ,X = x)
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The ERGM Framework for Network Modeling
Let Y be the sample space of Y e.g. {0, 1}Nand X be the sample space of X .Model the multivariate distribution of Y given X via:
P⌘(Y = y |X = x) =exp{⌘·g(y |x)}c(⌘, x ,Y)
y 2 Y, x 2 X
Frank and Strauss (1986)
⌘ 2 ⇤ ⇢ Rq q-vector of parameters
g(y |x) q-vector of graph statistics.) g(Y |x) are jointly su�cient for the model
c(⌘, x ,Y) distribution normalizing constant
c(⌘, x ,Y) =X
y2Yexp{⌘·g(y |x)}
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Simple model-classes for social networks
Homogeneous Bernoulli graph (Erdos-Renyi model)
Yij
are independent and equally likelywith log-odds ⌘ = logit[P⌘(Yij
= 1)]
P⌘(Y = y) =e⌘
Pi,j yij
c(⌘, x ,Y)y 2 Y
where q = 1, g(y) =P
i,j yij , c(⌘, x ,Y) = [1 + exp(⌘)]N
homogeneity means it is unlikely to be proposed as a model for realphenomena
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Dyad-independence models with attributes
Yij
are independent but depend on dyadic covariates {xk,ij}q
k=1
P⌘(Y = y |X = x) =eP
q
k=1
⌘k
g
k
(y |x)
c(⌘, x ,Y)y 2 Y
gk
(y |x) =X
i,j
xk,ijyij , k = 1, . . . , q
c(⌘, x ,Y) =Y
i,j
[1 + exp(qX
k=1
⌘k
xk,ij)]
Of course,logit[P⌘(Yij
= 1|X = x)] =X
k
⌘k
xk,ij
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Dyad-independence models with attributes
Yij
are independent but depend on dyadic covariates {xk,ij}q
k=1
P⌘(Y = y |X = x) =eP
q
k=1
⌘k
g
k
(y |x)
c(⌘, x ,Y)y 2 Y
gk
(y |x) =X
i,j
xk,ijyij , k = 1, . . . , q
c(⌘, x ,Y) =Y
i,j
[1 + exp(qX
k=1
⌘k
xk,ij)]
Of course,logit[P⌘(Yij
= 1|X = x)] =X
k
⌘k
xk,ij
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Dyad-independence models with attributes
Yij
are independent but depend on dyadic covariates {xk,ij}q
k=1
P⌘(Y = y |X = x) =eP
q
k=1
⌘k
g
k
(y |x)
c(⌘, x ,Y)y 2 Y
gk
(y |x) =X
i,j
xk,ijyij , k = 1, . . . , q
c(⌘, x ,Y) =Y
i,j
[1 + exp(qX
k=1
⌘k
xk,ij)]
Of course,logit[P⌘(Yij
= 1|X = x)] =X
k
⌘k
xk,ij
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Some history of exponential family models for socialnetworks
Holland and Leinhardt (1981) proposed a general dyad independencemodel
– Also an homogeneous version they refer to as the “p1” model
P⌘(Y = y) =exp{⇢
Pi<j
yij
yji
+ �y++
+P
i
↵i
yi+
+P
j
�j
y+j
}(⇢,↵,�,�)
where ⌘ = (⇢,↵,�,�).
– � controls the expected number of edges– ⇢ represent the expected tendency toward reciprocation– ↵
i
productivity of node i ; �j
attractiveness of node j
Much related work and generalizations
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Some history of exponential family models for socialnetworks
Holland and Leinhardt (1981) proposed a general dyad independencemodel
– Also an homogeneous version they refer to as the “p1” model
P⌘(Y = y) =exp{⇢
Pi<j
yij
yji
+ �y++
+P
i
↵i
yi+
+P
j
�j
y+j
}(⇢,↵,�,�)
where ⌘ = (⇢,↵,�,�).
– � controls the expected number of edges– ⇢ represent the expected tendency toward reciprocation– ↵
i
productivity of node i ; �j
attractiveness of node j
Much related work and generalizations
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Generative Theory for Network Structure
Actor Markov statistics
) Frank and Strauss (1986)
– motivated by notions of “symmetry” and “homogeneity”
– Yij
in Y that do not share an actor areconditionally independent given the rest of the network
) analogous to nearest neighbor ideas in spatial modeling
Degree distribution: dk
(y) = proportion of actors of degree k in y .
triangles: triangle(y) =number of triads that form a complete sub-graph in y .
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Generative Theory for Network Structure
Actor Markov statistics
) Frank and Strauss (1986)
– motivated by notions of “symmetry” and “homogeneity”– Y
ij
in Y that do not share an actor areconditionally independent given the rest of the network
) analogous to nearest neighbor ideas in spatial modeling
Degree distribution: dk
(y) = proportion of actors of degree k in y .
triangles: triangle(y) =number of triads that form a complete sub-graph in y .
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Generative Theory for Network Structure
Actor Markov statistics
) Frank and Strauss (1986)
– motivated by notions of “symmetry” and “homogeneity”– Y
ij
in Y that do not share an actor areconditionally independent given the rest of the network
) analogous to nearest neighbor ideas in spatial modeling
Degree distribution: dk
(y) = proportion of actors of degree k in y .
triangles: triangle(y) =number of triads that form a complete sub-graph in y .
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Generative Theory for Network Structure
Actor Markov statistics
) Frank and Strauss (1986)
– motivated by notions of “symmetry” and “homogeneity”– Y
ij
in Y that do not share an actor areconditionally independent given the rest of the network
) analogous to nearest neighbor ideas in spatial modeling
Degree distribution: dk
(y) = proportion of actors of degree k in y .
triangles: triangle(y) =number of triads that form a complete sub-graph in y .
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Generative Theory for Network Structure
Actor Markov statistics
) Frank and Strauss (1986)
– motivated by notions of “symmetry” and “homogeneity”– Y
ij
in Y that do not share an actor areconditionally independent given the rest of the network
) analogous to nearest neighbor ideas in spatial modeling
Degree distribution: dk
(y) = proportion of actors of degree k in y .
triangles: triangle(y) =number of triads that form a complete sub-graph in y .
!8
Classes of statistics used for modeling
1) Nodal Markov statistics ) Frank and Strauss (1986)
– motivated by notions of “symmetry” and “homogeneity”– edges in Y that do not share an actor are
conditionally independent given the rest of the network) analogous to nearest neighbor ideas in spatial statistics
• Degree distribution: dk
(y) = proportion of nodes of degree k in y.
• k-star distribution: sk
(y) = proportion of k-stars in the graph y.
• triangles: t1(y) = proportion of triangles in the graph y.
• •
•
i
j
h
....................................................................................................................................................................................................................................
..................................................................................................................
triangle= transitive triad
• •
•
j1 j2
i
..................................................................................................................
..................................................................................................................
two-star
• •••
j1 j2
i
j3
....................................................................................
..............................................................................................................................................................
three-star
( Mark S. Handcock Statistical Modeling With ERGM !
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More General mechanisms motivated by conditionalindependence
) Pattison and Robins (2002), Butts (2005)) Snijders, Pattison, Robins and Handcock (2006)
– Yuj and Yiv in Y are conditionallyindependent given the rest of the networkif they could not produce a cycle in the network
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More General mechanisms motivated by conditionalindependence
) Pattison and Robins (2002), Butts (2005)) Snijders, Pattison, Robins and Handcock (2006)
– Yuj and Yiv in Y are conditionallyindependent given the rest of the networkif they could not produce a cycle in the networkNew specifications for ERGMs
• •
• •
i v
u j
........
........
........
........
........
........
........
........
........
........
........
........
........
........
..
........
........
........
........
........
........
........
........
........
........
........
........
........
........
..
............. ............. ............. ............. .............
............. ............. ............. ............. .............
Figure 2: Partial conditional dependence when four-cycle is created
(see Figure 2). This partial conditional independence assumption states thattwo possible edges with four distinct nodes are conditionally dependent when-ever their existence in the graph would create a four-cycle. One substantiveinterpretation is that the possibility of a four-cycle establishes the structuralbasis for a “social setting” among four individuals (Pattison and Robins,2002), and that the probability of a dyadic tie between two nodes (here, iand v) is a�ected not just by the other ties of these nodes but also by otherties within such a social setting, even if they do not directly involve i and v.
A four-cycle assumption is a natural extension of modeling based on tri-angles (three-cycles), and was first used by Lazega and Pattison (1999) inan examination of whether such larger cycles could be observed in an empir-ical setting to a greater extent than could be accounted for by parametersfor configurations involving at most 3 nodes. Let us consider the four-cycleassumption alongside the Markov dependence. Under the Markov assump-tion, Yiv is conditionally dependent on each of Yiu, Yuv, Yij and Yjv, becausethese edge indicators share a node. So if yiu = yjv = 1 (the precondition inthe four-cycle partial conditional dependence), then all five of these possibleedges can be mutually dependent, and hence the exponential model (4) couldcontain a parameter corresponding to the count of such configurations. Weterm this configuration, given by
yiv = yiu = yij = yuv = yjv = 1 ,
a two-triangle (see Figure 3). It represents the edge yij = 1 as part of thetriadic setting yij = yiv = yjv = 1 as well as the setting yij = yiu = yju = 1.
Motivated by this approach, we introduce here a generalization of triadicstructures in the form of graph configurations that we term k-triangles. Fora non-directed graph, a k-triangle with base (i, j) is defined by the presenceof a base edge i � j together with the presence of at least k other nodesadjacent to both i and j. We denote a ‘side’ of a k-triangle as any edge thatis not the base. The integer k is called the order of the k-triangle Thus ak-triangle is a combination of k individual triangles, each sharing the sameedge i� j. The concept of a k-triangle can be seen as a triadic analogue of a
15
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This produces features on configurations of the form:
edgewise shared partner distribution: pk
(y) =proportion of edges between actors with exactly k shared partnersk = 0, 1, . . .
⇥ ⇤9
2) Other conditional independence statistics
⇧ Pattison and Robins (2002), Butts (2005)
⇧ Snijders, Pattison, Robins and Handcock (2004)
– edges in Y that are not tied are conditionally
independent given the rest of the network
• k-triangle distribution: tk(y) = proportion of k-triangles in the graph y.
• edgewise shared partner distribution:
pk(y) = propotion of nodes with exactly k edgewise shared partners in y.
•
•• • • • •
i
j
h1 h2 h3 h4 h5....................................................................................................................................................
............................................................................................................................
..........................................................................................................................
..................................................................................................................................................
................................................................
..................................................................................................................................................................................................................
..............................................................................................................................................................................................................
...........................................................................................................
.........................................................................................................................................................................................................................................................................................................................
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.................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................
........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
k-triangle for k = 5, i.e., 5-triangle
⌅ Mark S. Handcock Statistical Modeling With ERGM ⇤
Figure: The actors in the non-directed (i , j) edge have 5 shared partners
dyadwise shared partner distribution:dsp
k
(y) = proportion of dyads with exactly k shared partnersk = 0, 1, . . .
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Structural Signatures
– identify social constructs or features– based on intuitive notions or partial appeal to substantive theory
Clusters of edges are often transitive:Recall triangle(y) is the number of triangles amongst triads
triangle(y) =1�g
3
�X
{i,j,k}2(g3
)
yij
yik
yjk
A closely related quantity is theproportion of triangles amongst two-stars
C (y) =3⇥triangle(y)
two�star(y)
mean clustering coe�cient
Figure:
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Structural Signatures
– identify social constructs or features– based on intuitive notions or partial appeal to substantive theory
Clusters of edges are often transitive:Recall triangle(y) is the number of triangles amongst triads
triangle(y) =1�g
3
�X
{i,j,k}2(g3
)
yij
yik
yjk
A closely related quantity is theproportion of triangles amongst two-stars
C (y) =3⇥triangle(y)
two�star(y)
mean clustering coe�cient
Figure:
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Structural Signatures
– identify social constructs or features– based on intuitive notions or partial appeal to substantive theory
Clusters of edges are often transitive:Recall triangle(y) is the number of triangles amongst triads
triangle(y) =1�g
3
�X
{i,j,k}2(g3
)
yij
yik
yjk
A closely related quantity is theproportion of triangles amongst two-stars
C (y) =3⇥triangle(y)
two�star(y)
mean clustering coe�cient
Figure:
![Page 22: Exponential-family random network models for social networks · Exponential-family random network models for social networks Mark S. Handcock Ian E. Fellows Department of Statistics](https://reader035.vdocument.in/reader035/viewer/2022062605/5fcf522d0cb2204d182d487e/html5/thumbnails/22.jpg)
Structural Signatures
– identify social constructs or features– based on intuitive notions or partial appeal to substantive theory
Clusters of edges are often transitive:Recall triangle(y) is the number of triangles amongst triads
triangle(y) =1�g
3
�X
{i,j,k}2(g3
)
yij
yik
yjk
A closely related quantity is theproportion of triangles amongst two-stars
C (y) =3⇥triangle(y)
two�star(y)
mean clustering coe�cient
!8
Classes of statistics used for modeling
1) Nodal Markov statistics ) Frank and Strauss (1986)
– motivated by notions of “symmetry” and “homogeneity”– edges in Y that do not share an actor are
conditionally independent given the rest of the network) analogous to nearest neighbor ideas in spatial statistics
• Degree distribution: dk
(y) = proportion of nodes of degree k in y.
• k-star distribution: sk
(y) = proportion of k-stars in the graph y.
• triangles: t1(y) = proportion of triangles in the graph y.
• •
•
i
j
h
....................................................................................................................................................................................................................................
..................................................................................................................
triangle= transitive triad
• •
•
j1 j2
i
..................................................................................................................
..................................................................................................................
two-star
• •••
j1 j2
i
j3
....................................................................................
..............................................................................................................................................................
three-star
( Mark S. Handcock Statistical Modeling With ERGM !
Figure:
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Exponential-family Random Network ModelsJoint modeling of Y and X
Let N be the sample space of Y ,X
Model the multivariate distribution of Y ,Xvia the form:
P⌘(Y = y ,X = x) =exp{⌘·g(y , x)}
c(⌘,N )y , x 2 N
⌘ 2 ⇤ ⇢ Rq q-vector of parameters
g(y , x) q-vector of network statistics.) g(Y ,X ) are jointly su�cient for the model
c(⌘,N ) distribution normalizing constant
c(⌘,N ) =
Z
y , x2Nexp{⌘·g(y , x)}·dP
0
(y , x)
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Interesting model-classes of ERNM
Relationship to ERGM and Random Fields
Let N (x) = {y : (x , y) 2 N} and N (y) = {x : (x , y) 2 N}
ERGM P(Y = y |X = x ; ⌘) =1
c(⌘; x)e⌘·h(x,y) y 2 N (x)
Gibbs measure P(X = x |Y = y ; ⌘) =1
c(⌘; y)e⌘·h(x,y) x 2 N (y)
The first model is the ERGM for the network conditional on thenodal attributes.
The second model is an exponential-family for the field of nodalattributes conditional on the network.
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Relationship with ERGM
The model can be expressed as
P(X = x ,Y = y |⌘) = P(Y = y |X = x |⌘)P(X = x |⌘)where
P(Y = y |X = x ; ⌘) =1
c(⌘; x)e⌘·h(x,y) y 2 N (x)
P(X = x |⌘) =c(⌘; x)
c(⌘,N )x 2 X
The first sub-model is the ERGM for the network conditional on thenodal attributes.
The second sub-model is the marginal representation of the nodalattributes and is not necessarily an exponential-family with canonicalparameter ⌘.
This decomposition makes it clear why the conditional modeling ofY given X via ERGM di↵ers from the joint modeling of Y and Xvia ERNM.
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Separable ERGM and Field Models
Suppose the model can be expressed as
P(X = x ,Y = y |⌘1
, ⌘2
) =1
c(⌘1
, ⌘2
,N )e⌘1
·h(x)+⌘2
·g(y) (y , x) 2 N . (1)
where N = Y ⇥ X . Then
P(X = x |⌘1
) =1
c1
(⌘1
,X )e⌘1
·h(x)
P(Y = y |⌘2
) =1
c2
(⌘2
,Y)e⌘2
·g(y).
The first sub-model is a general exponential-family model for theattributes (e.g., generalized linear models)
The second sub-model is an ERGM for the graph that has nodependence on the nodal attributes.
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Example: Joint Ising Models
Suppose X is univariate and binary xi
2 {�1, 1}. One measure ofhomophily on x is
homophily(y , x) =nX
i=1
nX
j=1
xi
yi,jxj (2)
A simple model for the network is
P(X = x ,Y = y |⌘1
, ⌘2
) / e⌘1
density(y)+⌘2
homophily(y ,x) (y , x) 2 N .
where density(y) = 1
n
Pi
Pj
yi,j
GLM P(Yi,j = y
i,j |X = x , ⌘1
, ⌘2
) / e⌘1
1
n
y
i,j+⌘2
x
i
y
i,j xj y 2 {0, 1}, x 2 XIsing model P(X = x |Y = y , ⌘
2
) / e⌘2
Pi
Pj
x
i
y
i,j xj (y , x) 2 N
So we have a simple joint Ising model
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Model specification issues: Degeneracy
Joint Ising model: n = 20 with moderate homophily (mean-value=0.76):
# of edges within x = 1
Count
Frequency
0 50 100 150 200
020000
# of edges within x = -1
Count
Frequency
0 50 100 150 200
020000
# of edges between x = 1 and x = -1
Count
Frequency
0 20 40 60 80 100 120
06000
# of nodes with x = 1
Count
Frequency
0 5 10 15 20
04000
Figure: 100,000 draws from an Ising homophily network modelwith ⌘
1
= 0 and ⌘2
= 0.13. Mean values are marked in red.
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Model specification issues: Degeneracy
So standard homophily in ERGM leads to degeneracy in ERNM
Solution: regularized homophily
Suppose x is categorical with category labels 1, . . . ,K .
homophily
k,l(y , x) =nX
i=1
nX
j=1
I (xi
= k)yi,j I (xj = l). (3)
Let di,k(y , x) =
Pi<j
yij
I (xj
= k) be the number of edgesconnecting node i to nodes in category k .
rhomophily
k,l(y , x) =X
i :x
i
=k
qdi,l(y , x)� E??(
qdi,l(Y ,X )) (4)
where E??(·) is the expectation conditional on the number of nodes ineach category of x under the assumption that X and Y are independent.
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What happens when we fit the model with our new homophily?
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Comparison of Homophily Statistics
# of edges within group x = -1 # of edges within group x = 1
# of edges from x=-1 to x=1 # of edges from x=1 to x=-1
Regularized Homophily # of nodes with x = 1
0.00
0.01
0.02
0.03
0.00
0.01
0.02
0.03
0.00
0.02
0.04
0.06
0.08
0.00
0.02
0.04
0.06
0.08
0.0
0.1
0.2
0.3
0.4
0.0
0.2
0.4
0.6
0 50 100 150 200 0 50 100 150 200
0 20 40 60 0 20 40 60
0 10 5 10 15value
density
Model
Ising
Regularized Homophily
Figure:
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Fitting Models to Partially Observed Social Network Data
Focus on the joint distribution of Z = (Y ,X ).
Two types of data:Observed relations and covariates (z
obs
= (yobs
, xobs
)),and information about the observation mechanism (D)(e.g., indicators of relations and covariates sampled.
L(⌘, ) ⌘ P(Zobs
= zobs
,D|⌘, )
=X
z
unobs
P(Zobs
= zobs
,Zunobs
= zunobs
,D|⌘, )
=X
z
unobs
P(D|Zobs
= zobs
,Zunobs
= zunobs
, )P⌘(Zobs
= zobs
,Zunobs
= zunobs
)
⌘ is the model parameter
is the sampling parameter
When can we “ignore” the sampling process?
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Fitting Models to Partially Observed Social Network Data
Focus on the joint distribution of Z = (Y ,X ).
Two types of data:Observed relations and covariates (z
obs
= (yobs
, xobs
)),and information about the observation mechanism (D)(e.g., indicators of relations and covariates sampled.
L(⌘, ) ⌘ P(Zobs
= zobs
,D|⌘, )
=X
z
unobs
P(Zobs
= zobs
,Zunobs
= zunobs
,D|⌘, )
=X
z
unobs
P(D|Zobs
= zobs
,Zunobs
= zunobs
, )P⌘(Zobs
= zobs
,Zunobs
= zunobs
)
⌘ is the model parameter
is the sampling parameter
When can we “ignore” the sampling process?
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Fitting Models to Partially Observed Social Network Data
Focus on the joint distribution of Z = (Y ,X ).
Two types of data:Observed relations and covariates (z
obs
= (yobs
, xobs
)),and information about the observation mechanism (D)(e.g., indicators of relations and covariates sampled.
L(⌘, ) ⌘ P(Zobs
= zobs
,D|⌘, )
=X
z
unobs
P(Zobs
= zobs
,Zunobs
= zunobs
,D|⌘, )
=X
z
unobs
P(D|Zobs
= zobs
,Zunobs
= zunobs
, )P⌘(Zobs
= zobs
,Zunobs
= zunobs
)
⌘ is the model parameter
is the sampling parameter
When can we “ignore” the sampling process?
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Adaptive Sampling Designs
We call a sampling design adaptive if:
P(D = d |Zobs
,Zmis
, ) = P(D = d |Zobs
, ) 8z 2 Z.
that is, it uses information collected during the survey to directsubsequent sampling, but the sampling design depends only on theobserved data.
adaptive sampling designs satisfy a condition called “missing atrandom” by Rubin (1976) in the context of missing data.
Result: standard network sampling designs such as conventional,single wave and multi-wave link-tracing sampling designs areadaptive
) Thompson and Frank (2000), Handcock and Gile (2007).
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Adaptive Sampling Designs
We call a sampling design adaptive if:
P(D = d |Zobs
,Zmis
, ) = P(D = d |Zobs
, ) 8z 2 Z.
that is, it uses information collected during the survey to directsubsequent sampling, but the sampling design depends only on theobserved data.
adaptive sampling designs satisfy a condition called “missing atrandom” by Rubin (1976) in the context of missing data.
Result: standard network sampling designs such as conventional,single wave and multi-wave link-tracing sampling designs areadaptive
) Thompson and Frank (2000), Handcock and Gile (2007).
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When is sampling adaptive?
Examples of adaptive sampling:
Individual sample, sample based on observed things like race, sex,and age that we know.
Link-tracing sample starting with an adaptive sample with follow-upbased on observed relations with others in the sample, as well asthings like race and sex and age.
Link-tracing with probability proportional to number of partners isadaptive!
Examples of non-adaptive (not missing at random) sampling:
Individual sample based on unobserved properties ofnon-respondents - like infection status or illicit activity.
Link-tracing sample starting where links are followed dependent onunobserved properties of alters.
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Adaptive Sampling Designs and their Amenable ModelsDefinition: Consider a sampling design governed by parameter 2 and a stochastic network model P⌘(Y = y ,X = x) governed byparameter ⌘ 2 ⌅. We call the sampling design amenable to the model ifthe sampling design is adaptive and the parameters and ⌘ are distinct.
Result: If the sampling design is amenable to the model the likelihoodfor ⌘ and is
L[⌘, |Zobs
= zobs
,D = d ] / L[ |D = d ,Zobs
= zobs
]L[⌘|Zobs
= zobs
]
sampling design likelihood⇥face-value likelihood
sampling L[ |D = d ,Zobs
= zobs
] = P(D|Zobs
= zobs
)
network L[⌘|Zobs
= zobs
] =X
z
unobs
P⌘(Zobs
= zobs
,Zunobs
= zunobs
)
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Adaptive Sampling Designs and their Amenable ModelsDefinition: Consider a sampling design governed by parameter 2 and a stochastic network model P⌘(Y = y ,X = x) governed byparameter ⌘ 2 ⌅. We call the sampling design amenable to the model ifthe sampling design is adaptive and the parameters and ⌘ are distinct.
Result: If the sampling design is amenable to the model the likelihoodfor ⌘ and is
L[⌘, |Zobs
= zobs
,D = d ] / L[ |D = d ,Zobs
= zobs
]L[⌘|Zobs
= zobs
]
sampling design likelihood⇥face-value likelihood
sampling L[ |D = d ,Zobs
= zobs
] = P(D|Zobs
= zobs
)
network L[⌘|Zobs
= zobs
] =X
z
unobs
P⌘(Zobs
= zobs
,Zunobs
= zunobs
)
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Adaptive Sampling Designs and their Amenable Models
Result: If the sampling design is not amenable to the model thelikelihood for ⌘ and is
L(⌘, ) =X
z
unobs
P(D|Zobs
= zobs
,Zunobs
= zunobs
, )P⌘(Zobs
= zobs
,Zunobs
= zunobs
)
and the design will need to be represented.
Clearly P(D|Y ,X , ) can be modeled when it is unknown.
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Likelihood-based inference for ERNMwhen partially observed
Consider the conditional distribution of T given Tobs
:
P⌘(Tunobs
= t|Tobs
= tobs
) = exp [⌘·g(t + tobs
)� (⌘|tobs
)] t 2 T (tobs
)
where T (tobs
) = {t : t + tobs
2 T }
(⌘|tobs
) = logX
u2T (t
obs
)
exp [⌘·g(u + tobs
)].
Note thatL[⌘|T
obs
= tobs
] / exp [(⌘|tobs
)� (⌘)]
which can then be estimated by MCMC:
the first term by a chain on the complete data over T and;
the second by a chain conditional on tobs
over T (tobs
).
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Likelihood-based inference for ERNMwhen partially observed
Consider the conditional distribution of T given Tobs
:
P⌘(Tunobs
= t|Tobs
= tobs
) = exp [⌘·g(t + tobs
)� (⌘|tobs
)] t 2 T (tobs
)
where T (tobs
) = {t : t + tobs
2 T }
(⌘|tobs
) = logX
u2T (t
obs
)
exp [⌘·g(u + tobs
)].
Note thatL[⌘|T
obs
= tobs
] / exp [(⌘|tobs
)� (⌘)]
which can then be estimated by MCMC:
the first term by a chain on the complete data over T and;
the second by a chain conditional on tobs
over T (tobs
).
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Likelihood-based inference for ERNMwhen partially observed
In general, the observed data log likelihood ratio of (⌘, ✓) versus (⌘0
, ✓0
) is
`(⌘, ✓)� `(⌘0
, ✓0
) = log(c(t
obs
,w , ⌘, ✓)
c(tobs
,w , ⌘0
, ✓0
))� log(
c(⌘, T )
c(⌘0
, T ))
= log(X
t
miss
p(W = w |T = t, ✓)
p(W = w |T = t, ✓0
)e(⌘�⌘
0
)·g(t) p(W = w |T = t, ✓0
)e⌘0
·g(t)
c(tobs
,w , ⌘0
, ✓0
))
� log(X
t
miss
e(⌘�⌘0
)·g(t) e⌘0
·g(t)
c(⌘, T ))
= log(E⌘0
,✓0
(p(W = w |T , ✓)
p(W = w |T , ✓0
)e(⌘�⌘
0
)·g(T ))|W = w ,Tobs
= tobs
) (5)
� log(E⌘0
(e(⌘�⌘0
)·g(T )))
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Likelihood-based inference for ERNMwhen partially observed
In general, the observed data log likelihood ratio of (⌘, ✓) versus (⌘0
, ✓0
)may be approximated by
`(⌘, ✓)� `(⌘0
, ✓0
)
⇡ log(1
M
MX
i
p(w |t(i)m
, ✓)
p(w |t(i)m
, ✓0
)e(⌘�⌘
0
)·g(t(i)m
))� log(1
M
MX
i
e(⌘�⌘0
)·g(t(i))).
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Likelihood-based inference for ERNMwhen partially observed
ERGM case implemented in R package statnet (Handcock et al 2003)(http://statnet.org).
ERNM case implemented in ernm package (Fellows 2012).
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Adolescent Peer Networks
The National Longitudinal Study of Adolescent Health) www.cpc.unc.edu/projects/addhealth
– “Add Health” is a school-based study of the health-relatedbehaviors of adolescents in grades 7 to 12.
Each nominated up to 5 boys and 5 girls as their friends
160 schools: Smallest has 69 adolescents in grades 7–12
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Adolescent Peer Networks
The National Longitudinal Study of Adolescent Health) www.cpc.unc.edu/projects/addhealth
– “Add Health” is a school-based study of the health-relatedbehaviors of adolescents in grades 7 to 12.
Each nominated up to 5 boys and 5 girls as their friends
160 schools: Smallest has 69 adolescents in grades 7–12
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−10 −5 0 5 10
−10
−50
510
12
7 9
10
9
8
10
11
7
8
11
8
10
8
8
10
97
8
8
11
8
99
7
11
9
10
8
11
7
9
11
11
11
10
10
9
9
7
10
10
7
7 9
9
1111
8
12
9
9
10
7
7
9
7
11
9
7
12
7
8
9
11
11
7
8
12
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White (non-Hispanic)Grade 7
Black (non-Hispanic)
Hispanic (of any race)Asian / Native Am / Other (non-Hispanic)
Race NA
Grade 8
Grade 9
Grade 10
Grade 11
Grade 12
Grade NA
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Application to substance use in adolescent peer networksData: The National Longitudinal Study of Adolescent Health– Model friendship network as a function of student characteristics
Form Name DefinitionY Mean Degree Average degree of studentsY Log Variance of Degree The log variance of the student degreesY In Degree = 0 # of students with in degree 0Y In Degree = 1 # of students with in degree 1Y Out Degree = 0 # of students with out degree 0Y Out Degree = 1 # of students with out degree 1Y Reciprocity # of reciprocated tiesX Grade = 9 # of freshmenX Grade = 10 # of sophomoresX Grade = 11 # of juniorsX ,Y Within Grade Homophily Pooled homophily within gradeX ,Y +1 Grade Homophily Pooled homophily between each grade
and the grade above it
Table: Model Terms
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Application to substance use in adolescent peer networks
⌘ Std. Error Z p�valueMean Degree -217.02 7.81 -27.80 <0.001
Log Variance of degree 25.07 9.06 2.77 0.006In-Degree 0 2.62 0.50 5.20 <0.001In-Degree 1 1.05 0.40 2.62 0.009
Out-Degree 0 4.09 0.52 7.91 <0.001Out-Degree 1 1.93 0.45 4.25 <0.001
Reciprocity 2.71 0.23 11.77 <0.001Grade = 9 1.46 0.62 2.37 0.018Grade = 10 1.93 0.71 2.72 0.007Grade = 11 2.08 0.59 3.54 <0.001
Grade Homophily 4.34 0.46 9.41 <0.001+1 Grade Homophily 0.63 0.21 2.98 0.003
Table: ERNM Model with Standard Errors Based on the Fisher Information
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Application to substance use in adolescent peer networks
We see that students in the same grade are much more likely to befriends
The positive coe�cient for ’+1 Grade Homophily’ indicates thatstudents also tend to form connections to the grades just below orjust above them.
Evaluating the goodness-of-fit?Simulate networks from the fitted model, and visually compare themto the observed network (Hunter, Goodreau, Handcock 2008).Simulate network statistics from the model and compare them to theobserved network.
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Model-Based Simulated High School
Simulated Network
9
11
9
9
12
9
9
10 11
12
10
11
12
10
1212
12
9
10
10
10
9 1111
11
11
12
9
10
12
11
10
9
11
1111
1112
11
9
12
910
9119
9
9
9
12
9
9
12
9
910
11
10
10
9
11
9
12
12
10
10
10
10
10
10 11
10
11
9
Observed Network
11
11
10
9
109
9
12 10
101210
11
12
99
10
99
11
910
11
99
9
10
9
10
10
1112 11
9
10
9
11
11
11
1111
11
11
10
11
9
10
11
11
12
11
9
9
11
11
1011
1012
9
12
9
10
10
10
9
9
1212
12
11
12
10
12
![Page 54: Exponential-family random network models for social networks · Exponential-family random network models for social networks Mark S. Handcock Ian E. Fellows Department of Statistics](https://reader035.vdocument.in/reader035/viewer/2022062605/5fcf522d0cb2204d182d487e/html5/thumbnails/54.jpg)
Model Diagnostics: Goodness-of-fit
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
0
5
10
15
20
25
30
In-Degree
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
0
5
10
15
20
25
30
Out-Degree
10-10
11-10
12-10
9-10
10-11
11-11
12-11
9-11
10-12
11-12
12-12
9-12
10-9
11-9
12-9 9-9
0
20
40
60
80
100
120
# of Edges Between Grades
9 10 11 12
5
10
15
20
25
30
Grade Counts
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Network Regression with endogenous nodal attributes
Let Z 2 {0, 1} be a binary outcome variable (e.g., substance use) andconsider:
P(Z = z ,X = x ,Y = y |⌘,�,�) = 1
c(�, ⌘,�)ez·x�+⌘·g(x,y)+�·h(z,y). (6)
So, conditional on Y = y ,�,�:
logodds(zi
= 1|z�i
,Xi
= xi
)� logodds(zi
= 1|z�i
,Xi
= x⇤i
)
= �(xi
� x⇤i
)
where z�i
represents the set of z not including zi
, xi
represents the ithrow of X .
So � have their usual interpretations (conditional on the rest of thenetwork)
The usual independence assumptions do not hold
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Simple logistic regression of substance use on gender
What is the relationship between gender and substance use?
Logistic regression of substance use on gender:
� Std. Error Z p�valueIntercept -1.70 0.44 -3.84 <0.001Gender 1.18 0.57 2.09 0.037
Table: Simple Logistic Regression Model Ignoring Network Structure
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Logistic Regression using Network Data
Bootstrap Asymptotic⌘ Std. Error Std. Error Z p�value
Mean Degree -215.50 8.32 8.15 -26.44 <0.001Log Variance of degree 24.46 8.80 8.91 2.75 0.006
In-Degree 0 2.68 0.55 0.48 5.55 <0.001In-Degree 1 1.07 0.43 0.41 2.60 0.009
Out-Degree 0 4.15 0.54 0.52 8.03 <0.001Out-Degree 1 1.94 0.50 0.45 4.31 <0.001
Reciprocity 2.71 0.25 0.23 11.96 <0.001Grade Homophily 4.28 0.44 0.47 9.18 <0.001
+1 Grade Homophily 0.62 0.21 0.21 2.99 0.003Gender Homophily 0.78 0.24 0.24 3.27 0.001
Substance Homophily 0.76 0.25 0.25 3.02 0.003Intercept -1.72 0.50 0.44 -3.91 <0.001Gender 0.92 0.55 0.51 1.79 0.073
Table: ERNM Model Inference
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Model diagnostics for network regression
# of edges within substance categories
Count
Frequency
100 150 200 250 300 350
0100
# of edges between users and non-users
Count
Frequency
50 100 1500
100
# of non-substance users
Count
Frequency
45 50 55 60
0150
Figure: Substance Use Homophily Diagnostics.The values of the observed statistics are marked in red.
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Model diagnostics for network regression
# of edges within substance categories
Count
Frequency
100 150 200 250 300 350
0100
# of edges between users and non-users
Count
Frequency
50 100 1500
100
# of non-substance users
Count
Frequency
45 50 55 60
0150
Figure: Substance Use Homophily Diagnostics.The values of the observed statistics are marked in red.
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Latent Class Modeling using ERNM
We observe the relational ties Y
Postulate the existence of a categorical nodal covariate X
Build an ERNM model based on complex statistics
Treat all X values as missing
A new variant of the stochastic block model (Wang and Wong 1987)
![Page 61: Exponential-family random network models for social networks · Exponential-family random network models for social networks Mark S. Handcock Ian E. Fellows Department of Statistics](https://reader035.vdocument.in/reader035/viewer/2022062605/5fcf522d0cb2204d182d487e/html5/thumbnails/61.jpg)
Latent Class Modeling using ERNM
We observe the relational ties Y
Postulate the existence of a categorical nodal covariate X
Build an ERNM model based on complex statistics
Treat all X values as missing
A new variant of the stochastic block model (Wang and Wong 1987)
![Page 62: Exponential-family random network models for social networks · Exponential-family random network models for social networks Mark S. Handcock Ian E. Fellows Department of Statistics](https://reader035.vdocument.in/reader035/viewer/2022062605/5fcf522d0cb2204d182d487e/html5/thumbnails/62.jpg)
Latent Class Modeling using ERNMExample: Latent Cluster Model of Sampson’s Monks
Expressed “liking” between 18 monks within an isolated monastery) Sampson (1969)
A directed relationship aggregated over a 12 month period before thebreakup of the cloister.
Sampson identified three groups plus:(T)urks, (L)oyal Opposition, (O)utcasts and (W)averers
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Latent Class Modeling using ERNM
We observe the relational ties Y
Postulate the existence of a categorical nodal covariate X
Assume the sample space of X has K = 18 categories
Build a model based on count, density and relative homophily terms
Treat all X values as missing using the above ideas
Term ⌘ µ s.e.(⌘) s.e.(µ)# of edges -0.58 88.23 0.14 7.48Homophily 7.28 15.30 0.91 1.33# in group 2 -0.02 6.95 1.31 0.99# in group 3 -2.50 3.95 1.44 1.08
Table: Latent class model of Sampson’s monks.
p(X |Y = yobs
, ⌘) assign each monk with probability one to thecorrect clusters.
Almost all mass goes on three groups.
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Conclusions
Exponential-family random network models area powerful new way to model network data
Leads to a new approach to network regression
Leads to a new approach to latent variable modeling
Model specification and development are extensions of that forERGM, but require new perspectives