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Electronic copy available at: http://ssrn.com/abstract=1960262
063/"-0'"3,&5*/(&4&"3$)
Vol. XLVIII (August 2011), 713 –728
*Asim Ansari is the William T. Dillard Professor of Marketing (e-mail:
maa48@ columbia.edu), and Oded Koenigsberg is Barbara & Meyer Feld-
berg Associate Professor (e-mail: ok2018@columbia.ed), Columbia Busi-
ness School, Columbia University. Florian Stahl is Assistant Professor,
Department of Business Economics, University of Zurich (e-mail: florian.
stahl@ uzh.ch). Christophe Van den Bulte served as associate editor for this
article.
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© 2011, American Marketing Association
ISSN: 0022-2437 (print), 1547-7193 (electronic) 713
The rapid growth of online social networks has led to a
resurgence of interest in the marketing field in studying the
structure and function of social networks. A better under-
standing of social networks can enable managers to compre-
hend and predict economic outcomes (Granovetter 1985)
and, in particular, to interface with both external and internal
actors. Online brand communities, which are composed of
users interested in particular products or brands, allow such
an external interface with customers. Such communities not
only help firms interact with customers and prospects but
also enable customers to communicate and exchange infor-
mation with each other, consequently increasing the value
that can be derived from a firm’s products.
Similarly, firms forge alliances and enter into collabora-
tive relationships with other firms for coproduction and
social commerce (Stephen and Toubia 2010; Van den Bulte
and Wuyts 2007) using interorganizational networks.
Within the firm, intraorganizational networks of managers
play a crucial role in cross-functional integration, as is the
case with networks of marketing and organizational profes-
sionals engaged in new product development (Van den
Bulte and Moenaert 1998).
As Van den Bulte and Wuyts (2007) point out, network
structure has implications for power, knowledge dissemina-
tion, and innovation within firms and for contagion and dif-
fusion among customers. Thus, understanding and predict-
ing the patterns of interactions and relationships among
network members is an important first step in using them
effectively for marketing purposes. The focus of social net-
work analysis is on (1) explaining the determinants of rela-
tionship formation; (2) identifying well connected actors;
and (3) capturing structural characteristics of the network as
described by reciprocity, clustering, transitivity, and other
measures of local and global structure using a combination
of assortative, relational, and proximity perspectives (Rivera,
Soderstrom, and Uzzi 2010).
Actors belonging to a social network connect with each
other using multiple relationships, possibly of different
Reprinted with permission from the Journal of Marketing Research, published by the American Marketing Association,
Asim Ansari, Oded Koenigsberg, and Florian Stahl, vol. 48, no. 4, August 2011, pp. 713-728.
Electronic copy available at: http://ssrn.com/abstract=1960262
types. In this article, we develop statistical models of multi-
ple relationships that yield an understanding of the drivers
of multiplex relationships and predict the connectivity
structure of such multiplex networks.
Multiplexity of relationships can arise from different
modes of interaction or because of different roles people
play within a network setting. For example, in many online
networks, members can form explicit friendship and busi-
ness relations, exchange content and communicate with one
another. The relationships that connect a group of actors can
differ not only in their substantive content but also in their
directionality and intensity. For example, some relation-
ships are symmetric in nature, whereas others can be
directed. Some relationships involve the flow of resources,
thus necessitating a focus on the intensity of such weighed
connections. Finally, multivariate patterns of connections
can also arise from viewing the same relationship at differ-
ent points, as is the case of sequential networks.
An understanding of multiplex patterns in network struc-
tures is important for marketers. For example, Tuli, Bharad-
waj, and Kohli (2010) find that in a business-to-business
setting, increasing multiplexity in relationships leads to an
increase in sales and to a decrease in sales volatility to a
customer. Multiplexity contributes to the total strength of a
tie and increases the number of ways one can reciprocate
favors (Van den Bulte and Wuyts 2007). Thus, it is relevant
for identifying influential actors such as opinion leaders in
diffusion contexts and powerful executives within intra-
organizational networks. In analyzing sequential relation-
ships, a multivariate analysis can help investigate the impact
of managerial interventions on the relationship structure of
a network across points. Sequential dyadic interactions are
also useful for understanding the dynamics of power and
cooperation in intraorganizational networks and in model-
ing long-term relationships between buyers and sellers in
business markets (Iacobucci and Hopkins 1992). Finally,
when marketers are interested in predicting relationship pat-
terns, multiplexity allows leveraging information from one
relationship to predict connections on other relationships.
Researchers can obtain a substantive understanding of
multiplex relationships by simultaneously analyzing the
multiple connections among the network actors. They can
investigate whether these multiple relationships exhibit
multiplex patterns that are characterized by the flow of mul-
tiple relationships in the same direction or whether they rep-
resent patterns of generalized exchange in which a tie in one
direction on one relationship is reciprocated with a connec-
tion in the other direction using different relationships.
However, most models of social networks analyze a single
relationship among network members. When the lens is
trained on a single relationship, an incomplete understand-
ing of the nature of linkages can result. For example, it is
unclear whether people play a similar role across multiple
relationships. A joint analysis can also help uncover com-
mon antecedents that affect relationships. Moreover, if
some relationships exhibit unique patterns, such uniqueness
can emerge only when multiple relationships are contrasted.
In this article, we develop an integrated latent-variable
framework for modeling multiple relationships. We make a
methodological contribution to the social networking litera-
ture in both marketing and the wider social sciences by
offering a rich framework for modeling multiple relation-
ships of different types. Our modeling framework has sev-
eral novel features when compared with previously pro-
posed models for multirelational network data. Specifically,
our framework can (1) model multiple relationships of dif-
ferent types (i.e., weighted, unweighted, undirected, and
directed), (2) model sequential relationships, (3) leverage
partial information from one network (or relationship) to
predict connectivity in another relationship, (4) accommo-
date missing data in a natural way, (5) capture sparseness in
weighted relationships, (6) incorporate sources of dyadic
dependence, (7) account for higher order effects such as tri-
adic effects, and (8) include continuous covariates. Although
previous models have incorporated some of these aspects,
we do not know of any research in the social network analy-
sis literature that simultaneously incorporates all of them.
We illustrate the benefits of our approach using two
applications. Our first application involves sequential net-
work data that studies network structure over two points.
We reanalyze data from Van den Bulte and Moenaert (1998)
involving a network of research-and-development (R&D),
marketing, and operations managers who are engaged in
new product development. The data contain communica-
tions among these managers both before and after colloca-
tion of R&D teams. The results show that substantive con-
clusions can be affected if the full generality of our
framework is not utilized. We also show how our methods
can be used to leverage information from one relationship
to predict the connections in another relationship.
In our second application, we use data from an online
social networking site involving the interactions among a
set of musicians. We model friendship, communications,
and music download relationships within this network to
show how a combination of directed and undirected, and
weighted and unweighted, relationships can be modeled
jointly. We analyze the determinants of these relationships
and assess the importance of our model components in cap-
turing different facets of the network structure. Our results
show that artists exhibit similar network roles across the
three relationships and that these relationships are mostly
influenced by common antecedents. We also show that
when dealing with weighted relationships (e.g., music
downloads), it is crucial to jointly model both the incidence
and the intensity of such relationships, rather than simply
focusing on the intensity; otherwise, prediction and recov-
ery of structural characteristics suffers appreciably.
We organize the rest of the article as follows: The next
section provides a brief review of the marketing and statisti-
cal literature on social networks. Then, we present the com-
ponents of our modeling framework and describe inference
and identification of model parameters. The following two
sections describe the two applications. Finally, we conclude
with a discussion of our contributions and model limitations
and outline future research possibilities.
Social network data offer considerable opportunities for
research in marketing, as Van den Bulte and Wuyts (2007)
identify in their expansive survey of the role and importance
of social networks in the marketing field. Most research in
marketing on social networks falls into one of two streams.
In the first stream, researchers explore the impact of word
of mouth on the behavior of others and thus are primarily
# " ! ##!"
Electronic copy available at: http://ssrn.com/abstract=1960262
5*+2/4-;2:/62+ +2':/549./69/4!5)/'2+:=5819
concerned about the role of social interactions and conta-
gion (Iyengar, Van den Bulte, and Valente 2011; Nair, Man-
chanda, and Bhatia 2010; Trusov, Bodapati, and Bucklin
2010; Watts and Dodds 2007).
Research in the second stream focuses on modeling net-
work structure. Iacobucci and Hopkins’s (1992) work is a
pioneering contribution in this area. The focus here is on
understanding the antecedents of relationship formation and
in studying how interventions influence future connectivity
(Van den Bulte and Moenaert 1998). The current study adds
to this second stream of research by offering a comprehen-
sive framework for modeling multivariate or sequential
relations.
There is a rich history of statistical modeling of network
structure within sociology and statistics that spans more
than 70 years. More recent approaches stem from the log-
linear p1model developed by Holland and Leinhardt (1981),
which assumes independent dyads. Because the p1model is
incapable of representing many structural properties of the
data, the literature has proposed two general ways of cap-
turing the dependence among the relationships. The first
approach uses exponential random graph models or p*
models (Frank and Strauss 1986; Pattison and Wasserman
1999; Robins et al. 2007; Snijders et al. 2006; Wasserman
and Pattison 1996) that capture the dependence pattern in
the network using a set of statistics that embody important
structural characteristics of the network. However, care is
necessary when using these models because parameter esti-
mation sometimes suffers from model degeneracy, and how
to handle this degeneracy is an active area of research. The
second approach handles the dependence among the dyads
using correlated random effects and latent positions in a
Euclidean space for the individual people (Handcock,
Raftery, and Tantrum 2007; Hoff 2005; Hoff, Raftery, and
Handcock, 2002 ). In addition to these two approaches, mul-
tiple regression quadratic assignment procedure (MRQAP)
methods (Dekker, Krackhardt, and Snijders 2007) have also
been used in network analysis to account for dependence
among dyads.
Exponential random graph models describe the network
using a set of summary statistics, such as the total number
of ties, the number of triangles, and the degree of distribu-
tion, among others. This is good for describing “global”
properties and in assessing particular hypotheses of substan-
tive interest, such as the extent of reciprocity or clustering
and triadic closure. In contrast, latent space models capture
the “local” structure by estimating a latent variable for each
node in the network, which describes a person’s position in
the network. These models are thus suitable when the focus
is on understanding the determinants of connectivity using
covariates and in identifying influential people. The latent
variable framework is capable of recovering the structural
characteristics using a small set of model parameters (simi-
lar to nuisance parameters) and thus can be parsimonious in
some contexts.
When researchers are interested in specific substantive
hypotheses and when all relationships are binary in nature,
they may prefer exponential random graph models. How-
ever, the latent variable framework can accommodate mul-
tiplex relationships of different types, including weighted
relationships, and can also handle missing data in a straight-
forward fashion using data augmentation. Thus, it is prefer-
able when interest is in analyzing such complex multivari-
ate data structures.
Whereas the preceding methods explicitly model the net-
work structure, MRQAP methods offer a nonparametric
alternative for conducting permutation tests to assess
covariate effects using multiple regression while correcting
for the dependency and autocorrelation present in network
data. The MRQAP approach is useful for continuous data. It
can be used for binary relations using a linear probability
model, and to a certain extent for count data; however, its
effectiveness for multivariate relations of different types is
not clear.
Sequential data can also be modeled using two
approaches: a multivariate approach such as ours and the
conditional, continuous-time approach popularized by Sni-
jders (2005). The multivariate approach models the network
at each point in time and thus is useful when one is inter-
ested in assessing the impact of interventions that occur
between these discrete times. In contrast, the continuous-
time approach is inherently dynamic, focusing on either
edge-oriented or node-oriented dynamics, and can model
the evolution of the network one edge at a time. However,
this approach is limited to binary relations, whereas the
multivariate approach that we use can handle both weighted
and binary relationships.
In contrast to the current study, most models of social net-
work structure analyze a single relationship, and to the best
of our knowledge, none have incorporated the entire con-
stellation of desirable model characteristics that we outlined
in the introduction. In particular, there has been no work on
using the latent space framework for modeling multivariate
relationships or sequential data.
Although some researchers have modeled multiple rela-
tionships, these models either assume independence across
dyads, which is restrictive, or limit attention to binary rela-
tions (Fienberg, Meyer and Wasserman 1985; Iacobucci
1989; Iacobucci and Wasserman 1987; Pattison and Wasser-
man 1999; Van den Bulte and Moenaert 1998). Thus, there
is a need for an integrative framework for modeling multi-
ple relationships of different types (i.e., binary or weighted)
in a flexible way. The latent space framework offers such
flexibility, and using it, we develop an integrated approach
for multiple relationships in the following section.
We develop a modeling framework for the simultaneous
analysis of multiple relationships among a set of network
actors. Our framework accommodates multiple relation-
ships of different types and also enables us to simultane-
ously model the determinants of the relationships as well as
the structural characteristics such as the extent of reci-
procity or transitivity within each relationship and across
relationships. When analyzing multiple relationships, struc-
tural characteristics of interest include those that account for
.6-5*1-&9 patterns (i.e., flow of multiple relationships in the
same direction) and &9$)"/(&, in which a flow in one direc-
tion on one relationship is reciprocated with a flow in the
other direction using a different relationship. Similarly, pat-
terns of transitivity that involve more than one relationship
can also be investigated. When focusing on the determi-
nants of relationships, we can infer how the attributes of the
network actors influence the formation of relationships
between them. Here the interest is in understanding whether
actors exhibit similar popularity and expansiveness across
different relationships, and whether homophily governs
relationship formation.
The multiple relationships describing a common set of
actors can vary along different facets, such as existence,
intensity, and directionality. A relationship is directed if we
can distinguish the sender and receiver of the tie. For exam-
ple, a communication relationship typically has a sender and
a receiver. In contrast, relationships could be undirected,
such as a collaboration relationship. In modeling both
directed and undirected relationships, the focus of the analy-
sis could be on modeling the existence of a relationship (i.e.,
the presence or absence of a tie) or on the intensity of a
weighted tie (e.g., the intensity of the flow of resources
between a pair of people). Our objective is to show how
such disparate relationships can be jointly modeled within a
common framework. In the following section, we describe
formally our model.
0%&-&4$3*15*0/
We describe our model using two relationships. Although
these two relationships could represent a single relationship
observed over different time periods, for the sake of gener-
ality, we describe a model for two distinct, directed relation-
ships of different types.1 These two relationships are
observed over the same set of n actors.
*3&$5 &% #*/"3: 3&-"5*0/4 )*1. The first relationship is
directed and binary. Thus, we can distinguish between the
sender and the receiver of the tie and the sociomatrix
matrix, X1, which shows that the incidence of ties among
actors can, therefore, be asymmetric. We use the ordered
pair of binary dependent variables {Xij1, Xji1} to represent
the presence or absence of ties for a pair of actors i and j.
The variable Xij1 specifies the existence of interaction in the
direction from i to j (i.e., i Æj), whereas Xji1 represents the
presence of a tie in the opposite direction from j to i (i.e.,
i ¨j).
*3&$5&%8&*()5&%3&-"5*0/4)*1. The second relationship
is directed and weighted (i.e., valued). The entries in the
associated matrix, X2, indicate the bidirectional intensity of
interaction between the different pairs of people. Here, we
assume a count variable for the intensity, because this is
consistent with our second application presented in the sec-
tion “Online Social Network.” However, our model can be
adapted for continuous or ordinal measures of intensity. An
ordered pair of count variables (Xij2, Xji2) can represent the
observed intensity of interaction in the dyad, where the
variable Xij2 specifies the strength of the interaction from i
to j (i.e., i Æj), and Xji2 specifies the intensity in the reverse
direction.
In modeling this weighted relationship, we deviate from
the previous literature on social networks by jointly model-
ing both the existence and the intensity of the relationship.
This allows us to distinguish between the mechanisms that
drive the incidence from those that affect the intensity of
relationships. In addition, it also accommodates a prepon-
derance of zeros due to sparseness of ties. We can then
ascertain whether a specification that directly models the
intensity (such as a Poisson specification; e.g., Hoff 2005)
is sufficient for weighted relationships. Therefore, we use a
multivariate correlated hurdle count specification to jointly
model both the incidence of the relationship within a dyad
and the intensity of the relationship conditional on the exis-
tence of the tie. We model the incidence using the ordered
pair of binary variables (Xij2, I, Xji2, I). Then, the magnitude
of the relationship, conditional on its existence in a given
direction, can be modeled using the positively valued trun-
cated count variables Xij2, S or Xji2, S.
:"%*$.6-5*(3"1). Bringing together the two relation-
ships, we can then specify the nC2dyads in the multirela-
tional social network using the dyad-specific random
variables:
The relationships can be further specified in terms of under-
lying latent variables. The latent variable specification
enables us to model these random variables in terms of
dyad- and actor-specific covariates.
"5&/57"3*"#-&41&$*'*$"5*0/. We use underlying latent
utilities uij1 for modeling the existence of a tie in the direc-
tion (i Æj) and uji1 in the reverse direction, for the first
relationship:
For the second relationship, let uij2 and uji2 represent the
underlying utilities that characterizes the existence of the
relationship. Again, we assume that
We model the truncated counts, conditional on a tie in a
given direction, using a Poisson distribution truncated at
zero; in other words,
(3) Xij2, S ~tPoisson(lij) if uij2 > 0, and
Xji2, S ~tPoisson(lji) if uji2 > 0,
where the lij and lji are the rate parameters of the Poisson.
07"3*"5&4"/%)0.01)*-:. Each latent utility is composed
of a systematic part involving the observed covariates and a
stochastic part that incorporates unobserved variables. We
distinguish between dyad-specific covariates and individual-
specific covariates. Let xd
ij1 and xd
ij2 be vectors of dyad-
specific covariates that influence the two relationships.
These allow for homophily, which implies that people who
Dij
ij ji
ij I ji I
ij S ji S
XX
XX
XX
=
11
22
22
,
,
,
,,
,,
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,,
,
110
0
1
1
1
1
Xif u
Xif u
ij
ij
ji
j
=>
=
otherwise
ii1 0
0
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,
,.otherwise
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,,
,
,
,
210
0
1
2
1
2
Xif u
X
ij I
ij
ji I
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otherwise
otherwise
if uji2 0
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,
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# " ! ##!"
1We limit our model description to two relationships for clarity of pres-
entation. Our approach, however, can be extended readily to more relation-
ships, including undirected ones, as we do in our second application.
5*+2/4-;2:/62+ +2':/549./69/4!5)/'2+:=5819
share observable characteristics tend to form connections.
We also use individual-specific covariates for modeling
directed relationships. Let xi1, and xi2be the vector of indi-
vidual i’s covariates, respectively, for the two relationships.
The vector xij1 = (xd
ij1, xi1, xj1) then represents all the covari-
ates that affect the tie in the direction (i Æj) for the first
relationship, and xji1 = (xd
ji1, xj1, xi1) represents the covari-
ates for the reverse direction (i ¨j). The sender and
receiver effects are important to model the asymmetry in the
two directions. For the weighted relationship, we can further
distinguish between the incidence and intensity components.
Therefore, we use covariate vectors xij2, I = (xd
ij2, I, xi2, I, xj2, I)
and xji2 , I = (xd
ji2, I, xj2, I, xi2, I) for the binary component,
where, for example, xd
ij2, I contains a subset of the dyad-
specific variables in xd
ij2 that affect intensity, and xi2 , I is
similarly a subset of xi2. We use analogously defined vec-
tors xij2, S and xji2, S to model the Poisson rate parameters lij
and lji.
&5&30(&/&*5:. The dyads cannot be considered inde-
pendently, because multiple dyads share a common actor
either as a sender or receiver. Accounting for such depend-
ence is important for obtaining proper inferences about sub-
stantive issues. Therefore, we use heterogeneous and corre-
lated random effects to account for the dependence
structure. Whereas for undirected relationships, a single ran-
dom effect is needed, for directed relationships, we can use
two distinct random effects per actor to distinguish between
&91"/4*7&/&44, which is the propensity to “send” ties and
popularity, or "553"$5*7&/&44, which is the propensity to
“receive” ties. The expansiveness parameter aicaptures the
outdegree, which is the number of connections emanating
from an individual i, and the attractiveness parameter bi
captures the indegree, which is the number of connections
impinging on an individual. Thus, for the directed binary
relationship, we use random effects ai1 and bi1. We simi-
larly use ai2, I and bi2, I for the incidence component and ai2,
Sand bi2, S for the intensity equations of the weighted
directed relationship.
Let qi= {ai1, bi1, ai2, I, bi2, I, ai2, S, bi2, S}. We allow these
random effects to be correlated across the relationships and
assume that qiis distributed multivariate normal2N(0, Sq),
where Sqis an unrestricted covariance matrix consisting of
the following submatrices:
The diagonal submatrices capture the within-relationship
covariation in the random effects within a relationship. A
positive correlation between the random effects for a rela-
tionship implies that popular individuals also tend to reach
out more to others. The off-diagonal submatrices capture
correlation across relationships and help determine whether
individuals exhibit similar tendencies across relationships.
ΣΣΣ
ΣΣ
θ
θθ
θθ
=
,,
,,
.
11 12
21 22
If the off-diagonal submatrices in Sqindicate positive corre-
lations, this could possibly be a result of a latent trait gov-
erning commonality in behavior. In contrast, if these sub-
matrices are zero, the relationships can be modeled
separately as the attractiveness and expansiveness parame-
ters will be independent across the different relationships.
Other patterns of correlations are also possible, and their
meaning and relevance depend on the empirical context of a
particular application.
"5&/5 41"$& Social networks also exhibit patterns of
higher-order dependence involving triads of actors. There is
a potential for misleading inferences if such extradyadic
effects are ignored. Hoff and his colleagues demonstrate
how transitivity and other triad-specific structural charac-
teristics such as balance and clusterability can be modeled
using a latent space framework (Handcock, Raftery, and
Tantrum 2007; Hoff 2005; Hoff, Raftery, and Handcock
2002). We employ a latent space for each relationship. We
assume that individual i has a latent position zir in a Euclid-
ean space associated with each relationship r, where r Œ{1,
2}. The latent space framework stochastically models tran-
sitivity; if i is located close to individual j and if j is located
close to individual k, then, because of the triangle inequal-
ity, i will also be close to k.
In the current study, we follow Hoff (2005) and use
the inner-product kernel z¢
irzjr.3In particular, for a two-
relationship model, we use two kernels, one for each rela-
tionship, represented generically as z¢
irzjr. The latent vectors
for each relationship are assumed to come from a relation-
ship-specific multivariate-normal distribution zir ~N(0,
Szr). The dimensionality of the latent space can be deter-
mined using a scree plot of the sum of the mean absolute
prediction error of the entire triad census versus the dimen-
sionality, as is usually done in the multidimensional scaling
literature.4
6--.0%&-Bringing together all the components of the
model, we can write the latent utilities and response propen-
sities as follows:
We assume that the vector of all errors eij is distributed mul-
tivariate normal N(0, S). Also, qi~N(0, Sq) and zir ~N(0,
Szr), "r.
%&/5*'*$"5*0/. Not all parameters of the model are identi-
fied. The error variance matrix Shas a special structure
ue
u
ij ij i j i j ij
ji ji j
11111111
111
=++++
=+
′
x
x
µµ
µµ
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2Even though we assume a symmetric distribution for heterogeneity,
when it is combined with the data from the individuals, the resulting poste-
rior random effects can mimic skewed degree distributions that can arise
from a preferential attachment mechanism. We verified this using a simu-
lation that generated data from a preferential attachment mechanism and
were able to recover highly skewed degree distributions. Details of this
simulation are available on request.
3Other kernels such as those based on the Euclidean norm can also be
used. We leave a detailed examination of the pros and cons of using differ-
ent kernel forms for further research.
4The Bayes factor can also be used to determine dimensionality. How-
ever, it is difficult to compute in our model given the high dimensional
numerical integration that is involved in obtaining the likelihood for each
observation. Therefore, we opt for the predictive MAD criterion that
focuses on triad census recovery.
because of scale restrictions on the binary utilities and
because of exchangeability considerations stemming from
the fact that the labels i and j are arbitrary within a pair. As
the scale of the utilities of the binary responses cannot be
determined from the data, the error variances associated
with the binary components are set to 1. In addition, sym-
metry restrictions on the correlations stem from the
exchangeability considerations, and the resulting variance
matrix can be written as follows:
The correlation parameter r1captures the impact of common
unobserved variables affecting the binary relationship and
also accounts for reciprocity. Similarly, r6and r9capture
correlations for the weighted relationship and also account
for reciprocity within this relationship. The correlation
parameters r7and r8reflect common unobserved variables
that influence both the incidence and intensity equations of
the weighted component of the second relationship and are
akin to selectivity parameters. Note that the intensity equa-
tions have a common variance.
The expansiveness and attractiveness random effects are
individual specific and are thus separately identifiable from
the equation errors that are dyad specific. The latent posi-
tions also are individual specific, but because they enter the
equations as interactions, they can be separately identified
from the random effects. However, because they appear in
bilinear form, we can only identify these subject to rotation
and reflection transformations. Finally, the covariance
matrices Sz1 and Sz2 associated with the latent space
parameters are restricted to be diagonal as their covariance
terms are not identified. Moreover, each matrix has a com-
mon variance term across all the dimensions within a latent
space.
Table 1 summarizes how the different model parameters
can be related to substantive issues of interest. Given that the
model components work in tandem, a parameter may also
be related to other aspects apart from the one shown in the
table. For example, r1is needed to capture reciprocity, but
may also represent the impact of other shared unobservables.
()
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":&4*"/45*."5*0/
We now describe briefly our inference procedures. The
likelihood for the model is computationally complex. Con-
ditional on the random effects and latent positions, the
dyad-specific likelihood requires numerical integration to
obtain the multivariate normal cumulative distribution func-
tion. Moreover, the unconditional likelihood for the entire
network requires additional multiple integration of very
high dimensionality because of the crossed nature of the
random effects. The dependency structure of our model is
considerably more intricate than what is typically encoun-
tered in typical panel data settings in marketing, because we
cannot assume independence across individuals or dyads for
computing the unconditional likelihood. Therefore, we use
Markov chain Monte Carlo (MCMC) methods involving a
combination of data augmentation and the Metropolis–
Hastings algorithm to handle the numerical integration. The
data augmentation step allows us to leverage information
from one relation to predict missing data on other relation-
ships. The complexities involved in modeling multiple rela-
tionships and the identification restrictions on the covari-
ance matrix Smean that the methods of inference for
existing latent space models, as outlined, for example, in
Hoff (2005) and Hoff, Raftery, and Handcock (2002), can-
not be used directly for our model. Therefore, we provide a
full derivation of the posterior full conditionals in Appendix
A.
In the first application, we illustrate our modeling frame-
work on sequential network data. We start with a special
case of our general modeling framework and handle the
simpler situation of a single directed binary relationship
observed at two points in time. Therefore, we do not need
the intensity component in this application. We use the same
data as in Van den Bulte and Moenaert (1998; hereinafter,
VdBM) on communications among members of different
R&D teams and marketing and operations professionals
who are all involved in new product development activities.
Here, we briefly analyze this data set to revalidate the
results in VdBM and to investigate whether our modeling
framework (which differs significantly from that in VdBM)
is better able to recover the structural characteristics of the
network and whether it generates different conclusions or
additional insights.
Data are available about communication patterns both
before and after the R&D teams were collocated into a new
facility. The data set used in VdBM and the one we reana-
lyze here come from a survey conducted in the Belgian sub-
sidiary of a large U.S. telecommunication cooperation. The
data consist of two 22 ¥22 binary (who talks to whom)
sociomatrices, X1and X2, one for 1990 (before collocation)
and one for 1992 (after the R&D teams were collocated in a
separate facility). The actors in both years are the same, 13
R&D professionals spread over four teams and nine mem-
bers of a steering committee consisting of seven positions
in marketing and sales and two in operations. The charac-
teristic element xij, t in each of the two matrices is 1 if i
reports to talk to j at least once a week in year t and 0 if oth-
erwise. The specific area VdBM study is the impact of the
collocation intervention on the communication and coop-
# " ! ##!"
"'(2+
! " " !
"3*"#-& ''&$54
μ Covariate effects, homophily, and heterophily
α
~iExpansiveness, productivity
βiAttractiveness, popularity, or prestige
ziTransitivity, balance, and clusterability
r
~1, r6, r9Reciprocity
r7, r8Selectivity
r3, r5Generalized exchange
r2, r4Multiplexity
ΣθHeterogeneity
5*+2/4-;2:/62+ +2':/549./69/4!5)/'2+:=5819
eration patterns among the different R&D teams and
between R&D and marketing/operations by contrasting
these patterns before and after collocation.
Barriers between R&D and marketing professionals
resulting from differences in personality, training, or depart-
ment culture imply that intrateam and intrafunctional com-
munication will be more prevalent than cross-team and
cross-functional communication. The idea of collocation
was to foster communication among the different R&D
teams. Of the five hypotheses VdBM propose, the first two
test the communicative implications of the barriers between
R&D and marketing and relate to team- and function-specific
homophily effects. The third and fourth hypotheses focus on
effects of collocating R&D teams to foster communication
among these teams. Finally, VdBM posit that collocating
R&D groups may not just foster between-group communi-
cation, but may even annihilate any difference between
within- and between-group communication.
VdBM use Wasserman and Iacobucci’s (1988) p1log-linear
models to test their hypotheses. Our approach differs from
that of these previous studies on several counts. First, in
contrast to the p1model, we do not assume dyadic inde-
pendence. In our model, the dyads are independent condi-
tional on the random effects but are dependent uncondition-
ally. Second, we allow for individual-specific expansiveness
and attractiveness parameters in contrast to group-specific
parameters to yield a richer specification of heterogeneity.
Finally, we account for higher-order effects using a latent
space. The added generality of our model is consistent with
VdBM’s (p. 16) call for “a new generation of models better
able to handle triadic effects and other dependency issues.”
0%&-4"/%"3*"#-&4
We estimated four models on the data set:
1. The full model involves all the components that form part of
our modeling framework. These components include dyad-
specific variables, attractiveness and expansiveness random
effects that are correlated both within and across years, sepa-
rate latent spaces for the two years, and correlated error
terms for the utility equations of the four binary variables
characterizing a dyad.
2. The Uncorr model is a restriction of the full model. It
assumes that the random effects and the utility errors are cor-
related within a year but are uncorrelated across years. This
is akin to having a separate model for each year, and this
offers limited leeway in modeling muliplexity.
3. The NoZiZj model is a restriction of the full model such that
the higher-order terms that characterize the latent space, (i.e.,
the z¢
izjterms are not included. We use this model to assess
whether using the latent space results in better recovery of the
triadic structure of the network and whether it substantively
affects conclusions.
4. The team model closely mimics the VdBM article within our
modeling framework. In this model, we restrict the expan-
siveness and attractiveness parameters to be the same for all
individuals within a group and also do not include the latent
space.
"3*"#-&4
The following variables used in our investigation are the
same as those VdBM use:
INTEAMij = 1 if i and j are R&D professionals on the
same team and 0 if otherwise,
BETWTEAMij = 1 if i and j are R&D professionals but in dif-
ferent teams and 0 if otherwise,
INRDij = 1 if i and j are R&D professionals and 0 if
otherwise, and
INMKTOPSij = 1 if i and j are both marketing or both opera-
tions executives and 0 if otherwise.
&46-54
We estimated the four models using MCMC methods.
Each MCMC run is for 250,000 iterations, and the results
are based on 200,000 iterations after discarding a burn-in of
50,000.
&$07&3:0'4536$563"-$)"3"$5&3*45*$4. We begin by com-
paring the previously described models in their ability to
recover the structural characteristics of the network. Given
our interest in modeling sequential relationships, we focus
on aspects of the network structure that pertain to the two
relationships simultaneously. In particular, we compute sta-
tistics involving the dyadic as well as transitivity patterns of
interactions that span both years.
We can describe dyadic relationships in each year as
belonging to one of the following three types: mutual (M),
asymmetric (A), and null (N). Observing across the two
years, we can construct the following ten possible combina-
tions (Fienberg, Meyer, and Wasserman 1985): NN, AN,
NA, MN, NM, AA, AA, AM, MA, and MM. The names are
self-explanatory for most pairs. For example, NN refers to
the number of dyads that are null in both years. Two pat-
terns that require greater explanation are AA and AA. The
pair AA represents a dyad in which one actor is connected
to the other in both years but neither relationship is recipro-
cated. The pair AA represents a dyad in which one actor ini-
tiates communication with the other in the first year and the
other actor reciprocates by initiating in the second year, a
kind of generalized exchange.
Table 2 reports the recovery of the sequential dyadic pat-
terns. The columns report the absolute deviations between
the actual frequencies and those predicted by the different
models. The last row of the table reports the mean absolute
deviations (MAD) across all the patterns for each model. It
is clear that the uncorrelated model (Uncorr), which ignores
cross-year linkages and models the two years separately, is
significantly worse in recovering the cross-relationship dyadic
patterns compared with the other models. All other models
are roughly similar in their recovery, with the team model
being the best. This indicates that it is important to model the
"'(2+
" $ & !! "&
#"!
"55&3/ 6-- 0!*!+ /$033 &".
NN 7.51 6.44 12.67 7.04
AN 3.00 4.17 4.24 2.70
NA 4.25 4.90 4.82 5.50
MN 3.79 1.53 7.45 2.15
NM .81 2.65 6.07 1.43
AA .92 1.89 1.89 .80
AA 1.97 .38 4.84 .02
AM 2.14 2.26 1.67 1.92
MA .10 .80 .84 .77
MM 5.33 4.48 9.46 3.80
MAD 2.983 2.950 5.394 2.612
two relations jointly so that we can recover the cross-relation
dyadic patterns, as all the models, and, except Uncorr,
accommodate correlations across the two relationships.
We also investigate transitive patterns spanning both
years to understand how well the model recovers
extradyadic effects. This is necessary to assess whether
adding the latent space is important. Eight such transitivity
effects are possible: {Xij1, Xjk1, Xik2}, {Xij1, Xjk2, Xik1},
{Xij1, Xjk2, Xik2}, {Xij2, Xjk1, Xik2}, {Xij2, Xjk2, Xik1},{Xij2,
Xjk1, Xik1}, {Xij1, Xjk1, Xik1}, and {Xij2, Xjk2, Xik2}. For the
sake of brevity, we do not include a full table of results
(available on request). We find that the full model performs
significantly better that all other models in recovering these
transitive patterns (MAD = 34.43). The full and Uncorr
models (MAD = 47.09), both of which include the latent
space, perform significantly better than NoZiZj (MAD =
109.46) and team (MAD = 164.69), which do not include
the higher-order effects.
In summary, observing across all dyadic and triadic
measures, we find that the full model always does better
than Uncorr, thus highlighting the need for joint modeling.
The full model also performs better than all the other mod-
els in recovering the transitivity patterns, indicating that in
this application, the latent space is important for handling
extradyadic patterns. We can conclude that, on the whole,
the full model recovers best the structural characteristics of
the network.
"3".&5&345*."5&4"/%:105)&4&4
We begin by investigating whether the assumption of
individual-level expansiveness and attractiveness parame-
ters in our models has support. Figures 1 and 2 report the a
and bvalues for the 22 managers for Years 1 and 2, respec-
tively, for our full model. The labels for each point in the
figures represent the team name. It is apparent from these
figures that although some members of a group are clus-
tered together, many groups exhibit considerable within-
group heterogeneity. This is particularly noticeable for the
marketing group (m) and the R&D groups (r3, r1, and r4),
which exhibit greater within-group variability. This sug-
gests that the data support a richer characterization of
heterogeneity than what is possible with group-specific
effects.
Table 3 summarizes the parameter estimates. It is evident
from the table that the parameters values differ substantially
across the models in their magnitude and significance, indi-
cating that the different model components influence sub-
stantive conclusions. Focusing on the bottom part of the
table, we see that models that do not include the latent space
yield higher estimates of the error correlations, possibly due
to the confounding of variances across levels. We can use
the coefficients to infer the degree of support for the differ-
ent hypotheses studied in VdBM. Table 4 reports the extant
of support for each hypothesis according to the different
models. All other entries are computed from our reanalysis.
Each entry for a particular model represents the probability
that the corresponding hypothesis is true under that model.
Several significant differences across the models are evident
from this table.
The first two hypotheses pertain to within- and between-
team homophily effects. Comparing the full model with
VdBM indicates that our model supports both H1and H2,
whereas VdBM find mixed support for these. In particular,
we find that H1 b has significant probability across all our
models, indicating that R&D professionals tended to com-
municate predominantly with other R&D professionals
before the move. The full, NoZiZj, and Uncorr models sug-
gest strong support for H2a and H2b in contrast to team and
VdBM. It seems that differences in support for this hypothe-
# " ! ##!"
/-;8+
"ab$#! " !&
r1
r1
r1
r1
r2
r2
r2
r3
r3
r4
r4
r4
r4
m
m
m
m
m
m
m
o
o
-3
-2
-1
1
2
3
Α
-1.5
-1.0
-0.5
0.5
1.0
1.5
Β
Notes: The point labels indicate team membership.
/-;8+
"ab$#! " !&
r1
r1
r1
r1
r2
r2
r2
r3
r3
r4
r4
r4
r4
m
m
m
m
m
m
m
o
o
-3
-2
-1
1
2
3
Α
-1.5
-1.0
-0.5
0.5
1.0
1.5
Β
Notes: The point labels indicate team membership.
"'(2+
" " !""! "
!
"55&3/ 6-- 0!*!+ /$033 &".
INTERCEPT1 –1.48 –.87 –1.23 .48
INTEAM1 5.03 3.02 4.03 3.42
BETWTEAM1 1.60 .88 .84 1.42
INMKTOPS1 1.03 1.40 1.02 .49
INTERCEPT2 –1.18 –.99 –1.12 .78
INTEAM2 3.51 3.01 3.17 3.79
BETWTEAM2 2.15 1.73 1.80 2.71
INMKTOPS2 1.23 1.61 1.02 .07
r1.59 .83 .60 .79
r2.43 .64 —.63
r3.26 .57 —.53
r6.48 .72 .53 .67
Notes: Bold indicates that the 95% posterior interval does not span 0.
5*+2/4-;2:/62+ +2':/549./69/4!5)/'2+:=5819
sis are driven by the extent of within-group heterogeneity
captured by the model. Recall that both team and VdBM
assume that all members within a group share the same
attractiveness and expansiveness parameters. Figures 1 and
2 show, however, that there is considerable heterogeneity in
the recovered aand bparameters in the marketing group,
and we find that failure to model this heterogeneity compre-
hensively can substantively affect conclusions.
We also find some differences in the support for the col-
location hypotheses (H3and H4) across the models. The full
and Uncorr models have a lower probability associated with
these hypotheses compared with VdBM, team, and NoZiZj.
These differences can be explained by the presence or
absence of the latent space for capturing higher-order
effects and are also consistent with the strong support for
H1b in our model. Finally, all models yield no support for
H5.5These differences in supported hypotheses (H1b, H2a,
H2b, H3, and H4) across models demonstrate that the differ-
ent model components can affect the theoretical and sub-
stantive conclusions and that it is important to account for
dyadic-dependence and higher-order effects.
&7&3"(*/(/'03."5*0/$3044&-"5*0/4)*14
We now illustrate how our framework can be used to
leverage information in one relationship to predict relation-
ships in another. For example, we assume that the data
involving the entire marketing groups are missing in the
second year. In such a situation, we cannot readily use the
log-linear modeling framework previous researchers have
employed, because the group-specific parameters used in
such models will not be available for the marketing group.
However, for our models, the natural reliance on data aug-
mentation to obtain the utilities and the individual-specific
random effects when estimating parameters ensures that
such missing data can be handled seamlessly. In particular,
the covariance matrix of the random effects can be used to
leverage information from Year 1 to Year 2 about these indi-
vidual-specific parameters. Therefore, we estimate our full
and Uncorr models on such a data set to determine whether
incorporating cross-relationship linkages improves predic-
tions in such situations. Note that in the full model, the data
on Year 1 for the individuals in the marketing group can be
leveraged to predict relationships in Year 2. This is not pos-
sible in the Uncorr model, in which the two years are mod-
eled separately. Tables WA 1 and WA 2 of the Web Appendix
(see http:// www. marketingpower.com/jmraug11) report how
well the cross-year dyadic and transitivity patterns are recov-
ered on such a data set with missing values for the marketing
group. These tables report the absolute deviations between
the actual frequencies and those predicted by the full and
Uncorr models and illustrate that the full model does sig-
nificantly better in recovering these cross-year relationships.6
In this application, our focus is on modeling relation-
ships of different types. We use a combination of undirected
and directed binary relationships and a directed weighted
relationship. In particular, we show how it is important to
model both the incidence and intensity of weighted relation-
ships, because conclusions and predictions depend crucially
on this distinction. The data for this application come from
a Swiss online social network on which members can create
a profile as either a user or an artist and can then connect
with other registered members through friendship relation-
ships. The social networking site offers different services to
these two distinct user groups. Whereas both groups can
publish user-generated content such as blogs, photos, or
"'(2+
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:105)&4*4 03."-&45 6-- &". % 0!*!+ /$033
H1: Both before and after collocation, R&D professionals tend to
communicate with other R&D professionals rather than with
marketing or operations executives.
H1a INTEAM1 > 0
H1b BETWTEAM1 > 0
H1c INTEAM2 > 0
H1d BETWTEAM2 > 0
.99
.99
.99
.99
.99
.99
.99
.99
.9
n.s.
.99
.99
.99
.99
.99
.99
.99
.91
.99
.99
H2: Both before and after collocation, marketing and operations
executives tend to communicate with members of their own
department.
H2a INMKTOPS1 > 0
H2b INMKTOPS2 > 0
.93
.97
.85
.55
n.s.
n.s.
.99
.99
.92
.94
H3: R&D professionals have a higher probability of communicating
with members of other R&D teams after collocation than before.
H3BETWTEAM2 > BETWTEAM1 .82 .99 .99 .99 .86
H4: When collocating R&D teams implies increasing the physical
distance with other departments, R&D professionals’ tendency
to communicate with other R&D people rather than executives
from other departments increases.
H4INRD2 > INRD1 .86 .99 .99 .99 .82
H5: Before collocation, R&D professionals have a higher probability
of communicating with members of their own team rather than
with members of other R&D teams. After collocation, the
tendency to communicate among R&D people is as strong
between as within teams.
H5a INTEAM1 > BETWTEAM1
H5b INTEAM2 = BETWTEAM2
.99
n.s.
.99
n.s.
.99
n.s.
.99
n.s.
.99
n.s.
Notes: Entries represent the probabilities of a hypothesis being true; n.s. indicates that the corresponding hypothesis is not supported.
5For H5b, we found that the probability associated with INTEAM2 >
BETWTEAM2 is .99 for all our models.
6We also investigated the role of demographics that were part of the data
but did not find any significant impact. Our models can handle such continu-
ous covariates, something that is not possible with a log-linear specification.
videos on their profiles, only artists can, in addition, publish
up to a maximum of 30 songs on their profile.
Artists use the different services offered by the platform
to promote their music and concerts and to seek collabora-
tion with other musicians and bands. They establish friend-
ship relationships with other artists, send personal mes-
sages, and write public comments on other artists’ profiles.
For entertainment and informational reasons, users, as well
as artists, visit profiles of artists and download songs.
Artists engage in active promotion and relationship effort in
the hope that it will result in increased collaboration, com-
munication, popularity, and song downloads. In summary,
the online networking site offers a platform that combines
social networking services with entertainment and commu-
nication services.
There are four components of the data set: member data,
friendship data, communication data, and music download
data. The member data contain information collected at reg-
istration and include stable variables such as the registration
date, date of birth, gender, and city or, in the case of artists,
their genre and information about their offline concerts and
performances before joining the network. In addition, the
data also contain information on the number of page views
of each member’s page on the network during a given time
period. The other data components pertain to our three
dependent variables and are described in greater detail in
the following section.
"5""/%536$563"-)"3"$5&3*45*$4
Our sample consists of 230 artists who created a profile
on the network between February 1, 2007, and March 31,
2007, provided information about their activities and char-
acteristics, and uploaded at least one song on their profile.
We model three types of relationships among these artists
over the course of the six months between April 1, 2007,
and September 30, 2007.7These relationships include
friendship (f), communications (c) and music downloads
(m). The data set thus contains three 230 ¥230 matrices (Yf,
Yc, and Ym, respectively) for these relationships.
536$563"-$)"3"$5&3*45*$4. We now briefly describe the
structural characteristics of the three relationships for our
set of artists.
1. 3*&/%4)*1 The friendship relation yf
ij is binary and undirected.
It indicates whether a friendship is formed between the pair
{i, j} before the end of our data period. The network has
3564 friendship relations among a maximum possible 26,335
connections, yielding a network density of 13.53%.
2. 0..6/*$"5*0/4 The communication relation yc
ij is binary
and directed and indicates whether artist i sent a communica-
tion (direct message or comment) to artist j within the time
period of the data. We observed 4575 communication rela-
tions, yielding a density of 8.68%. The relation exhibits con-
siderable reciprocity or mutuality (defined as the ratio of the
number of pairs with bidirectional relations to the number of
pairs having at least one tie) equal to 30.9%. Artists vary in
their level of expansiveness (or outdegree), as measured by
the number of artists they communicate with and their popu-
larity or receptivity( i.e., the number of artists communicat-
ing with a given artist [indegree]). The indegree and outdegree
distributions are highly skewed. The mean degree is 19.97.
The maximum and minimum for the indegree distribution are
203 and 0, respectively, whereas the maximum and minimum
for the outdegree distribution are 182 and 0, respectively.
3. 64*$ 08/-0"%4 The music downloads represent a
directed and weighted relationship. Each song download
entry ym
ij is a count of the number of times that artist j listens
to a song on artist i’s profile and may include multiple down-
loads of the same song. As discussed in the “Modeling
Framework” section, we distinguish between the incidence
and intensity of music downloads. Focusing first on inci-
dence, we find that of the possible 52,670 ties, our data con-
tain only 17,912 binary ties, implying a density of 34%. The
reciprocity is 39.1%. The outdegree of an artist is the number
of other artists who download from that artist, and the inde-
gree is the number of other artists from whom the artist
downloads music. For the binary component, the mean
degree is 17.89, and the maximum indegree and outdegree
are 213 and 135, respectively. Because this is a weighted rela-
tion, we can also study the intensity of connections. On aver-
age, each artist downloads songs on 59.98 occasions (includ-
ing multiple song downloads). The maximum weighted
indegree (i.e., the number of songs downloaded from) an
artist is 2899, whereas the maximum weighted outdegree
(i.e., the maximum number of times a single artist listens to
songs is 612). We also find that the artist-specific degree sta-
tistics are highly correlated across the three relationships.
0%&-1&$*'*$"5*0/4"/%"3*"#-&4
We estimate several different variants of the full model
(hereinafter, we refer to this as “full model”) that we outlined
in the “Modeling Framework” section. Our null models
impose different restriction on the full model; we con-
structed them to investigate how crucial the different com-
ponents of the full model are in capturing important aspects
of the data generating process. The models are as follows:
;6-- 0%&-. The full model includes dyad-specific covariates
(xij) to accommodate homophily and heterophily; artist-specific
covariates (xi) and (xj) to account for asymmetry in responses;
artist- and relationship-specific sender and receiver parameters
(qi), to model heterogeneity in expansiveness and receptivity;
correlations in these random effects across relationships (Sq);
latent spaces of random locations for the three relations (zim,
zic, and zif) to capture higher order effects; and correlations in
errors (Sm, Sc, and Sf), to incorporate reciprocity within each
relationship. In addition, the full model uses a correlated hur-
dle count model to account for the sparse nature of many net-
work data sets.
;0*440/. The Poisson model uses a Poisson distribution for the
music download relationships, rather than the correlated hurdle
Poisson. Thus, the counts are modeled directly, and we do not
distinguish between the incidence and intensity of counts. The
model is otherwise identical to the full model in all other respects.
;/$033. In the Uncorr model, we treat the three relationships as
independent. Thus, we assume the artist-specific random effects
in qito be uncorrelated across the relationships, and therefore,
Sqhas a block-diagonal structure. This model is thus equiva-
lent to running three separate models on the three relationships.
;0!*!+. For the NoZiZj model, we do not include the higher
order terms z¢
izj that characterize the latent space.
Each artist can be described using the variables detailed
in Table 5. We use these artist-specific variables to compute
dyad-specific variables. We use different combinations of
# " ! ##!"
7Our data are not entirely representative of the whole network, which
consists of other artists who joined the network subsequent to our data
period. In addition, we focus only on the subnetwork of connections
involving artists, rather than also considering fans, because this is consis-
tent with the primary focus of the network in offering a platform for artists
to present themselves and seek collaborations.
5*+2/4-;2:/62+ +2':/549./69/4!5)/'2+:=5819
the artist- and dyad-specific variables in explaining the
three relationships. (Appendix B gives details of the covari-
ate specifications for the three relationships.) Table 6 shows
descriptive statistics associated with these covariates. The
last two rows of Table 6 contain dyad-specific binary
variables. The variable “common region” is equal to 1 when
both artists in a dyad are from the same region; otherwise, it
is 0. We code the last variable, “common genre,” analogously.
&46-54
We estimated the models using MCMC methods
(described in Appendix A). Each MCMC run is for 250,000
iterations, and the results are based on a sample of 200,000
iterations after a burn-in of 50,000.
0%&-"%&26"$:. Before discussing parameter estimates,
we use a combination of predictive measures and posterior
predictive model checking (Gelman, Meng, and Stern 1996)
to compare the adequacy of our models. This involves gen-
erating G hypothetical replicated data sets from the model
using the MCMC draws and comparing these data sets with
the actual data set. These comparisons are made using vari-
ous test quantities that represent different structural charac-
teristics of the network. If the replicated data sets differ sys-
tematically from the actual data on a given test quantity, the
model does not adequately mimic the structural characteris-
tic that the test quantity represents. The discrepancy
between the replicated data sets and the actual data can be
assessed using posterior predictive 1-value. This 1-value is
the proportion of the G replications in which the simulated
test quantity exceeds the realized value of that quantity in
the observed data. An extreme 1-value, (either close to 0 or
1; i.e., £.05 or ≥.95) suggests inadequate recovery of the
corresponding test quantity.
We use several test statistics associated with the weighted
relationship to assess whether the distinction between inci-
dence and intensity is crucial. Table 7 shows the model ade-
quacy results for the music download relationship based on
G = 10,000 MCMC draws. Column 2 of the table reports the
value of the test statistics for the observed data, and the
other columns report the posterior predictive mean and 1-
values for the different models. A few conclusions can be
readily drawn from the table. First, Poisson, which models
the intensity directly using a Poisson specification (as in
Hoff 2005), does not recover any of the test statistics ade-
quately, because almost all 1-values in Column 6 are
extreme. Furthermore, the posterior predictive mean values
for the test statistics (Column 5) are appreciably different
from their counterparts in the actual data. Second, we
observe that /0/& of the 1-values associated with the other
models are extreme, and this indicates that modeling inci-
"'(2+
"! "$!""!"! "!"
$ !
"3*"#-& #4&37"5*0/4 %/ */ "9
Page views 230 781.2 467 1069.8 43 10,931
Songs 230 4.108 3 3.26 1 29
Audience 230 .482 0 .5 01
Band 230 .704 1 .457 01
Years active 230 7.24 6 5.82 1 27
Common region 26,335 .628 1 .483 01
Common genre 26,335 .564 1 .495 01
"'(2+
"! "$ !
"3*"#-& &4$3*15*0/
Songs The number of songs available on the artist’s profile.
Band Whether the artist belongs to a band. Equal to 1 if artist
belongs to a band and 0 if otherwise.
Audience The audience size of the largest concert by the artist. A
median split yields 1 when the audience is > 700 and 0
otherwise.
Genre Genre of the artist. We distinguish between rock genres and
nonrock genres.
Region The geographical region to which the artist belongs. Artists
belong to one of the 26 cantons in Switzerland. These are
aggregated into three regions: French, German, and Italian.
PageViews The number of page views of the artist’s profile during the
data period. These page views originate from other
registered members, including the fans, or from Internet
users from outside the social network.
YActive The number of years of activity of the artist in the music
industry.
"'(2+
"!" "$ #!%! "# !
6-- /$033 0!*!+ 0*440/
"5" p"-6& p"-6& p"-6& p"-6&
*/"3:
Null dyads 23,376 23,420 .79 23,416 .77 23,420 .80 23,275 .03
Asymmetrical dyads 1802 1783 .35 1785 .36 1777 .30 2134 1
Mutual dyads 1157 1132 .24 1134 .26 1137 .28 925 0
Reciprocity .39 .39 .44 .39 .45 .39 .51 .30 0
Transitive triads 51,692 50,523 .3 50,710 .32 49,534 .16 44,303 0
Intransitive triads 136,230 133,398 .27 133,321 .26 135,012 .39 117,494 0
Mean degree 17.89 17.59 .18 17.62 .2 17.62 .21 17.32 .04
Standard indegree 17.924 18.1 .65 18.07 .62 18.16 .69 16.61 0
Standard outdegree 31.61 31.06 .14 31.09 .16 31.07 .14 27.74 0
Degree correlation .898 .893 .3 .89 .31 .893 .32 .896 .45
6"/5*5:
Mean strength 59.58 58.83 .25 58.69 .23 58.86 .26 209.4 1
Standard indegree 213.2 208.44 .34 206.34 .29 209.66 .38 1327.9 1
Standard outdegree 74.13 70.06 .15 69.47 .12 69.51 .13 751.1 1
dence and intensity separately is important for adequately
capturing the structural network characteristics associated
with weighted relations. Finally, because the NoZiZj model
performs almost as well as the full model, we conclude that
the latent space is not crucial for the recovery of structural
characteristics in this application.
The preceding discussion focuses on model adequacy for
music downloads to highlight the contribution of the corre-
lated multivariate hurdle Poisson in modeling weighted
relationships. The results from the other two relationships
show that all models (including Poisson) are similar in their
recovery of structural characteristics for these relationships.
The model adequacy results are based on in-sample simula-
tions. (We report the predictive performance of our model
in the Web Appendix at http://www.marketingpower. com/
jmraug11.) We find that the Full model outperforms other
models on almost all measures. We also find that the Pois-
son model does very poorly in predicting future activity,
indicating that there are significant gains in modeling the
incidence and intensity separately.
"3".&5&345*."5&4
We now discuss the parameter estimates based on the
entire sample of six months. We focus on the parameter esti-
mates from the full model. Estimates from other models
mostly yield similar qualitative conclusions, and we do not
include these for the sake of brevity.
07"3*"5&&''&$54. Table 8 reports the posterior means and
standard deviations of the coefficient estimates for the three
relationships. In interpreting this table, recall that all
variables associated with the friendship relation are dyad
specific and binary, whereas, for the other relations, some
variables are dyad specific and some are artist specific.
Also, note that both components of the music relation have
the same set of covariates. (The definitions for these covari-
ates are available in Appendix B.)
For the sake of brevity, we synthesize the results across
all the relationships. There is clear evidence of homophily
and proximity: For all three relationships, the dyad-specific
variables CRegion and CGenre have a positive and signifi-
cant impact on the likelihood of forming connections. The
positive coefficient for CRegion is consistent with the
notion that a common language and geographical proximity
can enhance the likelihood of collaborative effort. The posi-
tive coefficient for CGenre means that pairs of artists who
produce music in the same genre have a higher propensity
to form friendship connections, communicate with each
other, and download music from each other. We also find
that artists belonging to a band have a higher chance of
forming friendships (BothBand) and a higher probability of
sending and receiving communications (SBand and RBand).
We find that measures of online popularity that are based
on the total number of page views for an artist (BothPopu-
lar, SPviews, DPviews, and PPviews) positively influence
relationship formation. For example, the positive coeffi-
cients for DPviews in the music relation indicate that artists
with greater online popularity in this network have a greater
likelihood of downloading songs from other artists and that
they tend to download more music. In contrast, most measures
of prior and offline popularity or experience (indicating
audience size of concerts or years of activity) do not seem
to affect the formation of online relationships within the net-
work.8Finally, the data indicate that for the music relation-
ship, different coefficients influence the incidence and
intensity components of the music relationship. Thus, we
conclude that these two facets are not isomorphic and need
to be modeled separately.
07"3*"/$&4536$563&0'5)&3&-"5*0/4)*14. The covariance
matrix Sqcaptures the linkages among the expansiveness
and attractiveness parameters across the relationships. Table
9 reports the elements of Sq. A striking feature of Table 9 is
that all the covariances are significantly positive. The posi-
tive correlation within a relationship implies that attractive-
ness goes hand-in-hand with expansiveness (i.e., artists who
are sought by others also tend to be active in seeking rela-
tionships with others). Moreover, for the music relationship,
the attractiveness and expansiveness parameters for the
intensity equations are also positively correlated with the
corresponding random effects for the incidence component.
The positive correlations across relationships imply that an
artist who is popular in one relationship is also likely to be
both popular and productive in other relationships. Simi-
larly, an artist who is productive in one relationship is also
likely to be productive and popular in other relationships.
The utility errors for the two incidence equations of the
music download relationship are positively correlated (.74),
owing to shared unobservable influences and reciprocity.
The intensity equations are also positively correlated (.497).
Furthermore, the errors for the incidence equations are posi-
tively correlated with the errors for the intensity equations
(.395 and .411) implying selectivity through shared unob-
served factors driving both incidence and intensity. This
corroborates the need for a multivariate correlated hurdle
Poisson specification. Finally, the communication utilities
also exhibit positive correlation (.511) driven by reciprocity
and other shared unobservables.
As interest in social networks and brand communities
grows, marketers are becoming increasingly focused on
understanding and predicting the connectivity structure of
such networks. In this article, we developed a methodologi-
cal framework for jointly modeling multiple relationships
that vary in their directionality and intensity. Our integrated
approach for social network analysis is unique in that it
weaves together several distinct model components needed
for capturing multiplexity in networks.
We applied our framework to two distinct applications
that showcased different benefits of our approach. In the
first application, we investigated the impact of an organiza-
tional intervention (R&D collocation) on the patterns of
communications among professionals involved in new
product development activities. In this application, we
specifically investigated the gains from modeling relation-
ships jointly. Our results clearly indicate how the different
components of our framework are needed for a clear assess-
ment of substantive hypotheses. We found that the hetero-
# " ! ##!"
8We thank an anonymous reviewer for pointing out that covariates relat-
ing to popularity could be potentially endogenous. However, given that we
include both online and offline correlates of popularity, as well as actor-
specific random effects that capture attractiveness, it is unclear whether
additional unobserved variables relating to popularity are part of the utility
error. However, caution is still needed in interpreting the results.
5*+2/4-;2:/62+ +2':/549./69/4!5)/'2+:=5819
geneity specification, the latent space, and the correlations
across relationships can affect both substantive conclusions
and the recovery of structural characteristics. Finally, as the
Web Appendix shows (see http://www.marketingpower.
com/ jmraug11), our approach can be used to leverage infor-
mation from one relation to predict connectivity in another.
In our second application, we focused on modeling mul-
tiple relationships of different types. In particular, we
showed how it is critical to model both the incidence and
intensity of weighted relationships such as music down-
loads; otherwise, recovery of structural characteristics and
predictive performance suffers appreciably. On the substan-
tive front, our results show that the friendship, communica-
tions, and music download relationships share common
antecedents and exhibit homophily and reciprocity. We
found that offline proximity is relevant for all online rela-
tionships, and this is consistent with our understanding that
these connections are formed to forge collaborative relation-
ships. We also found that the artists exhibit similar roles
across relationships and that popular artists seem to be more
productive regardless of the relationship being studied.
Across the two applications, we found mixed evidence
regarding the benefits from incorporating a latent space. In
the first application, the latent space improved the recovery
of cross-relationship transitivity patterns and affected the
parameter estimates and the substantive findings. However,
higher-order effects did not seem to be important in the sec-
ond application. The latter result is consistent with Faust’s
(2007) conclusion that much of the variation in the triad cen-
sus across networks could be explained by simpler local
structure measures. These results suggest that the impact of
extradyadic effects could be application specific.
On the theoretical and substantive front, our framework
facilitates a detailed description of antecedents of relation-
ship formation and allows for theory testing taking into
account systematic variations in degree arising from
homophily and heterophily, local structuring, as well as
temporal or cross-relationship carryover (Rivera, Soder-
strom and Uzzi 2010). Our enquiry can be extended in many
directions. Our applications involved small networks. Most
online networks are much larger, and statistical methods
cannot scale directly to the level of these large networks.
However, recent research has shown that while online net-
works can have millions of members, communities within
such networks are relatively small, with sizes in the vicinity
of the 100–200 member range (Leskovec et al. 2008). This
implies that these very large networks can be broken down
into clusters of tightly knit communities, and when such
communities are identified, our methodological framework
can then be used on such communities to further understand
"'(2+
""!""! ""
"!! "#
"3".&5&3
3*&/%4)*1&-"5*0/4)*1
Intercept –2.732 .167
CRegion .341 .050
CGenre .150 .033
BothPopular .163 .063
BothNotPopular .015 .087
BothBigSongs .061 .061
BothSmallSongs .016 .065
BothBand .640 .153
BothNoBand –.257 .163
BothBigAudience .203 .143
BothSmallAud –.134 .143
BothLongActive –.036 .075
BothShortActive .034 .071
0..6/*$"5*0/4&-"5*0/4)*1
Intercept –3.446 .270
CRegion .321 .042
CGenre .144 .028
SPviews .015 .004
SSongs .009 .010
SBand .321 .106
SAudience .135 .093
SYActive –.003 .006
RPviews .002 .003
RSongs .013 .009
RBand .464 .154
RAudience .043 .138
RYActive –.004 .006
64*$&-"5*0/4)*1/$*%&/$&
Intercept –3.007 .272
CRegion .183 .040
CGenre .118 .031
PPviews .021 .004
PSongs –.006 .010
PBand .106 .104
PAudience .100 .091
PYActive –.013 .006
DPviews .031 .005
DSongs .022 .014
DBand .062 .141
DAudience –.113 .120
DYActive –.015 .009
64*$&-"5*0/4)*1/5&/4*5:
Intercept –2.024 .269
CRegion .235 .050
CGenre .152 .038
PPviews .011 .003
PSongs .023 .009
PBand .169 .086
PAudience –.005 .074
PYActive –.003 .006
DPviews .034 .006
DSongs .021 .017
DBand –.055 .149
DAudience –.157 .120
DYActive –.012 .011
Notes: Bold indicates that the 95% posterior interval does not span 0.
"'(2+
"$ "" "!Sq
a
.
*
b
.
*
a
.
*
b
.
*
a
*
$
b
*
$
a
'
*
am
i1 .397 .378 .234 .222 .268 .500 .499
(.051) (.056) (.037) (.049) (.040) (.064) (.065)
bm
i1 .707 .238 .441 .329 .617 .593
(.091) (.042) (.073) (.053) (.086) (.086)
am
i2 .192 .146 .195 .315 .319
(.034) (.038) (.033) (.050) (.052)
bm
i2 .489 .188 .313 .288
(.082) (.048) (.079) (.081)
ai
c.407 .478 .513
(.052) (.065) (.067)
bi
c1.000 .988
(.109) (.104)
af
i1.031
(.111)
Note: Posterior standard deviations are in parentheses.
the nature of linkages within these subcommunities. How-
ever, such a divide-and-conquer approach is unlikely to pro-
vide a complete picture of the nature of link formation in
such large networks.
We focused on modeling static relationships or on
sequential relationships observed over a few time periods.
However, networks are dynamic entities in which connec-
tions are formed over time. Incorporating such dynamics
would be interesting. We used a parametric framework based
on the normal distribution for the latent variables and ran-
dom effects, and this was sufficient for recovery of skewed
degree distributions. However, in other situations, Bayesian
nonparametrics (Sweeting 2007) may be more useful.
1. The full conditional for precision matrix S–1
qof the actor-
specific random effects is a Wishart distribution given by
1. where the prior for S–1
qis Wishart(rq, Rq). The quantities rq
and Rqrefer to the scalar degree of freedom and the scale
matrix for the Wishart, respectively, and N is the number of
actors in the network.
2. The covariance matrices Sr
z, for relationship r are diagonal,
because zr
iis a p-dimensional vector of independent compo-
nents. Let s2
z,r denote the common variance for the compo-
nents of zr
i. The full conditional for s2
z,r is an inverse gamma
distribution given by
3. The full conditional for the coefficients mis multivariate nor-
mal because we have a seemingly unrelated regression sys-
tem of equations conditional on knowing the latent variables.
Form the adjusted utilities (e.g., u
~ij1 = uij1 –ail –bj1 –z¢
i1zj1)
and adjusted log-rate parameters by subtracting terms that do
not involve mfrom the latent dependent variables. We then
have the system of equations, v
~{ij} = X{ij}m+ e{ij}, for an arbi-
trary pair {i, j}, where eij ~ N(0, S). We can write the full con-
ditional as follows:
1. where Wm
–1 = C–1 + S{ij}X¢
{ij}S–1X{ ij} and m
ˆ= Wm[C–1h+
S{ij}X¢
{ij}S–1v
~{ij}].
4. The full conditional for the heterogeneity parameter qiis a
multivariate normal. We again begin by creating adjusted
utilities and rate parameters, such as by subtracting all terms
that do not involve qi. Let be the vector of adjusted utilities
for the three relationships. Then we have the system. Again,
we can use standard Bayesian theory for the multivariate nor-
mal to obtain the resulting full conditional:
where Wq
–1 = Sq
–1 + (N – 1)S–1 and q
ˆi= S–1WqSj πiv
~{ij}q.
5. The full conditional for zithat contains all the latent space
vectors associated with an individual i is multivariate normal.
Creating adjusted utilities such as u
~ij1z = uij1 –x¢
ij1m1–ai1 –
bj1, we can form the vector of adjusted utilities and latent rate
()( |{}) ,
,
Ap z apN
pzb
zr irirk
k
p
i
N
r
21
242
11
σ=+ +
== ∑∑
IG −−
−
1
1
.
() {} ,ApWishartNR
ii
i
N
i
11
1
1
Σθθθ
θρθθ
−
=
−
()
=+
′+
∑
−1
,
() (|{ }) (, ),
{}
ApN
ij
3µµµµv
%
$
=Ω
µ
() (|{ }) (%,),
{}
ApN
iij i
4θθθθv
%
$
θθ
=Ω
parameters v
~{ij}z = Zjzi+ e{ij}, where Zjis an appropriately
constructed matrix from the latent space vector of actor j. This
is a seemingly unrelated regression system. Given the prior zi~
N(0, Sz), where Szis constructed from the different Szr matri-
ces, we can write the full conditional as N(z
ˆiWzi), where Wzi
–1 =
(Sz)–1 + Sj πiZ¢
jS–1Zjand z
ˆi= Wzi[Sj πiZ¢
j(S)–1v
~{ij}z]. The
model depends on the inner product of the latent space vec-
tors, which is invariant to rotations and reflections of the vec-
tors. Thus visual representations of these vectors require that
they be rotated to a common orientation, This can be done by
using a Procustrean transformation as outlined in Hoff (2005).
6. The variance–covariance matrix of the errors for the
weighted relationship, S, has a special structure as described
in the “Modeling Framework” section. Given this special
structure, we follow the separation strategy of Barnard,
McCulloch, and Meng (2000) in setting the prior in terms of
the standard deviations and correlations in S. The covariance
matrix Scan be decomposed into a correlation matrix, R, and
a vector, s, of standard deviations—that is, S= diag(s) ¥R¥
diag(s), where scontains the square roots of the standard
deviations. Let wcontain the logarithms of the elements in s.
We assume a multivariate normal distribution N(0, I) for the
nonredundant elements of R, such that it is constrained to the
subspace of the p-dimensional cube [–1, 1]p, where p is the
number of equations that yields a positive definite correlation
matrix. Finally, we assume a univariate standard normal prior
for the single log-standard deviation in w.
•The full conditional distribution for the free element in the
vector of log-standard deviations wof errors can only be writ-
ten up to a normalizing constant (recall that the terms asso-
ciated with the binary utilities in ware fixed to 0 for identi-
fication purposes). Given our assumption of a normal prior
for the single free element, we use a Metropolis–Hastings
step to simulate the standard deviation in w. A univariate
normal proposal density can be used to generate candidates
for this procedure. If is the current value of kth component
of w, a candidate value is generated using a random walk
chain wc
k= wk
(t – 1) + N(0, t), where tis a tuning constant
that controls the acceptance rate.
•Many different approaches can be used to sample the corre-
lation matrix R. Here, we use a multivariate Metropolis step
to sample a vector of nonredundant correlations in R. We
used adaptive MCMC (Atchade 2006) for obtaining the tun-
ing constant so as to ensure rapid mixing.
7. The full conditional distribution associated with the set of
latent utilities and latent rate parameters in uij is again
unknown. We sample the utilities and log-rate parameters
using univariate conditional draws. Sampling the utilities is
straightforward, because these are truncated univariate con-
ditional normal draws. The log-rate parameters log lij and
log lji are sampled such that these are univariate normal
draws if the corresponding observation involves a zero
count, and for an observation in which a positive count is
observed, we use a univariate Metropolis step that combines
the likelihood for a truncated Poisson distribution with a
conditional normal prior.
07"3*"5&4'035)&3*&/%4)*1&-"5*0/4)*1
CRegion: CRegion is equal to 1 if both artists in a pair are from
the same region; 0 otherwise.
CGenre: CGenre is equal to 1 if both artists in a pair are from
the same genre; 0 otherwise.
BothPopular: BothPopular is equal to 1 if both artists in a pair are
viewed (online) by more than the population median; 0 otherwise.
# " ! ##!"
5*+2/4-;2:/62+ +2':/549./69/4!5)/'2+:=5819
BothNotPopular: BothNotPopular is equal to 1 if both artists in a
pair are viewed by fewer than the population median; 0 otherwise.
BothBigSongs: BothBigSongs is equal to 1 if both artists in a
pair post more songs than the population median; 0 otherwise.
BothSmallSongs: BothSmallSongs is equal to 1 if both artists in
a pair post fewer songs than the population median; 0 otherwise.
BothBand: BothBand is equal to 1 if both artists in a pair post
are from a band; 0 otherwise.
BothNoBand: BothNoBand is equal to 1 if both artists in a pair
post are not from a band; 0 otherwise.
BothBigAudience: BothBigAudience is equal to 1 if both artists
in a pair had large concerts with more than 700 spectators; 0
otherwise.
BothSmallAudience: BothSmallAudience is equal to 1 if both
artists in a pair had small concerts with more than 700 spectators;
0 otherwise.
BothLongActive: BothLongActive is equal to 1 if both artists in
a pair had being active for more than six years; 0 otherwise.
BothShortActive: BothShortActive is equal to 1 if both artists
in a pair had being active for less than six years;0 otherwise.
07"3*"5&4'035)&0..6/*$"5*0/&-"5*0/4)*1
CRegion: CRegion is equal to 1 if both artists in a pair are from
the same region; 0 otherwise.
CGenre: CGenre is equal to 1 if both artists in a pair are from
the same genre; 0 otherwise.
SPviews: SPviews represents the number of sender paged views.
SSongs: SSongs represents the number of songs posted on the
sender web page.
SBand: SBand is equal to 1 if the sender belongs to a band; 0
otherwise.
SAudience: SAudience is equal to 1 if the sender performed in
front of an audience larger than 700 people; 0 otherwise.
SYActive: SYActive is equal to 1 if the sender was active for
more than six years; 0 otherwise.
RPviews: RPviews represents the number of receiver paged
views.
RSongs: RSongs represents the number of songs posted on the
receiver web page.
RBand: RBand is equal to 1 if the receiver belongs to a band; 0
otherwise.
RAudience: RAudience is equal to 1 if the receiver performed
in front of an audience larger than 700 people; 0 otherwise.
RYActive: RYActive is equal to 1 if the receiver was active for
more than six years; 0 otherwise.
07"3*"5&4'035)&64*$08/-0"%&-"5*0/4)*1
CRegion: CRegion is equal to 1 if both artists in a pair are from
the same region; 0 otherwise.
CGenre: CGenre is equal to 1 if both artists in a pair are from
the same genre; 0 otherwise.
PPviews: PPviews represents the number of provider paged
views.
PSongs: PSongs represents the number of songs posted on the
provider web page.
PBand: PBand is equal to 1 if the provider belongs to a band; 0
otherwise.
PAudience: PAudience is equal to 1 if the provider performed in
front of an audience larger than 700 people; 0 otherwise.
PYActive: PYActive is equal to 1 if the provider was active for
more than six years; 0 otherwise.
DPviews: DPviews represents the number of downloader paged
views.
DSongs: DSongs represents the number of songs posted on the
downloader web page.
DBand: DBand is equal to 1 if the downloader belongs to a
band; 0 otherwise.
DAudience: DAudience is equal to 1 if the downloader per-
formed in front of an audience larger than 700 people; 0 otherwise.
DYActive: DYActive is equal to 1 if the downloader was active
for more than six years; 0 otherwise.
REFERENCES
Atchade, Yves F. (2006), “An Adaptive Version for the Metropolis
Adjusted Langevin Algorithm with a Truncated Drift,” &5)0%
0-0(:"/%0.165*/(*/11-*&%30#"#*-*5:, 8 (June), 235–54.
Barnard, John, Robert McCulloch, and Xiao-Li Meng (2000),
“Modeling Covariance Matrices in Terms of Standard Devia-
tions and Correlations, with Application to Shrinkage,” 5"5*4
5*$"*/*$", 10, 1281–1311.
Dekker, David, David Krackhardt, and Tom A.B. Snijders (2007),
“Sensitivity of MRQAP Tests to Collinearity and Autocorrela-
tion Conditions,” 4:$)0.&53*,", 72 (December), 563–81.
DeSarbo, W.S., Y. Kim, and D. Fong (1998), “A Bayesian Multi-
dimensional Scaling Procedure for the Spatial Analysis of
Revealed Choice Data,” 063/"- 0' $0/0.& 53*$4, 89 (1/2),
79–108.
Faust, Katherine (2007), “Very Local Structure in Social Net-
works,” 0$*0-0(*$"-&5)0%0-0(:, 37 (December), 209–256.
Fienberg, Stephen E., Michael M. Meyer, and Stanley S. Wasser-
man (1985), “Statistical Analysis of Multiple Sociometric Rela-
tions,” 063/"- 0' 5)& .&3*$"/ 5"5*45*$"- 440$*"5*0/, 80
(March), 51–67.
Frank, Ove and David Strauss (1986), “Markov Graphs,” 063/"-
0'5)&.&3*$"/5"5*45*$"-440$*"5*0/, 81 (September), 832–42.
Gelman, Andrew, Xiao-li Meng, and Hal Stern (1996), “Posterior
Predictive Assessment of Model Fitness via Realized Discrep-
ancies,” 5"5*45*$"*/*$", 6 (October), 733–807.
Granovetter, Mark (1985), “Economic Action and Social Struc-
ture: The Problem of Embeddedness,” . &3*$"/ 063/"- 0'
0$*0-0(:, 91 (November), 481–510.
Handcock, Mark S., Adrian E. Raftery, and Jeremy Tantrum
(2007), “Model-Based Clustering for Social Networks,” 063
/"-0'5)&0:"-5"5*45*$"-0$*&5:, 170 (March), 301–354.
Hoff, Peter D. (2005), “Bilinear Mixed-Effects Models for Dyadic
Data,” 063/"- 0' 5)& .&3 *$"/ 5" 5*45*$" - 440$*" 5*0/, 100
(March), 286–95.
———, Adrian E. Raftery, and Mark S. Handcock (2002), “Latent
Space Approaches to Social Network Analysis,” 063/"-0'5)&
.&3*$"/5"5*45*$"-440$*"5*0/, 97 (December), 1090–1098.
Holland, Paul W. and Samuel Leinhardt (1981), “An Exponential
Family of Probability Distributions for Directed Graphs,” 063
/"-0'5)&.&3*$"/5"5*45*$"-440$*"5*0/, 76 (March), 33–50.
Iacobucci, Dawn (1989), “Modeling Multivariate Sequential
Dyadic Interactions,” 0$*"-&5803,4, 11 (December), 315–62.
——— and Nigel Hopkins (1992), “Modeling Dyadic Interactions
and Networks in Marketing,” 063/"-0'"3,&5*/(&4&"3$),
26 (February), 5–17.
——— and Stanley Wasserman (1987), “Dyadic Social Inter-
actions,” 4:$)0-0(*$"-6--&5*/, 102 (September), 293–306.
Iyengar, Raghuram, Christophe Van den Bulte, and Thomas W.
Valente (2011), “Opinion Leadership and Social Contagion in
New Product Diffusion,” "3,&5*/($*&/$&, 30 (2),195–212.
Leskovec, Jure, Kevin J. Lang, Anirban Dasgupta, and Michael W.
Mahoney (2008), “Statistical Properties of Community Structure
in Large Social and Information Networks,” in 0$*"-
&5803,4"/%&#*4$07&3:"/%70-65*0/0'0..6/*5*&4.
New York: Association for Computing Machinery, 695–704.
Li, H. and E. Loken (2002), “A Unified Theory of Statistical
Analysis and Inference for Variance Component Models for
Dyadic Data,” 5"5*45*$"*/*$", 12, 519–35.
McPherson, M., L. Smith-Lovin, and J. Cook (2001), “Birds of a
Feather: Homophily in Social Networks,” //6"- &7*&8 0'
0$*0-0(:, 27, 415–44.
Nair, Harikesh S., Puneet Manchanda, and Tulikaa Bhatia (2010),
“Asymmetric Social Interactions in Physician Prescription
Behavior: The Role of Opinion Leaders,” 063/"-0'"3,&5*/(
&4&"3$), 47 (October), 883–95.
Narayan, V. and S. Yang (2007), “Bayesian Analysis of Dyadic
Survival Data with Endogenous Covariates from an Online
Community,” working paper, Johnson Graduate School of Man-
agement, Cornell University.
Pattison, Philippa E. and Stanley Wasserman (1999), “Logit Models
and Logistic Regressions for Social Networks: II. Multivariate
Relations,” 3*5*4)063/"-0'"5)&."5*$"-"/%5"5*45*$"-4:
$)0-0(:, 52 (November), 169–93.
Rivera, Mark T., Sara B. Soderstrom, and Brian Uzzi (2010),
“Dynamics of Dyads in Social Networks: Assortative, Rela-
tional, and Proximity Mechanisms,” //6"-&7*&80'0$*0-
0(:, 36 (August), 91–115.
Robins, Garry L., Philippa E. Pattison, Y. Kalish, and D. Lusher
(2007), “An Introduction to Exponential Random Graph (p*)
Models for Social Networks,” 0$*"-&5803,4, 29 (2), 173–91.
———, Tom A.B. Snijders, Peng Wang, Mark S. Handcock, and
Philippa E. Pattison (2007), “Recent Developments in Exponen-
tial Random Graph (p*) Models for Social Networks,” 0$*"-
&5803,4, 29 (May), 192–215.
Snijders, Tom A.B. (2005), “Models for Longitudinal Network
Data,” in 0%&-4"/%&5)0%4*/0$*"-&5803,/"-:4*4, Peter
J. Carrington, John Scott, and Stanley Wasserman, eds. New
York: Cambridge University Press.
———, Philippa E. Pattison, Garry L. Robins, and Mark S. Hand-
cock (2006), “New Specifications for Exponential Random
Graph Models,” 0$*0-0 (*$"- & 5)0%0-0 (:, 36 (December),
99–153.
Stephen, Andrew T. and Olivier Toubia (2010), “Deriving Value
from Social Commerce Networks,” 063/"- 0' "3,&5*/(
&4&"3$), 47 (April), 215–28.
Sweeting, Trevor (2007), “Discussion on the Paper by Handcock,
Raftery and Tantrum,” 063/"-0'5)&0:"-5"5*45*$"-0$*&5:
, 170 (March), 327–28.
Trusov, Michael, Anand V. Bodapati, and Randolph E. Bucklin
(2010), “Determining Influential Users in Internet Social Net-
works,” 063/"-0'"3,&5*/(&4&"3$), 47 (August), 643–58.
Trusov, Michael, R.E., Bucklin, and K. Pauwels (2009), “Estimat-
ing the Dynamic Effects of Online Word-of-Mouth on Member
Growth of a Social Network Site,” 063/"-0' "3,&5*/(, 73
(September), 90–102.
Tuli, Kapil R., Sundar G. Bharadwaj, and Ajay K. Kohli (2010),
“Ties That Bind: The Impact of Multiple Types of Ties with a
Customer on Sales Growth and Sales Volatility,” 063/"- 0'
"3,&5*/(&4&"3$), 47 (February), 36–50.
Van den Bulte, Christophe and Rudy K. Moenaert (1998), “The
Effects of R&D Team Co-location on Communication Patterns
Among R&D, Marketing and Manufacturing,” "/"(&.&/5
$*&/$&, 44 (November), Part 2 of 2, 1–18.
——— and Stefan Wuyts (2007), “Social Networks and Market-
ing,” working paper, Marketing Science Institute.
Warner, R., D.A. Kenny, and M. Stoto (1979), “A New Round
Robin Analysis of Variance for Social Interaction Data,” 063
/"-0'&340/"-*5:"/%0$*"-4:$)0-0(:, 37 (10), 1742–57.
Wasserman, Stanley and Dawn Iacobucci (1988), “Sequential
Social Network Data,” 4:$)0.&53*,", 53 (June), 261–82.
——— and Philippa Pattison (1996), “Logit Models and Logistic
Regressions for Social Networks: I. An Introduction to Markov
Graphs and p*,” 4:$)0.&53*,", 61 (September), 401–425.
Watts, Duncan J., and Peter Sheridan Dodds (2007), “Influentials,
Networks, and Public Opinion Formation,” 063/"- 0 ' 0/
46.&3&4&"3$), 34 (May), 441–58.
Wedel, M. and W.S. DeSarbo (1996), “An Exponential Family
Scaling Mixture Methodology,” 063/"-0'64*/&44"/%$0
/0.*$45"5*45*$4, 14 (4), 447–59.
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