Asim AnsAri, Oded KOenigsberg, and FlOriAn stAhl *

Firms are increasingly seeking to harness the potential of social networks for marketing purposes. therefore, marketers are interested in understanding the antecedents and consequences of relationship formation within networks and in predicting interactivity among users. the authors develop an integrated statistical framework for simultaneously modeling the connectivity structure of multiple relationships of different types on a common set of actors. their modeling approach incorporates several distinct facets to capture both the determinants of relationships and the structural characteristics of multiplex and sequential networks. they develop hierarchical bayesian methods for estimation and illustrate their model with two applications: the first application uses a sequential network of communications among managers involved in new product development activities, and the second uses an online collaborative social network of musicians. the authors’ applications demonstrate the benefits of modeling multiple relations jointly for both substantive and predictive purposes. they also illustrate how information in one relationship can be leveraged to predict connectivity in another relation.

Keywords : social networks, online networks, bayesian, multiple relationships, sequential relationships

modeling multiple relationships in social networks

The rapid growth of online social networks has led to a social commerce (Stephen and Toubia 2010; Van den Bulte resurgence of interest in the marketing field in studying the and Wuyts 2007) using interorganizational networks. structure and function of social networks. A better under - Within the firm, intraorganizational networks of managers standing of social networks can enable managers to compre - play a crucial role in cross-functional integration, as is the hend and predict economic outcomes (Granovetter 1985) case with networks of marketing and organizational profes - and, in particular, to interface with both external and internal sionals engaged in new product development (Van den actors. Online brand communities, which are composed of Bulte and Moenaert 1998). users interested in particular products or brands, allow such As Van den Bulte and Wuyts (2007) point out, network an external interface with customers. Such communities not structure has implications for power, knowledge dissemina - only help firms interact with customers and prospects but tion, and innovation within firms and for contagion and dif - also enable customers to communicate and exchange infor - fusion among customers. Thus, understanding and predict - mation with each other, consequently increasing the value ing the patterns of interactions and relationships among that can be derived from a firm’s products. network members is an important first step in using them Similarly, firms forge alliances and enter into collabora - effectively for marketing purposes. The focus of social net - tive relationships with other firms for coproduction and 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 *Asim Ansari is the William T. Dillard Professor of Marketing (e-mail: described by reciprocity, clustering, transitivity, and other maa48@ columbia.edu), and Oded Koenigsberg is Barbara & Meyer Feld - berg Associate Professor (e-mail: [email protected]), Columbia Busi - measures of local and global structure using a combination ness School, Columbia University. Florian Stahl is Assistant Professor, of assortative, relational, and proximity perspectives (Rivera, Department of Business Economics, University of Zurich (e-mail: florian. Soderstrom, and Uzzi 2010). stahl@ uzh.ch). Christophe Van den Bulte served as associate editor for this article. Actors belonging to a social network connect with each other using multiple relationships, possibly of different

© 2011, American Marketing Association Journal of Marketing Research ISSN: 0022-2437 (print), 1547-7193 (electronic) 713 Vol. XLVIII (August 2011), 713 –728 714 JOurnAl OF mArKeting reseArch, August 2011 types. In this article, we develop statistical models of multi - ships of different types. Our modeling framework has sev - ple relationships that yield an understanding of the drivers eral novel features when compared with previously pro - of multiplex relationships and predict the connectivity posed models for multirelational network data. Specifically, structure of such multiplex networks. our framework can (1) model multiple relationships of dif - Multiplexity of relationships can arise from different ferent types (i.e., weighted, unweighted, undirected, and modes of interaction or because of different roles people directed), (2) model sequential relationships, (3) leverage play within a network setting. For example, in many online partial information from one network (or relationship) to networks, members can form explicit friendship and busi - predict connectivity in another relationship, (4) accommo - ness relations, exchange content and communicate with one date missing data in a natural way, (5) capture sparseness in another. The relationships that connect a group of actors can weighted relationships, (6) incorporate sources of dyadic differ not only in their substantive content but also in their dependence, (7) account for higher order effects such as tri - directionality and intensity. For example, some relation - adic effects, and (8) include continuous covariates. Although ships are symmetric in nature, whereas others can be previous models have incorporated some of these aspects, directed. Some relationships involve the flow of resources, we do not know of any research in the social network analy - thus necessitating a focus on the intensity of such weighed sis literature that simultaneously incorporates all of them. connections. Finally, multivariate patterns of connections We illustrate the benefits of our approach using two can also arise from viewing the same relationship at differ - applications. Our first application involves sequential net - ent points, as is the case of sequential networks. work data that studies network structure over two points. An understanding of multiplex patterns in network struc - We reanalyze data from Van den Bulte and Moenaert (1998) tures is important for marketers. For example, Tuli, Bharad - involving a network of research-and-development (R&D), waj, and Kohli (2010) find that in a business-to-business marketing, and operations managers who are engaged in setting, increasing multiplexity in relationships leads to an new product development. The data contain communica - increase in sales and to a decrease in sales volatility to a tions among these managers both before and after colloca - customer. Multiplexity contributes to the total strength of a tion of R&D teams. The results show that substantive con - tie and increases the number of ways one can reciprocate clusions can be affected if the full generality of our favors (Van den Bulte and Wuyts 2007). Thus, it is relevant framework is not utilized. We also show how our methods for identifying influential actors such as opinion leaders in can be used to leverage information from one relationship diffusion contexts and powerful executives within intra - to predict the connections in another relationship. organizational networks. In analyzing sequential relation - In our second application, we use data from an online ships, a multivariate analysis can help investigate the impact social networking site involving the interactions among a of managerial interventions on the relationship structure of set of musicians. We model friendship, communications, a network across points. Sequential dyadic interactions are and music download relationships within this network to also useful for understanding the dynamics of power and show how a combination of directed and undirected, and cooperation in intraorganizational networks and in model - weighted and unweighted, relationships can be modeled ing long-term relationships between buyers and sellers in jointly. We analyze the determinants of these relationships business markets (Iacobucci and Hopkins 1992). Finally, and assess the importance of our model components in cap - when marketers are interested in predicting relationship pat - turing different facets of the network structure. Our results terns, multiplexity allows leveraging information from one show that artists exhibit similar network roles across the relationship to predict connections on other relationships. three relationships and that these relationships are mostly Researchers can obtain a substantive understanding of influenced by common antecedents. We also show that multiplex relationships by simultaneously analyzing the when dealing with weighted relationships (e.g., music multiple connections among the network actors. They can downloads), it is crucial to jointly model both the incidence investigate whether these multiple relationships exhibit and the intensity of such relationships, rather than simply multiplex patterns that are characterized by the flow of mul - focusing on the intensity; otherwise, prediction and recov - tiple relationships in the same direction or whether they rep - ery of structural characteristics suffers appreciably. resent patterns of generalized exchange in which a tie in one We organize the rest of the article as follows: The next direction on one relationship is reciprocated with a connec - section provides a brief review of the marketing and statisti - tion in the other direction using different relationships. cal literature on social networks. Then, we present the com - However, most models of social networks analyze a single ponents of our modeling framework and describe inference relationship among network members. When the lens is and identification of model parameters. The following two trained on a single relationship, an incomplete understand - sections describe the two applications. Finally, we conclude ing of the nature of linkages can result. For example, it is with a discussion of our contributions and model limitations unclear whether people play a similar role across multiple and outline future research possibilities. relationships. A joint analysis can also help uncover com - mon antecedents that affect relationships. Moreover, if LiteRatuRe Review some relationships exhibit unique patterns, such uniqueness Social network data offer considerable opportunities for can emerge only when multiple relationships are contrasted. research in marketing, as Van den Bulte and Wuyts (2007) In this article, we develop an integrated latent-variable identify in their expansive survey of the role and importance framework for modeling multiple relationships. We make a of social networks in the marketing field. Most research in methodological contribution to the social networking litera - marketing on social networks falls into one of two streams. ture in both marketing and the wider social sciences by In the first stream, researchers explore the impact of word offering a rich framework for modeling multiple relation - of mouth on the behavior of others and thus are primarily modeling multiple relationships in social networks 715 concerned about the role of social interactions and conta - able when interest is in analyzing such complex multivari - gion (Iyengar, Van den Bulte, and Valente 2011; Nair, Man - ate data structures. chanda, and Bhatia 2010; Trusov, Bodapati, and Bucklin Whereas the preceding methods explicitly model the net - 2010; Watts and Dodds 2007). work structure, MRQAP methods offer a nonparametric Research in the second stream focuses on modeling net - alternative for conducting permutation tests to assess work structure. Iacobucci and Hopkins’s (1992) work is a covariate effects using multiple regression while correcting pioneering contribution in this area. The focus here is on for the dependency and autocorrelation present in network understanding the antecedents of relationship formation and data. The MRQAP approach is useful for continuous data. It in studying how interventions influence future connectivity can be used for binary relations using a linear probability (Van den Bulte and Moenaert 1998). The current study adds model, and to a certain extent for count data; however, its to this second stream of research by offering a comprehen - effectiveness for multivariate relations of different types is sive framework for modeling multivariate or sequential not clear. relations. Sequential data can also be modeled using two There is a rich history of statistical modeling of network approaches: a multivariate approach such as ours and the structure within sociology and statistics that spans more conditional, continuous-time approach popularized by Sni - than 70 years. More recent approaches stem from the log- jders (2005). The multivariate approach models the network linear p 1 model developed by Holland and Leinhardt (1981), at each point in time and thus is useful when one is inter - which assumes independent dyads. Because the p 1 model is ested in assessing the impact of interventions that occur incapable of representing many structural properties of the between these discrete times. In contrast, the continuous- data, the literature has proposed two general ways of cap - time approach is inherently dynamic, focusing on either turing the dependence among the relationships. The first edge-oriented or node-oriented dynamics, and can model approach uses exponential random graph models or p* the evolution of the network one edge at a time. However, models (Frank and Strauss 1986; Pattison and Wasserman this approach is limited to binary relations, whereas the 1999; Robins et al. 2007; Snijders et al. 2006; Wasserman multivariate approach that we use can handle both weighted and Pattison 1996) that capture the dependence pattern in and binary relationships. the network using a set of statistics that embody important In contrast to the current study, most models of social net - structural characteristics of the network. However, care is work structure analyze a single relationship, and to the best necessary when using these models because parameter esti - of our knowledge, none have incorporated the entire con - mation sometimes suffers from model degeneracy, and how stellation of desirable model characteristics that we outlined to handle this degeneracy is an active area of research. The in the introduction. In particular, there has been no work on second approach handles the dependence among the dyads using the latent space framework for modeling multivariate using correlated random effects and latent positions in a relationships or sequential data. Euclidean space for the individual people (Handcock, Although some researchers have modeled multiple rela - Raftery, and Tantrum 2007; Hoff 2005; Hoff, Raftery, and tionships, these models either assume independence across Handcock, 2002 ). In addition to these two approaches, mul - dyads, which is restrictive, or limit attention to binary rela - tiple regression quadratic assignment procedure (MRQAP) tions (Fienberg, Meyer and Wasserman 1985; Iacobucci methods (Dekker, Krackhardt, and Snijders 2007) have also 1989; Iacobucci and Wasserman 1987; Pattison and Wasser - been used in network analysis to account for dependence man 1999; Van den Bulte and Moenaert 1998). Thus, there among dyads. is a need for an integrative framework for modeling multi - Exponential random graph models describe the network ple relationships of different types (i.e., binary or weighted) using a set of summary statistics, such as the total number in a flexible way. The latent space framework offers such of ties, the number of triangles, and the degree of distribu - flexibility, and using it, we develop an integrated approach tion, among others. This is good for describing “global” for multiple relationships in the following section. properties and in assessing particular hypotheses of substan - tive interest, such as the extent of reciprocity or clustering ModeLing FRaMewoRk and triadic closure. In contrast, latent space models capture We develop a modeling framework for the simultaneous the “local” structure by estimating a latent variable for each analysis of multiple relationships among a set of network node in the network, which describes a person’s position in actors. Our framework accommodates multiple relation - the network. These models are thus suitable when the focus ships of different types and also enables us to simultane - is on understanding the determinants of connectivity using ously model the determinants of the relationships as well as covariates and in identifying influential people. The latent the structural characteristics such as the extent of reci - variable framework is capable of recovering the structural procity or transitivity within each relationship and across characteristics using a small set of model parameters (simi - relationships. When analyzing multiple relationships, struc - lar to nuisance parameters) and thus can be parsimonious in tural characteristics of interest include those that account for some contexts. multiplex patterns (i.e., flow of multiple relationships in the When researchers are interested in specific substantive same direction) and exchange , in which a flow in one direc - hypotheses and when all relationships are binary in nature, tion on one relationship is reciprocated with a flow in the they may prefer exponential random graph models. How - other direction using a different relationship. Similarly, pat - ever, the latent variable framework can accommodate mul - terns of transitivity that involve more than one relationship tiplex relationships of different types, including weighted can also be investigated. When focusing on the determi - relationships, and can also handle missing data in a straight - nants of relationships, we can infer how the attributes of the forward fashion using data augmentation. Thus, it is prefer - network actors influence the formation of relationships 716 JOurnAl OF mArKeting reseArch, August 2011 between them. Here the interest is in understanding whether ascertain whether a specification that directly models the actors exhibit similar popularity and expansiveness across intensity (such as a Poisson specification; e.g., Hoff 2005) different relationships, and whether homophily governs is sufficient for weighted relationships. Therefore, we use a relationship formation. multivariate correlated hurdle count specification to jointly The multiple relationships describing a common set of model both the incidence of the relationship within a dyad actors can vary along different facets, such as existence, and the intensity of the relationship conditional on the exis - intensity, and directionality. A relationship is directed if we tence of the tie. We model the incidence using the ordered can distinguish the sender and receiver of the tie. For exam - pair of binary variables (X ij2, I , X ji2, I ). Then, the magnitude ple, a communication relationship typically has a sender and of the relationship, conditional on its existence in a given a receiver. In contrast, relationships could be undirected, direction, can be modeled using the positively valued trun - such as a collaboration relationship. In modeling both cated count variables X ij2, S or X ji2, S . directed and undirected relationships, the focus of the analy - dyadic multigraph . Bringing together the two relation - n sis could be on modeling the existence of a relationship (i.e., ships, we can then specify the C2 dyads in the multirela - the presence or absence of a tie) or on the intensity of a tional social network using the dyad-specific random weighted tie (e.g., the intensity of the flow of resources variables: between a pair of people). Our objective is to show how   such disparate relationships can be jointly modeled within a XXij11, ji common framework. In the following section, we describe   Dij =  XXij22,, I, ji I  ,.i < j formally our model.    XXij22,, S, ji S  Model description We describe our model using two relationships. Although The relationships can be further specified in terms of under - these two relationships could represent a single relationship lying latent variables. The latent variable specification observed over different time periods, for the sake of gener - enables us to model these random variables in terms of ality, we describe a model for two distinct, directed relation - dyad- and actor-specific covariates. ships of different types. 1 These two relationships are Latent variable specification . We use underlying latent observed over the same set of n actors. utilities u ij1 for modeling the existence of a tie in the direc - directed binary relationship . The first relationship is tion (i Æ j) and u ji1 in the reverse direction, for the first directed and binary. Thus, we can distinguish between the relationship: sender and the receiver of the tie and the sociomatrix  matrix, X , which shows that the incidence of ties among  10,,if uij1 > 1 ()1 X = actors can, therefore, be asymmetric. We use the ordered ij1   0,,otherwise pair of binary dependent variables {X ij1 , X ji1 } to represent the presence or absence of ties for a pair of actors i and j.  1, if u ji1 > 0, The variable X ij1 specifies the existence of interaction in the Xji1 =   0,.otherwise direction from i to j (i.e., i Æ j), whereas X ji1 represents the  presence of a tie in the opposite direction from j to i (i.e., i ¨ j). For the second relationship, let u ij2 and u ji2 represent the directed weighted relationship . The second relationship underlying utilities that characterizes the existence of the is directed and weighted (i.e., valued). The entries in the relationship. Again, we assume that associated matrix, X2, indicate the bidirectional intensity of   10,,if uij1 > interaction between the different pairs of people. Here, we ()2 Xij2, I =  assume a count variable for the intensity, because this is  0,,otherwise consistent with our second application presented in the sec -  tion “Online Social Network.” However, our model can be  1, if u ji2 > 0, X = adapted for continuous or ordinal measures of intensity. An ji2, I   0,.otherwise ordered pair of count variables (X ij2 , X ji2 ) can represent the observed intensity of interaction in the dyad, where the We model the truncated counts, conditional on a tie in a variable X ij2 specifies the strength of the interaction from i given direction, using a Poisson distribution truncated at to j (i.e., i Æ j), and X ji2 specifies the intensity in the reverse zero; in other words, direction. In modeling this weighted relationship, we deviate from (3) Xij2, S ~ tPoisson( lij ) if u ij2 > 0, and the previous literature on social networks by jointly model - X ~ tPoisson( l ) if u > 0, ing both the existence and the intensity of the relationship. ji2, S ji ji2 This allows us to distinguish between the mechanisms that where the lij and lji are the rate parameters of the Poisson. drive the incidence from those that affect the intensity of Covariates and homophily . Each latent utility is composed relationships. In addition, it also accommodates a prepon - of a systematic part involving the observed covariates and a derance of zeros due to sparseness of ties. We can then stochastic part that incorporates unobserved variables. We distinguish between dyad-specific covariates and individual- specific covariates. Let xd and xd be vectors of dyad- 1We limit our model description to two relationships for clarity of pres - ij1 ij2 entation. Our approach, however, can be extended readily to more relation - specific covariates that influence the two relationships. ships, including undirected ones, as we do in our second application. These allow for homophily, which implies that people who modeling multiple relationships in social networks 717 share observable characteristics tend to form connections. If the off-diagonal submatrices in Sq indicate positive corre - We also use individual-specific covariates for modeling lations, this could possibly be a result of a latent trait gov - directed relationships. Let xi1 , and xi2 be the vector of indi - erning commonality in behavior. In contrast, if these sub - vidual i’s covariates, respectively, for the two relationships. matrices are zero, the relationships can be modeled d The vector xij1 = ( xij1 , xi1 , xj1 ) then represents all the covari - separately as the attractiveness and expansiveness parame - ates that affect the tie in the direction (i Æ j) for the first ters will be independent across the different relationships. d relationship, and xji1 = ( xji1 , xj1 , xi1 ) represents the covari - Other patterns of correlations are also possible, and their ates for the reverse direction (i ¨ j). The sender and meaning and relevance depend on the empirical context of a receiver effects are important to model the asymmetry in the particular application. two directions. For the weighted relationship, we can further Latent space. Social networks also exhibit patterns of distinguish between the incidence and intensity components. higher-order dependence involving triads of actors. There is d Therefore, we use covariate vectors xij2, I = ( xij2, I , xi2, I , xj2, I ) a potential for misleading inferences if such extradyadic d and xji2, I = ( x ji2, I , xj2, I , xi2, I ) for the binary component, effects are ignored. Hoff and his colleagues demonstrate d where, for example, x ij2, I contains a subset of the dyad- how transitivity and other triad-specific structural charac - d specific variables in x ij2 that affect intensity, and xi2, I is teristics such as balance and clusterability can be modeled similarly a subset of xi2 . We use analogously defined vec - using a latent space framework (Handcock, Raftery, and tors xij2, S and xji2, S to model the Poisson rate parameters lij Tantrum 2007; Hoff 2005; Hoff, Raftery, and Handcock and lji . 2002). We employ a latent space for each relationship. We Heterogeneity . The dyads cannot be considered inde - assume that individual i has a latent position zir in a Euclid - pendently, because multiple dyads share a common actor ean space associated with each relationship r, where r Œ {1, either as a sender or receiver. Accounting for such depend - 2}. The latent space framework stochastically models tran - ence is important for obtaining proper inferences about sub - sitivity; if i is located close to individual j and if j is located stantive issues. Therefore, we use heterogeneous and corre - close to individual k, then, because of the triangle inequal - lated random effects to account for the dependence ity, i will also be close to k. structure. Whereas for undirected relationships, a single ran - In the current study, we follow Hoff (2005) and use 3 dom effect is needed, for directed relationships, we can use the inner-product kernel z ¢ir zjr . In particular, for a two- two distinct random effects per actor to distinguish between relationship model, we use two kernels, one for each rela - expansiveness , which is the propensity to “send” ties and tionship, represented generically as z ¢ir zjr . The latent vectors popularity, or attractiveness , which is the propensity to for each relationship are assumed to come from a relation - “receive” ties. The expansiveness parameter ai captures the ship-specific multivariate-normal distribution zir ~ N(0, outdegree, which is the number of connections emanating Szr ). The dimensionality of the latent space can be deter - from an individual i, and the attractiveness parameter bi mined using a scree plot of the sum of the mean absolute captures the indegree, which is the number of connections prediction error of the entire triad census versus the dimen - impinging on an individual. Thus, for the directed binary sionality, as is usually done in the multidimensional scaling 4 relationship, we use random effects ai1 and bi1 . We simi - literature. larly use ai2, I and bi2, I for the incidence component and ai2, Full model. Bringing together all the components of the S and bi2, S for the intensity equations of the weighted model, we can write the latent utilities and response propen - directed relationship. sities as follows: Let qi = { ai1 , bi1 , ai2, I , bi2, I , ai2, S , bi2, S }. We allow these ue=++++x µ αβzz′ , random effects to be correlated across the relationships and ij11111111 ij i j i j ij 2 ′ assume that qi is distributed multivariate normal N(0, Sq), u ji111=+x jiµ α j111111++βiijjizz +e , where S is an unrestricted covariance matrix consisting of q ′ the following submatrices: uij2222=++x ij,, Iµµ+ Iαβ i , I j 2 , I + zzij22+ e ijI 2, , ′   u ji2222=+++x ji,, Iµµα I j , I β i 2 , Iz i222z jjiI+ e 2. , ΣΣθθ,,11 12 Σ =   . θ logλαβ=+++x µ z′′ z + e ,  ΣΣθθ,,21 22  ij ij22,, S S i 2 , S j 2 , S i 2jijS22, ′ logλαβji=+++x ji22,, Sµ S j 2 , S i 2 , Sz i 2z jjiS22+ e , . The diagonal submatrices capture the within-relationship covaria tion in the random effects within a relationship. A We assume that the vector of all errors eij is distributed mul - positive correlation between the random effects for a rela - tivariate normal N(0, S). Also, qi ~ N(0, Sq) and zir ~ N(0, tionship implies that popular individuals also tend to reach Szr ), "r. out more to others. The off-diagonal submatrices capture identification . Not all parameters of the model are identi - correlation across relationships and help determine whether fied. The error variance matrix S has a special structure individuals exhibit similar tendencies across relationships.

3Other kernels such as those based on the Euclidean norm can also be 2Even though we assume a symmetric distribution for heterogeneity, used. We leave a detailed examination of the pros and cons of using differ - when it is combined with the data from the individuals, the resulting poste - ent kernel forms for further research. rior random effects can mimic skewed degree distributions that can arise 4The Bayes factor can also be used to determine dimensionality. How - from a preferential attachment mechanism. We verified this using a simu - ever, it is difficult to compute in our model given the high dimensional lation that generated data from a preferential attachment mechanism and numerical integration that is involved in obtaining the likelihood for each were able to recover highly skewed degree distributions. Details of this observation. Therefore, we opt for the predictive MAD criterion that simulation are available on request. focuses on triad census recovery. 718 JOurnAl OF mArKeting reseArch, August 2011 because of scale restrictions on the binary utilities and Bayesian estimation because of exchangeability considerations stemming from We now describe briefly our inference procedures. The the fact that the labels i and j are arbitrary within a pair. As likelihood for the model is computationally complex. Con - the scale of the utilities of the binary responses cannot be ditional on the random effects and latent positions, the determined from the data, the error variances associated dyad-specific likelihood requires numerical integration to with the binary components are set to 1. In addition, sym - obtain the multivariate normal cumulative distribution func - metry restrictions on the correlations stem from the tion. Moreover, the unconditional likelihood for the entire exchangeability considerations, and the resulting variance network requires additional multiple integration of very matrix can be written as follows: high dimensionality because of the crossed nature of the random effects. The dependency structure of our model is XXXXXX ij112222 ji ij,,,, I ji I ij S ji S considerably more intricate than what is typically encoun - Xij1  1X j ρ123ρρσρσρ 45 tered in typical panel data settings in marketing, because we X  X 1 ρρρσρσρ ji1  ji2, S 3 254 cannot assume independence across individuals or dyads for ()5 Σ= XijjI2,  X ji2, S 1 ρσρσρ678 . computing the unconditional likelihood. Therefore, we use X ji2, I  X ji2, S 1 σρ87 σρ  Markov chain Monte Carlo (MCMC) methods involving a X  X σσρ2  combination of data augmentation and the Metropolis– ij2, S  ji2, S 9  2 X ji2, S  X ji2, S σ  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 - The correlation parameter r1 captures the impact of common unobserved variables affecting the binary relationship and ships. The complexities involved in modeling multiple rela - tionships and the identification restrictions on the covari - also accounts for reciprocity. Similarly, r6 and r9 capture correlations for the weighted relationship and also account ance matrix S mean that the methods of inference for for reciprocity within this relationship. The correlation existing latent space models, as outlined, for example, in Hoff (2005) and Hoff, Raftery, and Handcock (2002), can - parameters r7 and r8 reflect common unobserved variables that influence both the incidence and intensity equations of not be used directly for our model. Therefore, we provide a the weighted component of the second relationship and are full derivation of the posterior full conditionals in Appendix akin to selectivity parameters. Note that the intensity equa - A. tions have a common variance. new PRoduCt deveLoPMent netwoRk The expansiveness and attractiveness random effects are In the first application, we illustrate our modeling frame - individual specific and are thus separately identifiable from work on sequential network data. We start with a special the equation errors that are dyad specific. The latent posi - case of our general modeling framework and handle the tions also are individual specific, but because they enter the simpler situation of a single directed binary relationship equations as interactions, they can be separately identified observed at two points in time. Therefore, we do not need from the random effects. However, because they appear in the intensity component in this application. We use the same bilinear form, we can only identify these subject to rotation data as in Van den Bulte and Moenaert (1998; hereinafter, and reflection transformations. Finally, the covariance VdBM) on communications among members of different matrices Sz1 and Sz2 associated with the latent space R&D teams and marketing and operations professionals parameters are restricted to be diagonal as their covariance who are all involved in new product development activities. terms are not identified. Moreover, each matrix has a com - Here, we briefly analyze this data set to revalidate the mon variance term across all the dimensions within a latent results in VdBM and to investigate whether our modeling space. framework (which differs significantly from that in VdBM) Table 1 summarizes how the different model parameters is better able to recover the structural characteristics of the can be related to substantive issues of interest. Given that the network and whether it generates different conclusions or model components work in tandem, a parameter may also additional insights. be related to other aspects apart from the one shown in the Data are available about communication patterns both table. For example, r1 is needed to capture reciprocity, but before and after the R&D teams were collocated into a new may also represent the impact of other shared unobservables. facility. The data set used in VdBM and the one we reana - lyze here come from a survey conducted in the Belgian sub - table 1 sidiary of a large U.S. telecommunication cooperation. The descriptiOn OF pArAmeters data consist of two 22 ¥ 22 binary (who talks to whom) sociomatrices, X1 and X2, one for 1990 (before collocation) variable effects and one for 1992 (after the R&D teams were collocated in a μ Covariate effects, homophily, and heterophily separate facility). The actors in both years are the same, 13 ~ αi Expansiveness, productivity R&D professionals spread over four teams and nine mem - βi Attractiveness, popularity, or prestige bers of a steering committee consisting of seven positions z Transitivity, balance, and clusterability ~i in marketing and sales and two in operations. The charac - r1, r6, r9 Reciprocity teristic element x ij, t in each of the two matrices is 1 if i r7, r8 Selectivity r3, r5 Generalized exchange reports to talk to j at least once a week in year t and 0 if oth - r2, r4 Multiplexity erwise. The specific area VdBM study is the impact of the Σθ Heterogeneity collocation intervention on the communication and coop - modeling multiple relationships in social networks 719 eration patterns among the different R&D teams and BETWTEAM ij = 1 if i and j are R&D professionals but in dif - between R&D and marketing/operations by contrasting ferent teams and 0 if otherwise, these patterns before and after collocation. INRD ij = 1 if i and j are R&D professionals and 0 if Barriers between R&D and marketing professionals otherwise, and resulting from differences in personality, training, or depart - INMKTOPS ij = 1 if i and j are both marketing or both opera - ment culture imply that intrateam and intrafunctional com - tions executives and 0 if otherwise. munication will be more prevalent than cross-team and Results cross-functional communication. The idea of collocation We estimated the four models using MCMC methods. was to foster communication among the different R&D Each MCMC run is for 250,000 iterations, and the results teams. Of the five hypotheses VdBM propose, the first two are based on 200,000 iterations after discarding a burn-in of test the communicative implications of the barriers between 50,000. R&D and marketing and relate to team- and function-specific Recovery of structural characteristics . We begin by com - homophily effects. The third and fourth hypotheses focus on effects of collocating R&D teams to foster communication paring the previously described models in their ability to among these teams. Finally, VdBM posit that collocating recover the structural characteristics of the network. Given R&D groups may not just foster between-group communi - our interest in modeling sequential relationships, we focus cation, but may even annihilate any difference between on aspects of the network structure that pertain to the two within- and between-group communication. relationships simultaneously. In particular, we compute sta - tistics involving the dyadic as well as transitivity patterns of VdBM use Wasserman and Iacobucci’s (1988) p 1 log-linear models to test their hypotheses. Our approach differs from interactions that span both years. that of these previous studies on several counts. First, in We can describe dyadic relationships in each year as belonging to one of the following three types: mutual (M), contrast to the p 1 model, we do not assume dyadic inde - pendence. In our model, the dyads are independent condi - asymmetric (A), and null (N). Observing across the two tional on the random effects but are dependent uncondition - years, we can construct the following ten possible combina - ally. Second, we allow for individual-specific expansiveness tions (Fienberg, Meyer, and Wasserman 1985): NN, AN, and attractiveness parameters in contrast to group-specific NA, MN, NM, A A, AA, AM, MA, and MM. The names are parameters to yield a richer specification of heterogeneity. self-explanatory for most pairs. For example, NN refers to Finally, we account for higher-order effects using a latent the number of dyads that are null in both years. Two pat - space. The added generality of our model is consistent with terns that require greater explanation are AA and A A. The VdBM’s (p. 16) call for “a new generation of models better pair AA represents a dyad in which one actor is connected able to handle triadic effects and other dependency issues.” to the other in both years but neither relationship is recipro - cated. The pair A A represents a dyad in which one actor ini - Models and variables tiates communication with the other in the first year and the We estimated four models on the data set: other actor reciprocates by initiating in the second year, a kind of generalized exchange. 1. The full model involves all the components that form part of Table 2 reports the recovery of the sequential dyadic pat - our modeling framework. These components include dyad- specific variables, attractiveness and expansiveness random terns. The columns report the absolute deviations between effects that are correlated both within and across years, sepa - the actual frequencies and those predicted by the different rate latent spaces for the two years, and correlated error models. The last row of the table reports the mean absolute terms for the utility equations of the four binary variables deviations (MAD) across all the patterns for each model. It characterizing a dyad. is clear that the uncorrelated model (Uncorr ), which ignores 2. The Uncorr model is a restriction of the full model. It cross-year linkages and models the two years separately, is assumes that the random effects and the utility errors are cor - significantly worse in recovering the cross-relationship dyadic related within a year but are uncorrelated across years. This patterns compared with the other models. All other models is akin to having a separate model for each year, and this are roughly similar in their recovery, with the team model offers limited leeway in modeling muliplexity. 3. The NoZiZj model is a restriction of the full model such that being the best. This indicates that it is important to model th e the higher-order terms that characterize the latent space, (i.e., the z i¢zj terms are not included. We use this model to assess table 2 whether using the latent space results in better recovery of the ApplicAtiOn 1: recOVerY OF crOss-relAtiOn dYAdic triadic structure of the network and whether it substantively cOunts affects conclusions. 4. The team model closely mimics the VdBM article within our modeling framework. In this model, we restrict the expan - Pattern Full noZiZj uncorr team siveness and attractiveness parameters to be the same for all NN 7.51 6.44 12.67 7.04 individuals within a group and also do not include the latent AN 3.00 4.17 4.24 2.70 space. 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 variables AA .92 1.89 1.89 .80 AA 1.97 .38 4.84 .02 The following variables used in our investigation are the AM 2.14 2.26 1.67 1.92 same as those VdBM use: MA .10 .80 .84 .77 MM 5.33 4.48 9.46 3.80 INTEAM ij = 1 if i and j are R&D professionals on the same team and 0 if otherwise, MAD 2.983 2.950 5.394 2.612 720 JOurnAl OF mArKeting reseArch, August 2011 two relations jointly so that we can recover the cross-relation Figure 1 dyadic patterns, as all the models, and, except Uncorr, the a And b VAlues FOr the mAnAgers in YeAr 1 accommodate correlations across the two relationships. We also investigate transitive patterns spanning both Β years to understand how well the model recovers 1.5 m extradyadic effects. This is necessary to assess whether m adding the latent space is important. Eight such transitivity 1.0 o effects are possible: {X ij1 , X jk1 , X ik2 }, {X ij1 , X jk2 , X ik1 }, r 0.5 4 m {X ij1 , X jk2 , X ik2 }, {X ij2 , X jk1 , X ik2 }, {X ij2 , X jk2 , X ik1 },{X ij2 , r1 o Xjk1 , X ik1 }, {X ij1 , X jk1 , X ik1 }, and {X ij2 , X jk2 , X ik2 }. For the m r4 r4 mm Α sake of brevity, we do not include a full table of results -3 -2 -r1 1 2 3 1 rr34 r r1 (available on request). We find that the full model performs m -0.52 r significantly better that all other models in recovering these r2 1 transitive patterns (MAD = 34.43). The full and Uncorr -1.0 r models (MAD = 47.09), both of which include the latent r3 2 1.5 space, perform significantly better than NoZiZj (MAD = - 109.46) and team (MAD = 164.69), which do not include Notes: The point labels indicate team membership. the higher-order effects. In summary, observing across all dyadic and triadic Figure 2 measures, we find that the full model always does better than Uncorr, thus highlighting the need for joint modeling. the a And b VAlues FOr the mAnAgers in YeAr 2 The full model also performs better than all the other mod - els in recovering the transitivity patterns, indicating that in Β this application, the latent space is important for handling 1.5 extradyadic patterns. We can conclude that, on the whole, m 1.0 m the full model recovers best the structural characteristics of m m o the network. 0.5 o r4 rm1

Parameter estimates and Hypotheses Α r r4 -3 -2 -1 3 r4 r 1 2 3 r2m 4 We begin by investigating whether the assumption of m r1 -0.5 r1 r2 individual-level expansiveness and attractiveness parame - rr12 ters in our models has support. Figures 1 and 2 report the a -1.0 and b values for the 22 managers for Years 1 and 2, respec - tively, for our full model. The labels for each point in the r3 -1.5 figures represent the team name. It is apparent from these figures that although some members of a group are clus - Notes: The point labels indicate team membership. tered together, many groups exhibit considerable within- group heterogeneity. This is particularly noticeable for the table 3 marketing group (m) and the R&D groups (r 3, r 1, and r 4), ApplicAtiOn 1: pArAmeter estimAtes FrOm diFFerent which exhibit greater within-group variability. This sug - mOdels gests that the data support a richer characterization of heterogeneity than what is possible with group-specific Pattern Full noZiZj uncorr team effects. INTERCEPT1 –1.48 –.87 –1.23 .48 Table 3 summarizes the parameter estimates. It is evident INTEAM1 5.03 3.02 4.03 3.42 from the table that the parameters values differ substantially BETWTEAM1 1.60 .88 .84 1.42 across the models in their magnitude and significance, indi - INMKTOPS1 1.03 1.40 1.02 .49 INTERCEPT2 –1.18 –.99 –1.12 .78 cating that the different model components influence sub - INTEAM2 3.51 3.01 3.17 3.79 stantive conclusions. Focusing on the bottom part of the BETWTEAM2 2.15 1.73 1.80 2.71 table, we see that models that do not include the latent space INMKTOPS2 1.23 1.61 1.02 .07 yield higher estimates of the error correlations, possibly due r1 .59 .83 .60 .79 to the confounding of variances across levels. We can use r2 .43 .64 — .63 r3 .26 .57 — .53 the coefficients to infer the degree of support for the differ - r .48 .72 .53 .67 ent hypotheses studied in VdBM. Table 4 reports the extant 6 of support for each hypothesis according to the different Notes: Bold indicates that the 95% posterior interval does not span 0. models. All other entries are computed from our reanalysis. Each entry for a particular model represents the probability whereas VdBM find mixed support for these. In particular, that the corresponding hypothesis is true under that model. we find that H 1b has significant probability across all our Several significant differences across the models are evident models, indicating that R&D professionals tended to com - from this table. municate predominantly with other R&D professionals The first two hypotheses pertain to within- and between- before the move. The full, NoZiZj, and Uncorr models sug - team homophily effects. Comparing the full model with gest strong support for H 2a and H 2b in contrast to team and VdBM indicates that our model supports both H 1 and H 2, VdBM. It seems that differences in support for this hypothe - modeling multiple relationships in social networks 721

table 4 ApplicAtiOn 1: hYpOthesis suppOrted bY diFFerent mOdels

Hypothesis Formal test Full team vdBM noZiZj uncorr

H1: Both before and after collocation, R&D professionals tend to H1a INTEAM1 > 0 .99 .99 .9 .99 .99 communicate with other R&D professionals rather than with H1b BETWTEAM1 > 0 .99 .99 n.s. .99 .91 marketing or operations executives. H1c INTEAM2 > 0 .99 .99 .99 .99 .99 H1d BETWTEAM2 > 0 .99 .99 .99 .99 .99

H2: Both before and after collocation, marketing and operations H2a INMKTOPS1 > 0 .93 .85 n.s. .99 .92 executives tend to communicate with members of their own H2b INMKTOPS2 > 0 .97 .55 n.s. .99 .94 department.

H3: R&D professionals have a higher probability of communicating H3 BETWTEAM2 > BETWTEAM1 .82 .99 .99 .99 .86 with members of other R&D teams after collocation than before.

H4: When collocating R&D teams implies increasing the physical H4 INRD2 > INRD1 .86 .99 .99 .99 .82 distance with other departments, R&D professionals’ tendency to communicate with other R&D people rather than executives from other departments increases.

H5: Before collocation, R&D professionals have a higher probability H5a INTEAM1 > BETWTEAM1 .99 .99 .99 .99 .99 of communicating with members of their own team rather than H5b INTEAM2 = BETWTEAM2 n.s. n.s. n.s. n.s. n.s. with members of other R&D teams. After collocation, the tendency to communicate among R&D people is as strong between as within teams.

Notes: Entries represent the probabilities of a hypothesis being true; n.s. indicates that the corresponding hypothesis is not supported. sis are driven by the extent of within-group heterogeneity such missing data can be handled seamlessly. In particular, captured by the model. Recall that both team and VdBM the covariance matrix of the random effects can be used to assume that all members within a group share the same leverage information from Year 1 to Year 2 about these indi - attractiveness and expansiveness parameters. Figures 1 and vidual-specific parameters. Therefore, we estimate our full 2 show, however, that there is considerable heterogeneity in and Uncorr models on such a data set to determine whether the recovered a and b parameters in the marketing group, incorporating cross-relationship linkages improves predic - and we find that failure to model this heterogeneity compre - tions in such situations. Note that in the full model, the data hensively can substantively affect conclusions. on Year 1 for the individuals in the marketing group can be We also find some differences in the support for the col - leveraged to predict relationships in Year 2. This is not pos - location hypotheses (H 3 and H 4) across the models. The full sible in the Uncorr model, in which the two years are mod - and Uncorr models have a lower probability associated with eled separately. Tables WA 1 and WA 2 of the Web Appendix these hypotheses compared with VdBM, team, and NoZiZj. (see http:// www. marketingpower.com/jmraug11 ) report how These differences can be explained by the presence or well the cross-year dyadic and transitivity patterns are recov - absence of the latent space for capturing higher-order ered on such a data set with missing values for the marketing effects and are also consistent with the strong support for group. These tables report the absolute deviations between H1b in our model. Finally, all models yield no support for the actual frequencies and those predicted by the full and 5 H5. These differences in supported hypotheses (H 1b , H 2a , Uncorr models and illustrate that the full model does sig - H2b , H 3, and H 4) across models demonstrate that the differ - nificantly better in recovering these cross-year relationships. 6 ent model components can affect the theoretical and sub - stantive conclusions and that it is important to account for onLine SoCiaL netwoRk dyadic-dependence and higher-order effects. In this application, our focus is on modeling relation - ships of different types. We use a combination of undirected Leveraging information across Relationships and directed binary relationships and a directed weighted We now illustrate how our framework can be used to relationship. In particular, we show how it is important to leverage information in one relationship to predict relation - model both the incidence and intensity of weighted relation - ships in another. For example, we assume that the data ships, because conclusions and predictions depend crucially involving the entire marketing groups are missing in the on this distinction. The data for this application come from second year. In such a situation, we cannot readily use the a Swiss online social network on which members can create log-linear modeling framework previous researchers have a profile as either a user or an artist and can then connect employed, because the group-specific parameters used in with other registered members through friendship relation - such models will not be available for the marketing group. ships. The social networking site offers different services to However, for our models, the natural reliance on data aug - these two distinct user groups. Whereas both groups can mentation to obtain the utilities and the individual-specific publish user-generated content such as blogs, photos, or random effects when estimating parameters ensures that

6We also investigated the role of demographics that were part of the data 5 For H 5b , we found that the probability associated with INTEAM2 > but did not find any significant impact. Our models can handle such continu - BETWTEAM2 is .99 for all our models. ous covariates, something that is not possible with a log-linear specification. 722 JOurnAl OF mArKeting reseArch, August 2011 videos on their profiles, only artists can, in addition, publish the number of artists they communicate with and their popu - up to a maximum of 30 songs on their profile. larity or receptivity( i.e., the number of artists communicat - Artists use the different services offered by the platform ing with a given artist [indegree]). The indegree and outdegree to promote their music and concerts and to seek collabora - distributions are highly skewed. The mean degree is 19.97. tion with other musicians and bands. They establish friend - The maximum and minimum for the indegree distribution are 203 and 0, respectively, whereas the maximum and minimum ship relationships with other artists, send personal mes - for the outdegree distribution are 182 and 0, respectively. sages, and write public comments on other artists’ profiles. 3. Music downloads: The music downloads represent a For entertainment and informational reasons, users, as well directed and weighted relationship. Each song download m as artists, visit profiles of artists and download songs. entry y ij is a count of the number of times that artist j listens Artists engage in active promotion and relationship effort in to a song on artist i’s profile and may include multiple down - the hope that it will result in increased collaboration, com - loads of the same song. As discussed in the “Modeling munication, popularity, and song downloads. In summary, Framework” section, we distinguish between the incidence the online networking site offers a platform that combines and intensity of music downloads. Focusing first on inci - social networking services with entertainment and commu - 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 nication services. reciprocity is 39.1%. The outdegree of an artist is the number There are four components of the data set: member data, of other artists who download from that artist, and the inde - friendship data, communication data, and music download gree is the number of other artists from whom the artist data. The member data contain information collected at reg - downloads music. For the binary component, the mean istration and include stable variables such as the registration degree is 17.89, and the maximum indegree and outdegree date, date of birth, gender, and city or, in the case of artists, are 213 and 135, respectively. Because this is a weighted rela - their genre and information about their offline concerts and tion, we can also study the intensity of connections. On aver - performances before joining the network. In addition, the age, each artist downloads songs on 59.98 occasions (includ - data also contain information on the number of page views ing multiple song downloads). The maximum weighted indegree (i.e., the number of songs downloaded from) an of each member’s page on the network during a given time artist is 2899, whereas the maximum weighted outdegree period. The other data components pertain to our three (i.e., the maximum number of times a single artist listens to dependent variables and are described in greater detail in songs is 612). We also find that the artist-specific degree sta - the following section. tistics are highly correlated across the three relationships. data and Structural Characteristics Model Specifications and variables Our sample consists of 230 artists who created a profile We estimate several different variants of the full model on the network between February 1, 2007, and March 31, (hereinafter, we refer to this as “full model”) that we outlined 2007, provided information about their activities and char - in the “Modeling Framework” section. Our null models acteristics, and uploaded at least one song on their profile. impose different restriction on the full model; we con - We model three types of relationships among these artists structed them to investigate how crucial the different com - over the course of the six months between April 1, 2007, ponents of the full model are in capturing important aspects and September 30, 2007. 7 These relationships include of the data generating process. The models are as follows: friendship (f), communications (c) and music downloads •Full Model . The full model includes dyad-specific covariates (m). The data set thus contains three 230 ¥ 230 matrices (Y f, (x ) to accommodate homophily and heterophily; artist-specific Y , and Y , respectively) for these relationships. ij c m covariates ( xi) and ( xj) to account for asymmetry in responses; Structural characteristics . We now briefly describe the artist- and relationship-specific sender and receiver parameters structural characteristics of the three relationships for our (qi), to model heterogeneity in expansiveness and receptivity; set of artists. correlations in these random effects across relationships ( Sq); f latent spaces of random locations for the three relations ( zim , 1. Friendship: The friendship relation y ij is binary and undirected. z , and z ) to capture higher order effects; and correlations in It indicates whether a friendship is formed between the pair ic if errors ( Sm, Sc, and Sf), to incorporate reciprocity within each {i, j} before the end of our data period. The network has relationship. In addition, the full model uses a correlated hur - 3564 friendship relations among a maximum possible 26,335 dle count model to account for the sparse nature of many net - connections, yielding a network density of 13.53%. c work data sets. 2. Communications: The communication relation y ij is binary •Poisson . The Poisson model uses a Poisson distribution for the and directed and indicates whether artist i sent a communica - music download relationships, rather than the correlated hurdle tion (direct message or comment) to artist j within the time Poisson. Thus, the counts are modeled directly, and we do not period of the data. We observed 4575 communication rela - distinguish between the incidence and intensity of counts. The tions, yielding a density of 8.68%. The relation exhibits con - model is otherwise identical to the full model in all other respects. siderable reciprocity or mutuality (defined as the ratio of the •uncorr . In the Uncorr model, we treat the three relationships as number of pairs with bidirectional relations to the number of independent. Thus, we assume the artist-specific random effects pairs having at least one tie) equal to 30.9%. Artists vary in in qi to be uncorrelated across the relationships, and therefore, their level of expansiveness (or outdegree), as measured by Sq has a block-diagonal structure. This model is thus equiva - lent to running three separate models on the three relationships. •noZiZj . For the NoZiZj model, we do not include the higher 7Our data are not entirely representative of the whole network, which consists of other artists who joined the network subsequent to our data order terms z¢izj that characterize the latent space. period. In addition, we focus only on the subnetwork of connections Each artist can be described using the variables detailed 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 in Table 5. We use these artist-specific variables to compute to present themselves and seek collaborations. dyad-specific variables. We use different combinations of modeling multiple relationships in social networks 723 the artist- and dyad-specific variables in explaining the Results three relationships. (Appendix B gives details of the covari - We estimated the models using MCMC methods ate specifications for the three relationships.) Table 6 shows (described in Appendix A). Each MCMC run is for 250,000 descriptive statistics associated with these covariates. The iterations, and the results are based on a sample of 200,000 last two rows of Table 6 contain dyad-specific binary iterations after a burn-in of 50,000. variables. The variable “common region” is equal to 1 when Model adequacy . Before discussing parameter estimates, both artists in a dyad are from the same region; otherwise, it we use a combination of predictive measures and posterior is 0. We code the last variable, “common genre,” analogously. predictive model checking (Gelman, Meng, and Stern 1996) to compare the adequacy of our models. This involves gen - table 5 erating G hypothetical replicated data sets from the model ApplicAtiOn 2: descriptiOn OF VAriAbles using the MCMC draws and comparing these data sets with the actual data set. These comparisons are made using vari - variable description ous test quantities that represent different structural charac - Songs The number of songs available on the artist’s profile. teristics of the network. If the replicated data sets differ sys - Band Whether the artist belongs to a band. Equal to 1 if artist tematically from the actual data on a given test quantity, the belongs to a band and 0 if otherwise. model does not adequately mimic the structural characteris - Audience The audience size of the largest concert by the artist. A tic that the test quantity represents. The discrepancy median split yields 1 when the audience is > 700 and 0 between the replicated data sets and the actual data can be otherwise. assessed using posterior predictive p-value. This p-value is Genre Genre of the artist. We distinguish between rock genres and nonrock genres. the proportion of the G replications in which the simulated test quantity exceeds the realized value of that quantity in Region The geographical region to which the artist belongs. Artists belong to one of the 26 cantons in Switzerland. These are the observed data. An extreme p-value, (either close to 0 or aggregated into three regions: French, German, and Italian. 1; i.e., £ .05 or ≥ .95) suggests inadequate recovery of the PageViews The number of page views of the artist’s profile during the corresponding test quantity. data period. These page views originate from other We use several test statistics associated with the weighted registered members, including the fans, or from Internet relationship to assess whether the distinction between inci - users from outside the social network. dence and intensity is crucial. Table 7 shows the model ade - YActive The number of years of activity of the artist in the music industry. 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 table 6 other columns report the posterior predictive mean and p- ApplicAtiOn 2: descriptiVe stAtistics FOr Artist values for the different models. A few conclusions can be VAriAbles readily drawn from the table. First, Poisson, which models the intensity directly using a Poisson specification (as in variable observations M Mdn Sd Min Max Hoff 2005), does not recover any of the test statistics ade - Page views 230 781.2 467 1069 .8 43 10,931 quately, because almost all p-values in Column 6 are Songs 230 4.108 3 3.26 1 29 extreme. Furthermore, the posterior predictive mean values Audience 230 .482 0 .5 01 Band 230 .704 1 .457 01for the test statistics (Column 5) are appreciably different Years active 230 7.24 6 5.82 1 27 from their counterparts in the actual data. Second, we Common region 26,335 .628 1 .483 01observe that none of the p-values associated with the other Common genre 26,335 .564 1 .495 01models are extreme, and this indicates that modeling inci -

table 7 ApplicAtiOn 2: pOsteriOr predictiVe checKing FOr music dOWnlOAds FOr the FOur mOdels

Full uncorr noZiZj Poisson data M p-value M p-value M p-value M p-value Binary 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 Quantity 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 1 724 JOurnAl OF mArKeting reseArch, August 2011 dence and intensity separately is important for adequately work. 8 Finally, the data indicate that for the music relation - capturing the structural network characteristics associated ship, different coefficients influence the incidence and with weighted relations. Finally, because the NoZiZj model intensity components of the music relationship. Thus, we performs almost as well as the full model, we conclude that conclude that these two facets are not isomorphic and need the latent space is not crucial for the recovery of structural to be modeled separately. characteristics in this application. Covariance structure of the relationships . The covariance The preceding discussion focuses on model adequacy for matrix Sq captures the linkages among the expansiveness music downloads to highlight the contribution of the corre - and attractiveness parameters across the relationships. Table lated multivariate hurdle Poisson in modeling weighted 9 reports the elements of Sq. A striking feature of Table 9 is relationships. The results from the other two relationships that all the covariances are significantly positive. The posi - show that all models (including Poisson) are similar in their tive correlation within a relationship implies that attractive - recovery of structural characteristics for these relationships. ness goes hand-in-hand with expansiveness (i.e., artists who The model adequacy results are based on in-sample simula - are sought by others also tend to be active in seeking rela - tions. (We report the predictive performance of our model tionships with others). Moreover, for the music rela tionship, in the Web Appendix at http://www.marketingpower. com/ the attractiveness and expansiveness parameters for the jmraug11.) We find that the Full model outperforms other intensity equations are also positively correlated with the models on almost all measures. We also find that the Pois - corresponding random effects for the incidence component. son model does very poorly in predicting future activity, The positive correlations across relationships imply that an indicating that there are significant gains in modeling the artist who is popular in one relationship is also likely to be incidence and intensity separately. both popular and productive in other relationships. Simi - Parameter estimates larly, an artist who is productive in one relationship is also likely to be productive and popular in other relationships. We now discuss the parameter estimates based on the The utility errors for the two incidence equations of the entire sample of six months. We focus on the parameter esti - music download relationship are positively correlated (.74), mates from the full model. Estimates from other models owing to shared unobservable influences and reciprocity. mostly yield similar qualitative conclusions, and we do not The intensity equations are also positively correlated (.497). include these for the sake of brevity. Furthermore, the errors for the incidence equations are posi - Covariate effects . Table 8 reports the posterior means and tively correlated with the errors for the intensity equations standard deviations of the coefficient estimates for the three (.395 and .411) implying selectivity through shared unob - relationships. In interpreting this table, recall that all served factors driving both incidence and intensity. This variables associated with the friendship relation are dyad corroborates the need for a multivariate correlated hurdle specific and binary, whereas, for the other relations, some Poisson specification. Finally, the communication utilities variables are dyad specific and some are artist specific. also exhibit positive correlation (.511) driven by reciprocity Also, note that both components of the music relation have and other shared unobservables. the same set of covariates. (The definitions for these covari - ates are available in Appendix B.) ConCLuSionS For the sake of brevity, we synthesize the results across As interest in social networks and brand communities all the relationships. There is clear evidence of homophily grows, marketers are becoming increasingly focused on and proximity: For all three relationships, the dyad-specific understanding and predicting the connectivity structure of variables CRegion and CGenre have a positive and signifi - such networks. In this article, we developed a methodologi - cant impact on the likelihood of forming connections. The cal framework for jointly modeling multiple relationships positive coefficient for CRegion is consistent with the that vary in their directionality and intensity. Our integrated notion that a common language and geographical proximity approach for social network analysis is unique in that it can enhance the likelihood of collaborative effort. The posi - weaves together several distinct model components needed tive coefficient for CGenre means that pairs of artists who for capturing multiplexity in networks. produce music in the same genre have a higher propensity We applied our framework to two distinct applications to form friendship connections, communicate with each that showcased different benefits of our approach. In the other, and download music from each other. We also find first application, we investigated the impact of an organiza - that artists belonging to a band have a higher chance of tional intervention (R&D collocation) on the patterns of forming friendships (BothBand) and a higher probability of communications among professionals involved in new sending and receiving communications (SBand and RBand). product development activities. In this application, we We find that measures of online popularity that are based specifically investigated the gains from modeling relation - on the total number of page views for an artist (BothPopu - ships jointly. Our results clearly indicate how the different lar, SPviews, DPviews, and PPviews) positively influence components of our framework are needed for a clear assess - relationship formation. For example, the positive coeffi - ment of substantive hypotheses. We found that the hetero - 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 8We thank an anonymous reviewer for pointing out that covariates relat - they tend to download more music. In contrast, most measures ing to popularity could be potentially endogenous. However, given that we of prior and offline popularity or experience (indicating include both online and offline correlates of popularity, as well as actor- specific random effects that capture attractiveness, it is unclear whether audience size of concerts or years of activity) do not seem additional unobserved variables relating to popularity are part of the utility to affect the formation of online relationships within the net - error. However, caution is still needed in interpreting the results. modeling multiple relationships in social networks 725

table 8 table 9

ApplicAtiOn 2: cOeFFicient estimAtes FOr the three ApplicAtiOn 2: cOVAriAtiOn in the rAndOm eFFects Sq relAtiOnships FOr the Full mOdel m m m m i i f v ai1 bi1 ai2 bi2 ac bc ai Parameter M Sd m ai1 .397 .378 .234 .222 .268 .500 .499 Friendship Relationship (.051) (.056) (.037) (.049) (.040) (.064) (.065) m Intercept –2.732 .167 bi1 .707 .238 .441 .329 .617 .593 CRegion .341 .050 (.091) (.042) (.073) (.053) (.086) (.086) CGenre .150 .033 m a i2 .192 .146 .195 .315 .319 BothPopular .163 .063 (.034) (.038) (.033) (.050) (.052) BothNotPopular .015 .087 m BothBigSongs .061 .061 b i2 .489 .188 .313 .288 BothSmallSongs .016 .065 (.082) (.048) (.079) (.081) i BothBand .640 .153 ac .407 .478 .513 BothNoBand –.257 .163 (.052) (.065) (.067) BothBigAudience .203 .143 i bc 1.000 .988 BothSmallAud –.134 .143 (.109) (.104) BothLongActive –.036 .075 f BothShortActive .034 .071 ai 1.031 (.111) Communications Relationship Intercept –3.446 .270 Note: Posterior standard deviations are in parentheses. CRegion .321 .042 CGenre .144 .028 SPviews .015 .004 showed how it is critical to model both the incidence and SSongs .009 .010 intensity of weighted relationships such as music down - SBand .321 .106 loads; otherwise, recovery of structural characteristics and SAudience .135 .093 predictive performance suffers appreciably. On the substan - SYActive –.003 .006 tive front, our results show that the friendship, communica - RPviews .002 .003 RSongs .013 .009 tions, and music download relationships share common RBand .464 .154 antecedents and exhibit homophily and reciprocity. We RAudience .043 .138 found that offline proximity is relevant for all online rela - RYActive –.004 .006 tionships, and this is consistent with our understanding that Music Relationship: incidence these connections are formed to forge collaborative relation - Intercept –3.007 .272 CRegion .183 .040 ships. We also found that the artists exhibit similar roles CGenre .118 .031 across relationships and that popular artists seem to be more PPviews .021 .004 productive regardless of the relationship being studied. PSongs –.006 .010 Across the two applications, we found mixed evidence PBand .106 .104 regarding the benefits from incorporating a latent space. In PAudience .100 .091 PYActive –.013 .006 the first application, the latent space improved the recovery DPviews .031 .005 of cross-relationship transitivity patterns and affected the DSongs .022 .014 parameter estimates and the substantive findings. However, DBand .062 .141 higher-order effects did not seem to be important in the sec - DAudience –.113 .120 DYActive –.015 .009 ond application. The latter result is consistent with Faust’s Music Relationship: intensity (2007) conclusion that much of the variation in the triad cen - Intercept –2.024 .269 sus across networks could be explained by simpler local CRegion .235 .050 structure measures. These results suggest that the impact of CGenre .152 .038 extradyadic effects could be application specific. PPviews .011 .003 On the theoretical and substantive front, our framework PSongs .023 .009 PBand .169 .086 facilitates a detailed description of antecedents of relation - PAudience –.005 .074 ship formation and allows for theory testing taking into PYActive –.003 .006 account systematic variations in degree arising from DPviews .034 .006 homophily and heterophily, local structuring, as well as DSongs .021 .017 DBand –.055 .149 temporal or cross-relationship carryover (Rivera, Soder - DAudience –.157 .120 strom and Uzzi 2010). Our enquiry can be extended in many DYActive –.012 .011 directions. Our applications involved small networks. Most Notes: Bold indicates that the 95% posterior interval does not span 0. online networks are much larger, and statistical methods cannot scale directly to the level of these large networks. geneity specification, the latent space, and the correlations However, recent research has shown that while online net - across relationships can affect both substantive conclusions works can have millions of members, communities within and the recovery of structural characteristics. Finally, as the such networks are relatively small, with sizes in the vicinity Web Appendix shows (see http://www.marketingpower. of the 100–200 member range (Leskovec et al. 2008). This com/ jmraug11), our approach can be used to leverage infor - implies that these very large networks can be broken down mation from one relation to predict connectivity in another. into clusters of tightly knit communities, and when such In our second application, we focused on modeling mul - communities are identified, our methodological framework tiple relationships of different types. In particular, we can then be used on such communities to further understand 726 JOurnAl OF mArKeting reseArch, August 2011

~ the nature of linkages within these subcommunities. How - parameters v{ij}z = Zjzi + e{ij} , where Zj is an appropriately ever, such a divide-and-conquer approach is unlikely to pro - constructed matrix from the latent space vector of actor j. This vide a complete picture of the nature of link formation in is a seemingly unrelated regression system. Given the prior zi ~ such large networks. N(0, Sz), where Sz is constructed from the different Szr matri - ces, we can write the full conditional as N (zˆ W ), where W–1 = We focused on modeling static relationships or on i zi zi ( )–1 + Z –1Z and zˆ = [ Z ( )–1v~ ]. The Sz Sj π i j¢S j i Wzi Sj π i j¢ S {ij}z sequential relationships observed over a few time periods. model depends on the inner product of the latent space vec - However, networks are dynamic entities in which connec - tors, which is invariant to rotations and reflections of the vec - tions are formed over time. Incorporating such dynamics tors. Thus visual representations of these vectors require that would be interesting. We used a parametric framework based they be rotated to a common orientation, This can be done by on the normal distribution for the latent variables and ran - using a Procustrean transformation as outlined in Hoff (2005). dom effects, and this was sufficient for recovery of skewed 6. The variance–covariance matrix of the errors for the degree distributions. However, in other situations, Bayesian weighted relationship, S, has a special structure as described nonparametrics (Sweeting 2007) may be more useful. in the “Modeling Framework” section. Given this special structure, we follow the separation strategy of Barnard, aPPendix a: FuLL ConditionaL diStRiButionS McCulloch, and Meng (2000) in setting the prior in terms of the standard deviations and correlations in S. The covariance –1 1. The full conditional for precision matrix Sq of the actor- matrix S can be decomposed into a correlation matrix, R, and specific random effects is a Wishart distribution given by a vector, s, of standard deviations—that is, S = diag( s) ¥ R ¥ diag( s), where s contains the square roots of the standard −1   N   deviations. Let w contain the logarithms of the elements in s. −1  −1  ()A1 ()pWishartNRΣθ {}θρθθii=+θ , ′i + θ  , We assume a multivariate normal distribution N( 0, I) for the   ∑     i = 1   nonredundant elements of R, such that it is constrained to the subspace of the p-dimensional cube [–1, 1] p, where p is the –1 1. where the prior for S q is Wishart( rq, R q). The quantities rq number of equations that yields a positive definite correlation and R q refer to the scalar degree of freedom and the scale matrix. Finally, we assume a univariate standard normal prior matrix for the Wishart, respectively, and N is the number of for the single log-standard deviation in w. actors in the network. •The full conditional distribution for the free element in the r 2. The covariance matrices Sz, for relationship r are diagonal, vector of log-standard deviations w of errors can only be writ - r because z i is a p-dimensional vector of independent compo - ten up to a normalizing constant (recall that the terms asso - 2 nents. Let s z,r denote the common variance for the compo - ciated with the binary utilities in w are fixed to 0 for identi - r 2 nents of z i. The full conditional for sz,r is an inverse gamma fication purposes). Given our assumption of a normal prior distribution given by for the single free element, we use a Metropolis–Hastings step to simulate the standard deviation in w. A univariate −1   N p   normal proposal density can be used to generate candidates  1  2 42 −1 for this procedure. If is the current value of kth component ()(Ap2 σ=+zr, |{}) zi IG  apNr , zbirk +r .  p∑∑  of w, a candidate value is generated using a random walk   i = 1 k = 1   c (t – 1) chain wk = w k + N(0, t), where t is a tuning constant that controls the acceptance rate. 3. The full conditional for the coefficients m is multivariate nor - •Many different approaches can be used to sample the corre - mal because we have a seemingly unrelated regression sys - lation matrix R. Here, we use a multivariate Metropolis step tem of equations conditional on knowing the latent variables. ~ to sample a vector of nonredundant correlations in R. We Form the adjusted utilities (e.g., uij1 = u ij1 – ail – bj1 – z¢i1 zj1 ) used adaptive MCMC (Atchade 2006) for obtaining the tun - and adjusted log-rate parameters by subtracting terms that do ing constant so as to ensure rapid mixing. not involve m from the latent dependent variables. We then ~ 7. The full conditional distribution associated with the set of have the system of equations, v{ij} = X {ij} m + e{ij} , for an arbi - latent utilities and latent rate parameters in uij is again trary pair {i, j}, where eij ~ N(0, S). We can write the full con - unknown. We sample the utilities and log-rate parameters ditional as follows: using univariate conditional draws. Sampling the utilities is % $ straightforward, because these are truncated univariate con - ()ApN3 (|{})(,),µµµv{}ij =Ωµ µ ditional normal draws. The log-rate parameters log lij and –1 log are sampled such that these are univariate normal 1. where W = C–1 + S X¢ S–1X and mˆ = W [C –1h + lji m {ij} {ij} {ij} m draws if the corresponding observation involves a zero S X¢ S–1v~ ]. {ij} {ij} {ij} count, and for an observation in which a positive count is 4. The full conditional for the heterogeneity parameter qi is a multivariate normal. We again begin by creating adjusted observed, we use a univariate Metropolis step that combines utilities and rate parameters, such as by subtracting all terms the likelihood for a truncated Poisson distribution with a conditional normal prior. 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 - aPPendix B: ModeL SPeCiFiCationS FoR mal to obtain the resulting full conditional: aPPLiCation 2 % $ Covariates for the Friendship Relationship ()ApN4 (|{})(%,),θθθi v{}ij θ =Ωθ i θ CRegion: CRegion is equal to 1 if both artists in a pair are from –1 –1 –1 ˆ –1 ~ where Wq = Sq + (N – 1) S and qi = S WqSj π iv{ij} q. the same region; 0 otherwise. 5. The full conditional for zi that contains all the latent space CGenre: CGenre is equal to 1 if both artists in a pair are from vectors associated with an individual i is multivariate normal. the same genre; 0 otherwise. ~ Creating adjusted utilities such as uij1z = u ij1 – x¢ij1 m1 – ai1 – BothPopular: BothPopular is equal to 1 if both artists in a pair are bj1 , we can form the vector of adjusted utilities and latent rate viewed (online) by more than the population median; 0 otherwise. modeling multiple relationships in social networks 727

BothNotPopular: BothNotPopular is equal to 1 if both artists in a DAudience: DAudience is equal to 1 if the downloader per - pair are viewed by fewer than the population median; 0 otherwise. formed in front of an audience larger than 700 people; 0 otherwise. BothBigSongs: BothBigSongs is equal to 1 if both artists in a DYActive: DYActive is equal to 1 if the downloader was active pair post more songs than the population median; 0 otherwise. for more than six years; 0 otherwise. BothSmallSongs: BothSmallSongs is equal to 1 if both artists in a pair post fewer songs than the population median; 0 otherwise. REFERENCES BothBand: BothBand is equal to 1 if both artists in a pair post are from a band; 0 otherwise. Atchade, Yves F. (2006), “An Adaptive Version for the Metropolis BothNoBand: BothNoBand is equal to 1 if both artists in a pair Adjusted Langevin Algorithm with a Truncated Drift,” Method - post are not from a band; 0 otherwise. ology and Computing in applied Probability , 8 (June), 235–54. BothBigAudience: BothBigAudience is equal to 1 if both artists Barnard, John, Robert McCulloch, and Xiao-Li Meng (2000), in a pair had large concerts with more than 700 spectators; 0 “Modeling Covariance Matrices in Terms of Standard Devia - otherwise. tions and Correlations, with Application to Shrinkage,” Statis - BothSmallAudience: BothSmallAudience is equal to 1 if both tica Sinica , 10, 1281–1311. artists in a pair had small concerts with more than 700 spectators; Dekker, David, David Krackhardt, and Tom A.B. Snijders (2007), 0 otherwise. “Sensitivity of MRQAP Tests to Collinearity and Autocorrela - BothLongActive: BothLongActive is equal to 1 if both artists in tion Conditions,” Psychometrika , 72 (December), 563–81. a pair had being active for more than six years; 0 otherwise. DeSarbo, W.S., Y. Kim, and D. Fong (1998), “A Bayesian Multi - BothShortActive: BothShortActive is equal to 1 if both artists dimensional Scaling Procedure for the Spatial Analysis of in a pair had being active for less than six years;0 otherwise. Revealed Choice Data,” Journal of econometrics , 89 (1/2), 79–108. Covariates for the Communication Relationship Faust, Katherine (2007), “Very Local Structure in Social Net - works,” Sociological Methodology , 37 (December), 209–256. CRegion: CRegion is equal to 1 if both artists in a pair are from Fienberg, Stephen E., Michael M. Meyer, and Stanley S. Wasser - the same region; 0 otherwise. man (1985), “Statistical Analysis of Multiple Sociometric Rela - CGenre: CGenre is equal to 1 if both artists in a pair are from tions,” Journal of the american Statistical association , 80 the same genre; 0 otherwise. (March), 51–67. SPviews: SPviews represents the number of sender paged views. Frank, Ove and David Strauss (1986), “Markov Graphs,” Journal SSongs: SSongs represents the number of songs posted on the of the american Statistical association , 81 (September), 832–42. sender web page. Gelman, Andrew, Xiao-li Meng, and Hal Stern (1996), “Posterior SBand: SBand is equal to 1 if the sender belongs to a band; 0 Predictive Assessment of Model Fitness via Realized Discrep - otherwise. ancies,” Statistica Sinica , 6 (October), 733–807. SAudience: SAudience is equal to 1 if the sender performed in Granovetter, Mark (1985), “Economic Action and Social Struc - front of an audience larger than 700 people; 0 otherwise. ture: The Problem of Embeddedness,” american Journal of SYActive: SYActive is equal to 1 if the sender was active for Sociology , 91 (November), 481–510. more than six years; 0 otherwise. Handcock, Mark S., Adrian E. Raftery, and Jeremy Tantrum RPviews: RPviews represents the number of receiver paged (2007), “Model-Based Clustering for Social Networks,” Jour - views. nal of the Royal Statistical Society , 170 (March), 301–354. RSongs: RSongs represents the number of songs posted on the Hoff, Peter D. (2005), “Bilinear Mixed-Effects Models for Dyadic receiver web page. Data,” Journal of the american Statistical association , 100 RBand: RBand is equal to 1 if the receiver belongs to a band; 0 (March), 286–95. otherwise. ———, Adrian E. Raftery, and Mark S. Handcock (2002), “Latent RAudience: RAudience is equal to 1 if the receiver performed Space Approaches to Social Network Analysis,” Journal of the in front of an audience larger than 700 people; 0 otherwise. american Statistical association , 97 (December), 1090–1098. RYActive: RYActive is equal to 1 if the receiver was active for Holland, Paul W. and Samuel Leinhardt (1981), “An Exponential more than six years; 0 otherwise. Family of Probability Distributions for Directed Graphs,” Jour - Covariates for the Music download Relationship nal of the american Statistical association , 76 (March), 33–50. Iacobucci, Dawn (1989), “Modeling Multivariate Sequential CRegion: CRegion is equal to 1 if both artists in a pair are from Dyadic Interactions,” Social networks , 11 (December), 315–62. the same region; 0 otherwise. ——— and Nigel Hopkins (1992), “Modeling Dyadic Interactions CGenre: CGenre is equal to 1 if both artists in a pair are from and Networks in Marketing,” Journal of Marketing Research , the same genre; 0 otherwise. 26 (February), 5–17. PPviews: PPviews represents the number of provider paged ——— and Stanley Wasserman (1987), “Dyadic Social Inter - views. actions,” Psychological Bulletin , 102 (September), 293–306. PSongs: PSongs represents the number of songs posted on the Iyengar, Raghuram, Christophe Van den Bulte, and Thomas W. provider web page. Valente (2011), “Opinion Leadership and Social Contagion in PBand: PBand is equal to 1 if the provider belongs to a band; 0 New Product Diffusion,” Marketing Science , 30 (2),195–212. otherwise. Leskovec, Jure, Kevin J. Lang, Anirban Dasgupta, and Michael W. PAudience: PAudience is equal to 1 if the provider performed in Mahoney (2008), “Statistical Properties of Community Structure front of an audience larger than 700 people; 0 otherwise. in Large Social and Information Networks,” in www 08: Social PYActive: PYActive is equal to 1 if the provider was active for networks and web 2.0 -discovery and evolution of Communities . more than six years; 0 otherwise. New York: Association for Computing Machinery, 695–704. DPviews: DPviews represents the number of downloader paged Li, H. and E. Loken (2002), “A Unified Theory of Statistical views. Analysis and Inference for Variance Component Models for DSongs: DSongs represents the number of songs posted on the Dyadic Data,” Statistica Sinica , 12, 519–35. downloader web page. McPherson, M., L. Smith-Lovin, and J. Cook (2001), “Birds of a DBand: DBand is equal to 1 if the downloader belongs to a Feather: Homophily in Social Networks,” annual Review of band; 0 otherwise. Sociology , 27, 415–44. 728 JOurnAl OF mArKeting reseArch, August 2011

Nair, Harikesh S., Puneet Manchanda, and Tulikaa Bhatia (2010), Sweeting, Trevor (2007), “Discussion on the Paper by Handcock, “Asymmetric Social Interactions in Physician Prescription Raftery and Tantrum,” Journal of the Royal Statistical Society, Behavior: The Role of Opinion Leaders,” Journal of Marketing a, 170 (March), 327–28. Research , 47 (October), 883–95. Trusov, Michael, Anand V. Bodapati, and Randolph E. Bucklin Narayan, V. and S. Yang (2007), “Bayesian Analysis of Dyadic (2010), “Determining Influential Users in Internet Social Net - Survival Data with Endogenous Covariates from an Online works,” Journal of Marketing Research , 47 (August), 643–58. Community,” working paper, Johnson Graduate School of Man - Trusov, Michael, R.E., Bucklin, and K. Pauwels (2009), “Estimat - agement, Cornell University. ing the Dynamic Effects of Online Word-of-Mouth on Member Pattison, Philippa E. and Stanley Wasserman (1999), “Logit Models Growth of a Social Network Site,” Journal of Marketing , 73 and Logistic Regressions for Social Networks: II. Multivariate (September), 90–102. Relations,” British Journal of Mathematical and Statistical Psy - Tuli, Kapil R., Sundar G. Bharadwaj, and Ajay K. Kohli (2010), chology , 52 (November), 169–93. “Ties That Bind: The Impact of Multiple Types of Ties with a Rivera, Mark T., Sara B. Soderstrom, and Brian Uzzi (2010), Customer on Sales Growth and Sales Volatility,” Journal of “Dynamics of Dyads in Social Networks: Assortative, Rela - Marketing Research , 47 (February), 36–50. tional, and Proximity Mechanisms,” annual Review of Sociol - Van den Bulte, Christophe and Rudy K. Moenaert (1998), “The ogy , 36 (August), 91–115. Effects of R&D Team Co-location on Communication Patterns Robins, Garry L., Philippa E. Pattison, Y. Kalish, and D. Lusher Among R&D, Marketing and Manufacturing,” Management (2007), “An Introduction to Exponential Random Graph (p*) Science , 44 (November), Part 2 of 2, 1–18. Models for Social Networks,” Social networks , 29 (2), 173–91. ——— and Stefan Wuyts (2007), “Social Networks and Market - ———, Tom A.B. Snijders, Peng Wang, Mark S. Handcock, and ing,” working paper, Marketing Science Institute. Philippa E. Pattison (2007), “Recent Developments in Exponen - Warner, R., D.A. Kenny, and M. Stoto (1979), “A New Round tial Random Graph (p*) Models for Social Networks,” Social Robin Analysis of Variance for Social Interaction Data,” Jour - networks , 29 (May), 192–215. nal of Personality and Social Psychology , 37 (10), 1742–57. Snijders, Tom A.B. (2005), “Models for Longitudinal Network Wasserman, Stanley and Dawn Iacobucci (1988), “Sequential Data,” in Models and Methods in Social network analysis , Peter Social Network Data,” Psychometrika , 53 (June), 261–82. J. Carrington, John Scott, and Stanley Wasserman, eds. New ——— and Philippa Pattison (1996), “Logit Models and Logistic York: Cambridge University Press. Regressions for Social Networks: I. An Introduction to Markov ———, Philippa E. Pattison, Garry L. Robins, and Mark S. Hand - Graphs and p*,” Psychometrika , 61 (September), 401–425. cock (2006), “New Specifications for Exponential Random Watts, Duncan J., and Peter Sheridan Dodds (2007), “Influentials, Graph Models,” Sociological Methodology , 36 (December), Networks, and Public Opinion Formation,” Journal of Con - 99–153. sumer Research , 34 (May), 441–58. Stephen, Andrew T. and Olivier Toubia (2010), “Deriving Value Wedel, M. and W.S. DeSarbo (1996), “An Exponential Family from Social Commerce Networks,” Journal of Marketing Scaling Mixture Methodology,” Journal of Business and eco - Research , 47 (April), 215–28. nomics Statistics , 14 (4), 447–59. Copyright of Journal of Marketing Research (JMR) is the property of American Marketing Association and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.