Scalable Inference of Customer Similarities from Interactions Data using Dirichlet Processes

Scalable Infer ence of Customer Similarities fr om Interactions Data using Dirichlet Pr ocesses Michael Braun MIT Sloan School of Management Massachusetts Institute of T echnology Cambridge, MA 02139 braunm@mit.edu Andr ´ e Bonfrer College of Business and Economics Australian National University Canberra ACT 0200 andre.bonfr er@anu.edu.au ∗ December 21, 2010 Abstract Under the sociological theory of homophily , people who are similar to one another are mor e likely to interact with one another . Marketers often have access to data on interactions among customers from which, with homophily as a guiding principle, infer ences could be made about the underlying similarities. However , larger networks face a quadratic explosion in the number of potential interactions that need to be modeled. This scalability problem renders probabil- ity models of social interactions computationally infeasible for all but the smallest networks. In this paper we develop a probabilistic framework for modeling customer interactions that is both grounded in the theory of homophily , and is ﬂexible enough to account for random variation in who interacts with whom. In particular , we pr esent a novel Bayesian nonparamet- ric approach, using Dirichlet processes, to moderate the scalability problems that marketing resear chers encounter when working with networked data. W e ﬁnd that this framework is a powerful way to draw insights into latent similarities of customers, and we discuss how mar- keters can apply these insights to segmentation and targeting activities. ∗ The authors thank David Dahl, Daria Dzyabura , Pete Fader , Jacob Goldenberg, John Hauser , Barak Libai, Jon McAuliffe, Carl Mela, Adrian Raftery , David Schweidel, and Romain Thibaux for useful suggestions and helpful com- ments on previous versions of this paper , as well as Rico Bumbaca and Alex Riegler for resear ch assistance, and Jeong- wen Chiang and China Mobile for providing the dataset. Introduction Marketers have long been inter ested in the notion that interactions among customers will affect behavior . For example, knowledge of how customers relate to one another improves our under - standing on how prefer ences are formed (Reingen et al. 1984), how prefer ences are correlated within groups (W itt and Bruce 1972; Park and Lessig 1977; Ford and Ellis 1980; Bear den and Et- zel 1982), or how useful referrals are for marketers when developing new markets (Reingen and Kernan 1986). Connections among customers are opportunities for pr eference inﬂuence (e.g. con- tagion in diffusion, Bass 1969). Marketers can leverage ”word of mouth” to amplify the efﬁcacy of their communication campaigns (Goldenberg et al. 2001; Nam et al. 2007; Iyengar et al. 2008; Godes and Mayzlin 2009). Incorporating network information into marketing models has also been shown to improve forecasts of both new product adoption (Hill et al. 2006) and customer churn (Dasgupta et al. 2008). Similar customers ar e mor e likely to interact with one another , so given the need for marketers to ﬁnd ef ﬁcient ways to attract and cultivate customers, there exists vast opportunity in leverag- ing interactions data to infer similarity and connect this to marketing behavior (e.g., Y ang and Allenby 2003; Bell and Song 2007; Nam et al. 2007). This link between similarity and interactions is the sociological theory of homophily (Akerlof 1997; Blau 1977; Lazarsfeld and Merton 1954) and is the basis for many marketing studies that examine or accommodate interactions among cus- tomers (e.g. Gatignon and Robertson 1985; Brown and Reingen 1987; Choi et al. 2010). Put simply , homophily implies that customers who are similar to one another are mor e likely to interact with one another , and share information and inﬂuence, than customers who are not. There is a sub- stantial volume of literature that links similarities to interactions (see McPherson et al. 2001, for a review), but interactions and similarities are not the same thing. W e consider interactions to be the “data” that r ecords some observable action between two individuals, while similarities form a latent, unobserved construct (though possibly corr elated with other observed measurements) that determines which individuals are mor e likely to interact with others. In this paper , we pr esent an illustrative yet parsimonious model, grounded in the theory of homophily , that allows marketers to infer latent similarities fr om observed interactions. The idea is to develop a probability model that uses interactions data to infer latent similarities, and generates output that can help marketers 2 better understand why customers interact with whom they do, or why they behave the way they do, in terms that are useful to marketers. W e build on a class of pr obability models known as latent space models (Hoff et al. 2002; Handcock et al. 2007). The fundamental idea behind latent space models is that each individual is characterized as occupying some unobserved point on a multi-dimensional space. When esti- mated on relationship data (e.g. a list of self-r eported friendships, as in Reingen et al. 1984; Brown and Reingen 1987, or working relationships, as in Iyengar et al. 2008), the distance among points in latent space determines the probabilities for the incidence of these r elationships. 1 What is becom- ing increasingly available to marketers, however , are clean, observational data on the interactions among customers, such as phone call records or online social networking transactions, but with no observed information about the content of the interaction (who these people ar e and what they talk about), or the natur e of the r elationship between these individuals (what it is about these two particular people that generates an interaction between them). When latent space models are estimated on interactions data, we can interpret the distance among points as relative similarity . Homophily gives us the theor etical foundation on which we can make this claim. The managerial usefulness of estimating latent space models on interactions data comes from identifying and inferring these similarities. Sometimes, such as our application in telecommunication services, interactions generate r evenue directly . There are many examples, like those mentioned in the ﬁrst paragraph, where marketers deliberately target customers who will contact, and hopefully inﬂuence, others. But in other cases, the marketing activities themselves might have nothing at all to do with “following network links,” or generating “wor d of mouth.” Knowing how similar customers are to one another is of direct relevance to marketing practition- ers because it forms the basis of segmentation and targeting across a heterogeneous population. Once we have inferences about relative similarities of customers in hand (through posterior dis- tributions of latent distances), we can segment and target customers accordingly . Ordinarily , this segmentation is done based on observed characteristics of individuals. V ery little attention has been paid to how marketers might be able exploit the information contained in interactions data for traditional, non-networked marketing tactics, like deciding in which publications (online or 1 Latent space methods are, of course, not limited to examining social network data, and could be used to model sim- ilarities between units in two distinct groups (Bradlow and Schmittlein 2000), or to model the differ ence in knowledge by individuals (van Alstyne and Brynjolfsson 2005). Further applications are discussed in T oivonen et al. (2009). 3 otherwise) to advertise. Indeed, the company that uses interactions data for segmentation and targeting (e.g. an online r etailer) does not necessarily have to be the same company that collects it (e.g., the cell phone provider). One reason modelers have not been able to apply latent space models to marketing data in a general sense is that it can be a daunting computational challenge. One of the key tenets of probability modeling is that we need to take all data into account, including pairs of individu- als for whom we do not observe any interactions at all (the “zeros” in the data offer valuable information about relative similarities). Thus, there has been a formidable obstacle to using pr ob- ability models for larger observational network datasets. A dataset with N individuals involves ( N 2 ) dyads (the binomial coefﬁcient ( N x ) is deﬁned as N ! x ! ( N − x ) ! ). For the exemplar dataset that we use in this paper , there are 11,426,590 sets of dyad-speciﬁc parameters that we need to consider , and this is for a dataset of only 4,781 individuals. Unless we want to break the interdependencies among dyads, ignore unobserved heterogeneity , or make other assumptions that are similarly re- strictive, we need to compute all of these ( N 2 ) dyad-speciﬁc likelihoods, and the same number of dyad-speciﬁc parameters, at each iteration of our estimation algorithm. The problem with scale makes pr obability models of social interactions computationally intractable for all but the smallest datasets. The modeling challenge is therefore to r eveal similarities in heter ogeneous characteristics from customers’ interaction data, in a scalable and interpretable way . W e accomplish this by applying a Bayesian nonparametric prior , the Dirichlet process (DP), as the distribution of locations on the latent space. The DP is essentially a distribution over distributions (as opposed to over scalars or vectors), and for our purposes, its most salient characteristic is that each realized distribution is discrete . Consequently , individuals in the network ar e cluster ed on common locations on the latent space. So if this discrete distribution has k mass points, ther e are only ( k 2 ) + 1 distinct distances on the latent space (the + 1 comes fr om the zero distance between two individuals at the same latent coordinate). Since k must be smaller than N , there ar e substantially fewer distinct likelihoods to compute and parameters to estimate. In this resear ch, we show how marketers could use latent space models to segment customers based on posterior infer ences of latent similarities, using this more ef ﬁcient Bayesian nonparamet- ric appr oach. An output of our algorithm is a posterior estimate of the latent space that is inferr ed 4 from the interactions data. Our probabilistic approach to modeling these data allows for the fact that similar individuals may not interact, even though they may have similar characteristics and travel in the same social cir cles. Also, we recognize that while interactions typically occur among similar customers, there is also the possibility that dissimilar customers (who may have differ ent purchase patterns and pr eferences) may interact at some time. T o demonstrate the power and util- ity of this appr oach to modeling interactions data, we apply it to a dataset of observed interactions from a cellular communication network. W e propose a pr obability speciﬁcation for this particular dataset, in which the incidence and rates of interactions ar e functions of distances in latent space. W e validate the approach in two ways: by showing that adding the latent space structur e to the probability model improves the ﬁt of the model, with respect to several metrics commonly used in the social networking literature; and by showing that the latent space model can distinguish among pairs of individuals for whom the observed number of interactions ar e all identically zero during a calibration period, in terms of how well the model predicts which of those pairs will eventually interact in a future holdout period. These tests demonstrate that failing to account for the unobserved heter ogeneous interdependencies among individuals leads to a model that simply does not repr esent the observed patterns in interactions. W e then assess the computational improvements and scalability issues surrounding our Bayesian nonparametric appr oach, and the managerial insights that one can get from estimates of the latent space itself. By using a graphical representation of the latent space, we show how marketers can augment network based practices that follow observed interaction paths, with tactics that segment and target customers according to inferred latent similarities. The data that ar e available to us do not let us offer har d evidence of a correlation between similarities and purchase preferences but, given the ﬁndings in the marketing literature that show the importance of similarities and interac- tions in customer behavior , it is r easonable to expect that marketing mix ef forts beneﬁt fr om being able to distinguish interactions among similar customers from interactions among dissimilar cus- tomers. The computational improvements fr om using a DP prior for the latent space make these inferences attainable for the datasets that marketers typically encounter . 5 1 General model formulation 1.1 Intuitive description In a probability model of network data, each dyad in the network generates some vector of data, which can repr esent a wide variety of behavior . Examples include binary indicators of relation- ships, counts of transactions (among customers), times between interactions, or combinations thereof. However simple or complicated the data ar e, they should be treated as some output of a stochastic process that is governed by dyad-speciﬁc parameters (and possibly some addi- tional population-level parameters). Data that is generated from a network of customers differs from individually-generated data, such as household pur chase data, in that we can no longer as- sume that the data-generating pr ocesses are independent across dyads. For example, if we were to observe telephone calls between members of a dyad, the rate at which A calls B, and B calls C, can provide information about how often A calls C. However , we do assume that the dyad- level processes are conditionally independent, so the only correlation among dyads is what occurs because of similarities in parameters. This means that even though frequencies of phone calls might be dependent across dyads, the speciﬁc times at which those calls ultimately take place are independent, conditional on the rate of interactions. W e determine dyad-level parameters so that similar individuals will have higher incidence of interaction than dissimilar individuals. The characteristics upon which this similarity is based ar e likely unobservable by the r esearcher . Therefor e, we repr esent unobserved, exogenous charac- teristics of the individual (and thus, the individual himself), as a D -dimensional vector on some latent space (Hof f et al. 2002; Handcock et al. 2007; Bradlow and Schmittlein 2000; van Alstyne and Brynjolfsson 2005). Similarity between two individuals is measur ed by the distance between their latent coordinates acr oss this latent space, and we can expr ess the rates or probabilities of interac- tion between two people as a decreasing function of the latent distance between them. Note that these distances and locations do not directly repr esent physical or geographic locations in any way (although they may , of course, be incidentally corr elated with them). Instead, they ar e individual- level parameters to be estimated, based on observed patterns of interaction. For the purposes of this article, we treat the location of each latent coordinate as persistent and stationary . So even though interactions among people may appear and disappear periodically (a non-stationary ob- 6 served phenomenon that Kossinets and W atts (2006) describe as evolving ), the underlying rates and probabilities of these incidences remain the same. Thus, our stationary model can still cap- ture non-stationary behavior in the observed data. Also, we want to emphasize that a latent space model is an abstraction of reality , and we caution r esearchers not to place too much concrete mean- ing on any one dimension. It is the relative distances among individual latent coordinates, and not the absolute positioning in the latent space, that matter . 1.2 Formal model A more formal deﬁnition of the general model is as follows. Let y i j be a vector of observed data that is attributable to the dyad of person i and person j , and let f  y i j | θ i j  be the likelihood of observing y i j , given the dyad-speciﬁc parameter vector θ i j . Next, let θ i j be heterogeneous across dyads, with each θ i j drawn randomly from a dyad-speciﬁc prior distribution g  θ i j | φ i j  . A model in which φ i j is common acr oss all dyads, or itself distributed independently (drawn from its own mixing distribution), would imply cr oss-dyad independence of θ i j , which may not make sense in a network setting. T o incorporate some network-based dependence in the distribution of θ i j , we instill a pattern of heterogeneity of φ i j that allows for a useful, intuitive interpretation of the similarities. Thus, ther e ar e two sources of heter ogeneity that generate θ i j : independent dyad-level variation from g  θ i j | φ i j  , and network-induced interdependence in the distribution of φ i j . Before explaining how we model heterogeneity in φ i j , let us shift our focus from the level of the dyad to the level of the individual. Each dyad is made up of two individuals, each of whom has its own, mostly unobserved traits and characteristics. Let z i be a D − dimensional vector that is associated with person i , and let z be the collection of all N of these vectors. Since each z i is unobserved, we call it a “latent coordinate,” and the D -dimensional space on which it lies a “latent space,” as in Hof f et al. (2002) and Handcock et al. (2007). Even if the N vectors in z are distributed independently on the latent space, the distances between every pair of z i (the “latent distances”) ar e not. By expressing φ i j as a monotonic function of the distance between z i and z j , we induce dependency among all the φ i j and, in turn, all the θ i j . As an example, suppose that θ i j repr esents a rate of contact between i and j , and the distribution of θ i j depends positively on φ i j (for example, the mean of g  θ i j | φ i j  increases with φ i j ). W e determine φ i j by evaluating a 7 monotonically decr easing function of the latent distance, so as i and j are less similar (the distance between z i and z j goes up), the rate of interaction between i and j goes down. But we never need to estimate θ i j or φ i j directly . W e need only to estimate the locations of z i for all N people to get the values of φ i j for all ( N 2 ) dyads. 1.3 Mixtures of Dirichlet processes: what they are, and how to use them to model the latent space Even if we model φ i j as a function of the latent distance among the z i ’s, we still have the issue that there ar e many z i ’s, and thus a large number of latent distances, to model. This means that at each iteration of our estimation algorithm, we need to compute ( N 2 ) values of φ i j , and ( N 2 ) corre- sponding data likelihoods. When N is small, scalability becomes less of a pr oblem, and one could use the original parametric formulation of the latent space model. But as N becomes even mod- erately large, estimating the latent coordinates becomes computationally infeasible. W e reduce the number of distinct values of z i by using a discrete distribution, H ( z i | · ) for the distribution of z i on the latent space. If this discrete distribution has k mass points, then there are only ( k 2 ) + 1 distinct latent distances. For a given network size, a larger difference between k and N leads to a greater computational savings by having fewer distinct values of φ i j to consider . T o avoid having to estimate each θ i j directly , we choose f  y i j | θ i j  and g  θ i j | φ i j  such that we can integrate over θ i j analytically , and express the mar ginal distribution f  y i j | φ i j  in closed form. However , we do not want to prespecify the functional form of H , because we don’t know for certain what it is, nor do we want to prespecify k , since we do not know what the “correct” number of mass points for H is. Our appr oach is to use a mixtur e of Dirichlet pr ocesses as a Bayesian nonparametric prior dis- tribution for the points on the latent space. Although the properties of Dirichlet processes (DP) have been known for a while (back to Ferguson 1973), they are still relatively new to marketing. The few examples include Ansari and Mela (2003) (as a Bayesian alternative to collaborative ﬁl- tering), Kim et al. (2004) (identifying clusters of customers in discrete choice models), W edel and Zhang (2004) (analyzing brand competition across subcategories) and Braun et al. (2006) (esti- mating thresholds of claiming behavior for home owners’ insurance). In our context, a Dirichlet process is a probability distribution over distributions (as opposed to a distribution over a scalar 8 or vector). Accordingly , a single draw from a Dirichlet process is itself a random distribution, from which we can draw samples of a variable of interest. An important feature for our context is that each r ealization of a DP is a discr ete distribution, with its support having a ﬁnite number of mass points ( k in the previous paragraph), so the DP can be thought of as a prior distribution on discrete distributions. 2 There are, of course, many ways to model discrete points on a space; a traditional latent class model with a prespeciﬁed number of locations is an extreme example. What makes the DP more useful in this context is that it has a parsimonious r epresentation, with straightforward sampling properties, and does not requir e a prespeciﬁcation of the number of mass points. In our latent space framework, we let H be a r ealization fr om D P ( H 0 , α ) , and then have each z i be a draw fr om H . The ﬁrst parameter , H 0 , is itself a probability distribution, and is the “mean” of the distribu- tions that the DP generates. A scalar α controls the variance of the realizations of the DP around H 0 . This variance is low when α is high, so for high α , realizations from D P ( H 0 , α ) will look a lot like the distribution function of H 0 . This concentration of the DP towards H 0 results from a DP that generates a discrete distribution with a lot of mass points (a high k ). When α is low , realiza- tions from D P ( H 0 , α ) look much less like H 0 (high variance), because this DP generates discrete distributions with fewer mass points. So α plays an important role in determining just how dis- crete (i.e. value of k ), or clustered, a DP-generated distribution r eally is. Reasonable choices for H 0 are those distributions for z i that one might use in a purely parametric model (note that H 0 could have parameters of its own, with their own priors, that need to be estimated.) Depending on the application, one can either put a prior on α or set it directly . Given H 0 and α we need to know how to simulate H from D P ( H 0 , α ) and then each z i from H . Since H is nonparametric, even though we know it was generated by the D P ( H 0 , α ) , the posterior distribution of any new z i depends on all the other z − i . Consequently , ther e is no obvious way to draw a z i from H directly . The “trick” is to integrate out H analytically , and treat z i as if it were drawn fr om this mar ginal distribution, a mixtur e of Dirichlet processes (MDP , see Antoniak 1974). 2 The formal deﬁnition of what makes a stochastic distribution a DP is a more technical issue. Essentially , the prob- abilities of certain events occurring must follow a Dirichlet distribution with parameters that depend on H 0 and α ). There is an accessible and r eadable explanation in O’Hagan and Forster (2004, ch. 13). 9 The probability of any one z i , given the empirical distribution E D ( · ) of all the other z − i , is Prob ( z i | z − i , H 0 , α ) = α H 0 + E D ( z − i ) α + N − 1 (1) (Blackwell and MacQueen 1973; Escobar 1994). So in the estimation algorithm, α determines how likely it is that any new draw of z i comes from one of the existing, distinct values already possessed by another individual in the dataset (if this is likely , then there are few mass points, with lots of clustering), or from the baseline distribution H 0 , as a new value. T o illustrate how this works, Figur e 1 shows simulations from an MDP when H 0 is a univariate standard normal distribution, for differ ent values of α . In the ﬁgure, the heavy black line is the standard normal cdf, and each color ed line is a single r ealization fr om the MDP . W e see that when α is low , there ar e fewer mass points in each realization, and when α is high, the higher number of mass points allows the realizations to approximate the normal cdf. In Figure 1b, for each α we pr esent histograms fr om draws of a single realization of the MDP (so these are draws fr om a distribution that the MDP generated). Again, we see fewer distinct clusters (low k ) when α is low , more clusters when α is high. In our network model, we are dealing with more dimensions and a richer speciﬁcation of H 0 , but the basic idea remains the same. How we select H 0 and α , and the priors we place on them, is described in more detail in Appendix B. Selection of an appropriate distribution for H 0 requir es that we intr oduce some iden- tifying restrictions on the location vectors ( z i ). The concern is that we cannot simultaneously and uniquely identify both the scale of the latent space, and the parameters of the distance function determining φ i j . T o handle this problem, we constrain the prior distribution of z i so the mean distance of any z i from the origin is one. However , we need to do this without introducing too much “incorr ect” prior information. For example, a simple choice for H 0 could be a standard mul- tivariate normal distribution; setting the mean at the origin and the variance at one addresses the translation and scale identiﬁcation issues. The pr oblem with deﬁning H 0 as a multivariate normal is that it implies that our prior on the distribution of z i has a mode at the origin. This prior turns out to be informative, as it generates artifactual clusters of individuals around the origin in the pos- terior . An alternative speciﬁcation for H 0 could be a bounded uniform distribution (so the mean distance from the origin remains one), but that would constrain all z i to be inside a hypersphere, 10 x p 0.2 0.4 0.6 0.8 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● : alpha 0.4 −3 −2 −1 0 1 2 3 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● : alpha 5 −3 −2 −1 0 1 2 3 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● : alpha 20 0.2 0.4 0.6 0.8 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● : alpha 100 (a) Sample realizations x mass 0.0 0.1 0.2 0.3 0.4 0.5 0.6 −2 0 2 : alpha 0.4 0.00 0.05 0.10 0.15 −4 −2 0 2 4 : alpha 5 0.00 0.05 0.10 −4 −2 0 2 : alpha 20 0.000 0.010 0.020 −2 0 2 : alpha 100 (b) Histograms of draws from a single r ealization Figure 1: Illustration of realizations from a mixture of Dirichlet process with H 0 being a standar d normal. At left, the black line is the cdf of the H 0 , and each color ed line r epresents a single r ealiza- tion of a MDP . Each panel corresponds to a different value of α . At right, each panel is a histogram of draws from a single r ealization. effectively placing an upper bound on the latent distance between customers. This, too, seems like an unreasonable expr ession of prior information. Our solution involves using spherical coor- dinates for z i , consisting of two components: a radius repr esenting the distance from the origin, and the location on the surface of a hyperspher e that has that radius. W e show in Appendix B that H 0 can be factored into priors for these two components fr om which it is straightforward to draw samples. This prior on z i , combined with the data likelihood, leads to conditional posterior distributions that ar e easily incorporated into Gibbs samplers. Escobar (1994) and Escobar and W est (1998) describe some of the theory and derivations behind this, while Neal (2000) details step-by-step instructions on how to add MDPs to Gibbs samplers for both conjugate and nonconjugate mod- els. 3 Thus, MDPs allow marketing modelers to relax many of their distributional assumptions by adding only one additional step to the parametric Gibbs sampling algorithm. W e give details of our estimation algorithm in Appendix C. Our exploitation of the discr eteness pr operty of Dirichlet 3 This way of expressing the MDP (and the approach we took in our estimation is known as the “Polya urn” repr e- sentation. There is another , equally useful approach known as the “stick-breaking” repr esentation (Sethuraman 1994) that one can also use to build conditional posterior distributions for Gibbs samplers Ishwaran and James (2001). 11 processes also lets us reduce the computational burden substantially , as we demonstrate in Section 3. 2 Example: telephone calls W e now turn to a speciﬁc application of our model, using a dataset provided by Chongqing Mo- bile, a subsidiary of China Mobile, the largest cellular phone operator in China. Cellular phone networks have been reported to be highly r epresentative of self-reported friendships (Eagle et al. 2009), making such data ideal for studies of network based interdependencies among customers. The data consist of contact r ecord information (for phone calls and SMS messages) for a panel of 4, 781 r esidents of Chongqing who are members of the “silver tier ”, “gold tier ” or “diamond tier ” of the company’s pr eferred customer program. Each recor d contains the identiﬁers for both parties in the contact, and the date of when the contact takes place. For the purposes of this ex- ample, we ignore contacts with people outside this N -person network. 4 The observed geodesic distance is ﬁnite for all dyads (i..e, all customers are connected to every other customer in a ﬁnite number of steps). W e divide the observation period into a six-month calibration period and a six-month holdout period. Descriptive statistics for this dataset are summarized in T able 1. Of the 18, 078 nonempty dyads in the dataset, only 7, 559 appear in both the calibration and holdout samples. 5, 058 dyads are nonempty in calibration, but empty in holdout, and 5, 461 are empty in calibration, but nonempty in holdout. One way to describe the structure of the observed network is to compare it to the “small world” networks described in W atts and Strogatz (1998) and W atts (1999). Generally speaking, a small world network is one in which everyone in the network is connected to everyone else through a relatively small number of intermediaries (i.e., a low mean geodesic distance), and a relatively large number of common friends who are connected among themselves (i.e., a high clustering coefﬁcient). W e can assess the extent to which a network is “small world” comparing the mean geodesic distances and clustering coef ﬁcients to those that we would expect to see fr om a network 4 Our intent in using this dataset is to demonstrate the effectiveness of our estimation method, and to illustrate some of the issues that arise when modeling dyadic data. Therefor e, we treat our dataset as an entire population of individuals, and not as a random sample; our interest is only in contacts made among individuals in this population. If we were to generalize parameter estimates and pr edictions to a greater population, ignoring out-of-network calls could inﬂuence speciﬁc parameter estimates. There are an additional 209 silver , gold or diamond customers in the panel for whom there wer e no observed calls to other silver , gold or diamond customers during the observation period. 12 Calibration Holdout Full W eeks 26 26 52 Customers 4,781 4,781 4,781 Non-empty dyads 12,617 13,020 18,078 Proportion of empty dyads 0.9989 0.9989 0.9984 Clustering coefﬁcient 0.128 0.127 0.127 Mean (s.d.) degree distribution 5.3 (4.9) 5.4 (5.3) 7.6 (6.7) Mean (s.d.) geodesic distance 5.5 (1.4) 5.3 (1.3) 4.7 (1.1) Mean (s.d.) calls per non-empty dyad 7.5 (18.7) 7.6 (18.7) 15.1 (36.0) Mean (s.d.) shared friends in non-empty dyad 3.0 (2.7) 3.3 (2.8) 5.7 (2.8) T able 1: Descriptive statistics of China Mobile dataset. in which connections are determined at random for the same number of people (4,781) and aver - age number of “friends” per person (7.6). Using the asymptotic approximations in W atts (1999), the mean geodesic distance we would expect from a random graph of this size is about 4.2, and the expected clustering coefﬁcient is about 0.002. In the observed Chongqing Mobile network we observe quite a bit more clustering than we expect to see from a random graph, while the mean geodesic path is slightly longer what we would expect. One possible reason that our mean geodesic distance is not smaller is that we could have a large number of small clusters, and not all small clusters are connected to each other . In fact, our estimates of k (illustrated in Figure 4) will bear this out. W e also note that our network would not qualify as a “scale free” network, in that the degree distribution clearly does not follow a power law-type distribution (we show the observed degree distribution in Figur e 2). 2.1 Model speciﬁcs Using the notation introduced in Section 1, y i j is the vector of intercontact times, ending with the survival time (the duration between the last observed contact and the end of the observation period). If there ar e no observed contacts in the dyad, y i j is the length of the observation period, and we call that dyad “empty .” If ther e are observed calls, the dyad is “non-empty .” The deﬁnition of f  y i j | θ i j  follows the logic of the “exponential never-triers” model in (Fader et al. 2003), which in turn draws from the “hard-cor e never-buyers” model in Morrison and Schmittlein (1981) and Morrison and Schmittlein (1988). First, there is a probability p i j that a dyad will remain forever empty , no matter how long we wait. W e call dyads like this “closed”. Next, for dyads that are 13 “open,” (with probability 1 − p i j ), inter contact times follow an exponential distribution with rate λ i j . T o link these speciﬁcs with our general model, θ i j = { p i j , λ i j } . Note that there ar e two ways we could observe an empty dyad. The dyad is either closed, or it is open, but with a contact rate that is sufﬁciently low that we just happened to not observe any contacts during the observation period. Whether the exponential distribution is appr opriate for this dataset is ultimately an empirical question, but we choose it for four reasons. First, we do not need to make special pr ovisions for left-censoring because of the memorylessness property . Second, the number of contacts is a suf- ﬁcient statistic for the individual elements in y i j . W e were able to exploit these two features of the exponential distribution to gain computational savings without compr omising the fundamen- tal purpose of the resear ch. Third, we did run the model on a much smaller dataset where f ( · ) is governed by a “W eibull never-triers” model, in order to allow for duration dependence, and found that since the shape parameter of the W eibull was close to 1, it reduced to the exponential distribution anyway . Finally , we chose the exponential distribution because it forms a conjugate pair with our choice of g  θ i j | φ i j  , a gamma distribution for λ i j with dyad-speciﬁc mean µ i j and common variance v , and a degenerate distribution over p i j , so that at this level of the hierarchy , p i j is homogeneous for all dyads (we will add heter ogeneity to p i j later thr ough the latent space). The exponential-gamma pair lets us integrate over θ i j analytically , further easing computational effort. The vector φ i j therefor e contains three elements, p i j , µ i j and v ( p i j is contained in both θ i j and φ i j .) T o evaluate whether latent space is worth adding to a model of interactions data, we estimated the model with three differ ent deﬁnitions of the elements of φ i j . For a “Baseline” model, we let φ i j = φ , a common value for all dyads (note that we still maintain dyad-level heterogeneity in θ , but it does not appear explicitly in the data likelihood). For a second model, HMCR (for “Ho- mogeneous Mean Contact Rate”), µ i j and v r emain homogeneous across dyads, but p i j is now determined by the distribution on the latent space. Speciﬁcally , we deﬁne logit p i j = β 1 p − β 2 p d β 3 p i j (2) where the β ’s are coef ﬁcients to be estimated, and d i j is the latent distance between z i and z j . For a third model, named “Full,” p i j retains the same deﬁnition as in Equation 2, except that µ i j is now 14 heterogeneous acr oss dyads, deﬁned as log µ i j = β 1 µ − β 2 µ d β 3 µ i j (3) Equations (2) and (3) allow the respective relationships to latent distance to be concave, linear or convex. The parameters β 2 p , β 3 p , β 2 µ , and β 3 µ are constrained to be nonnegative, because as latent distance increases, the probability of contact, and the rate of contact, should decrease. W e selected Euclidean distance as our distance measur e, after experimenting with others that did not perform as well (van Alstyne and Brynjolfsson 2005). 5 Another candidate for this distance met- ric is the Mahabalonis distance (as used in Bradlow and Schmittlein 2000), which weights some dimensions more than others in the computation of the distance among individuals. However , the non-parametric nature of the estimated latent space means the dimensions ar e already dif- ferentially scaled. Also, the Euclidean distance is computationally more efﬁcient. As with the parametric speciﬁcation, the functions in Equations 2 and 3, and the distance measur e, are subject to empirical testing and may not be appropriate in all contexts. 2.2 Assessing contribution of the latent space So far , we have assumed that parameter inter dependence is an important characteristic of a model of customer interactions. However , one could falsify this claim by showing that models in which dyad-level parameters ar e independent ﬁt no worse than models that incorporate a latent space. W e ran our algorithm with latent spaces of differ ent dimensionality and, based on estimates of log marginal likelihoods, we decided that the parsimonious choice of D = 2 is most appropriate (see Appendix A). As evidence that the latent space models do better than independent models, we evaluate the contribution of latent space based on both posterior predictive checks (PPCs) and on forecasting interactions in empty dyads. Posterior predictive checks allow us to evaluate how well our model repr esents the data- generating process (Rubin 1984; Gelman et al. 1996). Three of our PPC test statistics are the same as those used by Hunter et al. (2008) to assess goodness-of-ﬁt for social networking data: the degree 5 Here, we are talking about distance between two individuals’ coordinates on the latent space. This concept of distance is different from when we talk about geodesic distance, which is the smallest number of observed connections along the shortest path between two individuals. 15 distribution, the dyad-wise shared partner distribution, and the distribution of geodesic distances. W e also examine the histogram of the number of calls made within non-empty dyads, the density of the network, and the clustering coefﬁcient for the network. All of our PPCs in this article are with r espect to the 26-week holdout sample. Figur e 2 shows the r esults for the distributional PPCs, and Figure 3 shows the PPCs for the density and clustering coef ﬁcients. The x-axis in each panel is the count of individuals or dyads, and the y-axis is the log proportion of those individuals with each count. The dark dots r epresent the log pr obabilities generated fr om the actual dataset, and the box-and-whisker plot represents the distribution of log probabilities acr oss the simulated datasets. Figure 3 shows the PPCs for the network density and clustering coefﬁcients; the vertical line is the observed value. At ﬁrst glance, it might appear that all of the models replicate the actual datasets rather well. The reason that even the Baseline model does as well as it does is that most of the value from posterior prediction comes from inferring whether a dyad is open or closed. Simply looking at whether a dyad is empty or non-empty pr ovides a lot of information about the likelihood of futur e emptiness, because non-empty dyads must be open. However , closer examination r eveals that the Baseline model is not well calibrated at all. The “actual” dots lie far outside the whiskers for the predictive distributions for many of the counts. The two models that involve some kind of latent space structur e ﬁt better on these test statistics. However , we do not see much dif ference between the HMCR and Full models. This suggests that the value of the latent space is more in predicting the potential existence of an interaction (whether the dyad is open or closed), than in predicting the contact rate. In addition to assessing model ﬁt in aggregate, we also care about how well the model per- forms at the dyad level. Our appr oach her e is to predict which of the dyads that ar e empty during the calibration period become nonempty in the holdout period. Empty dyads all have the same observed data pattern, so there is no obvious way to differentiate among them. W e can, however , use the latent space structure and a straightforward application of Bayes’s Theorem to compute posterior distributions of unobserved parameters, and then use those probabilities to rank dyads in terms of those most likely to generate interactions during some future period of any duration we want. T o assess the pr edictive ability of a model, we ﬁrst identify , individual by individual, the top Q % lift (or the top q lift, wher e q = Q / 100) most likely , pr eviously uncontacted, individ- 16 degree count propor tion of individuals − log scale 0.025 0.05 0.1 0 2 4 6 8 10 12 ● ● ● ● ● ● ● ● ● ● ● ● Baseline 0 2 4 6 8 10 12 ● ● ● ● ● ● ● ● ● ● ● ● HMCR 0 2 4 6 8 10 12 0.025 0.05 0.1 ● ● ● ● ● ● ● ● ● ● ● ● Full (a) Degree Distribution number of shared partners propor tion of dy ads − log scale 5e−06 5e−05 5e−04 0.005 0.05 0.5 0 2 4 6 8 10 12 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Baseline 0 2 4 6 8 10 12 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● HMCR 0 2 4 6 8 10 12 5e−06 5e−05 5e−04 0.005 0.05 0.5 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Full (b) Dyadwise Shared Partners Distribution length of geodesic path propor tion of dy ads − log scale 5e−06 5e−05 5e−04 0.005 0.05 0.5 0 2 4 6 8 10 12 ● ● ● ● ● ● ● ● ● ● ● ● ● ● Baseline 0 2 4 6 8 10 12 ● ● ● ● ● ● ● ● ● ● ● ● ● ● HMCR 0 2 4 6 8 10 12 5e−06 5e−05 5e−04 0.005 0.05 0.5 ● ● ● ● ● ● ● ● ● ● ● ● ● ● Full (c) Geodesic distance distribution number of calls number of dy ads − log scale 10 100 1000 5 15 25 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Baseline 5 15 25 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● HMCR 5 15 25 10 100 1000 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Full (d) Number of calls Figure 2: Posterior pr edictive checks for holdout sample. For each sub-ﬁgure, observed data are repr esented using dots, and the posterior predictive distributions ar e repr esented by the “box- and-whisker ” symbols. 17 0.00112 0.00114 Baseline Net Density 0.00112 0.00114 HMCR Net Density 0.00112 0.00114 Full Net Density 0.115 0.120 0.125 Baseline Clustering Coef 0.120 0.125 0.130 HMCR Cluster ing Coef 0.120 0.125 0.130 Full Cluster ing Coef Figure 3: PPCs for the density and clustering coefﬁcients. uals observed to contact during the holdout period 6 . For a completely random or naive model, the per centage of the top Q % most likely empty dyads to become non-empty should be equal to q . For any other model, if the value for the top Q % metric is greater than q , then the model pr o- vides some ”better -than-chance” predictive value. The use of the Q % lift metric ensures that the maximum of this value is always equal to one. T able 2 pr esents these lift metrics for dif ferent models and values of Q %, and dif ferent calibra- tion/holdout samples. Results are pr esented for all thr ee model variants, with D = 2 for the latent space models. In addition, we present results for a “Condition on Observed” prediction rule, un- der which empty dyads are to r emain empty in holdout, and non-empty dyads r emain non-empty in holdout. The “Geodesic Distance” model ranks dyads according to their geodesic distances, as in Kossinets and W atts (2006) (we break ties in two different ways: randomly , or based on the total number of observed interactions along the path). The lift metrics suggest that both the Base- line model and the “Condition on Observed” rule do exactly as well as one would expect from random selection. This is because they both assumes that there is no network structure among individuals in the dataset, and thus all empty dyads are considered to be identical. In contrast, 6 For any individual i , the top q lift requires rank ordering all potential customers j 6 = i based on the predicted probability of interaction. For a holdout sample, the top q lift of this rank or dered list is equal to the pr oportion of the top q customers for whom we observe interactions with i , divided by the proportion of total interactions made by customer i . 18 Duration of calibration (holdout) period 13 (39) weeks 26 (26) weeks 39 (13) weeks Q%= 0.1 0.2 1.0 0.1 0.2 1.0 0.1 0.2 1.0 Baseline .001 .002 .010 .001 .002 .010 .001 .002 .010 HMCR .100 .112 .141 .133 .138 .178 .153 .154 .177 Full .100 .109 .146 .132 .135 .157 .154 .159 .197 Condition on Observed .001 .002 .010 .001 .002 .010 .001 .002 .010 Geodesic - random tiebreak .050 .088 .166 .036 .067 .149 .025 .051 .198 Geodesic - # calls tiebreak .064 .110 .186 .056 .098 .190 .046 .083 .206 T able 2: Percentage of the top Q % of the empty dy ads (in calibration period) that ar e most likely to become nonempty during holdout period, that actually did become nonempty during the holdout period. Reported values are posterior means; cr edible intervals ar e removed for space and clarity . in the two latent space models, some dyads are more likely to contact each other than others. By sorting the empty dyads according to their posterior latent distances, we no longer assume that all empty dyads are the same. Thus, we can improve on dyad-level prediction dramatically . W e do not, however , see any substantive dif ferences between the HMCR and Full models, suggesting that, in this application, all of the action is on the open/closed pr obability and not on the contact rates. Nevertheless, our results indicate that the use of the latent space structur e for networked data is a better model than assuming independence across dyads. 7 3 Scalability and computation Having demonstrated the contribution of latent space models, we now turn to the issue of scal- ability and computation. The amount of computational improvement one can expect fr om using DP priors on the latent space depends on how well we can cluster dyads into groups that have the same data and parameters. In networks in which every dyad generates a dif ferent observed outcome (e.g., if the network is dense and the observed value is continuous), the likelihood for each dyad will have to be computed separately , and a discr ete repr esentation of the latent space 7 There are of course many differ ent ways to predict link formation (known in the machine learning community as “link mining”), such as the Katz Score (Katz 1953) and the SimRank algorithm (Jeh and W idom 2003). Getoor and Diehl (2005) provides a detailed review of link mining methods, and Liben-Nowell and Kleinberg (2007) compare the performance of some of them. W e compared the predictive ability of our latent space approach against some of these methods, and found that while our model did best when tests were more discriminating (low Q ), the other models “caught up” when Q was increased. However , our model offers behavioral intuition (see Section 4) that machine learning algorithms cannot provide, and we are willing trade off some pr edictive power for managerial interpretability . Nevertheless, our objective in pr edicting future link formation is only to demonstrate the value of accounting for latent network structure when modeling interactions data. 19 will have little ef fect. However , the density of many (if not most) social networks tends to be very low . Even if non-empty dyads generate data on a continuous domain (as in our China Mobile example), there are so many empty dyads, all with the same data, that the number of distinct likelihoods to compute is much lower than the total number of dyads in the network. If the ob- served data is discrete, then even mor e aggr egation is possible. Of course, aggregation according to observed data is standard practice when a model is homogeneous, or marginal likelihoods ar e available in closed form. The DP prior lets us group observations with similar latent parameters as well. The number of likelihood evaluations at each MCMC iteration depends on two factors: i) the number of groups with distinct data patterns (which in turn depends on the size and density of the network); and ii) the number of mass points for each realization from the Dirichlet process. Dyads with the same z i , z j pair , and the same value of y i j , must have the same likelihood, since they have the same data and same parameters. As long as we keep track of the number of dyads with each z i , z j pair , we can compute the log likelihood for that pair once for each y , and multiply by the number of dyads with that pair and that y . Among all the data zeros, ther e are only ( k 2 ) + 1 possible likelihood values. If k is less than N , there is a computational saving, even if all of the non-zero values of y are differ ent (as happens when y is continuous). If y is discrete (so C y is the number of distinct values of y ), there ar e at most  ( k 2 ) + 1  C y possible likelihoods. For a continuous y , but with a large number of zer os, the number of possible likelihoods is ( k 2 ) + 1, plus the number of nonzero y ’s. Clearly , the mor e distinct observed data patterns there ar e, the less one can take advantage of the discr etization of the latent space that is generated by the DP . But in the social networking applications that are common in marketing, networks ar e often very sparse, so we have at least one very large gr oup of dyads with the same data. T o assess just how much computational savings there is, consider the Full model in the tele- phone call example. The calibration dataset has 12, 617 non-empty dyads; likelihoods for each of these dyads must be computed individually . The mean of k is 530, so there are 140, 186 distinct distances between mass points on the latent space. Instead of computing 11, 413, 973 separate like- lihoods for each of the empty dyads, we only need to compute 140, 186 of them. Thus, the number of likelihoods to compute at each MCMC iteration is 152, 803. This r epresents a 98.7% r eduction in computational requir ements. 20 Size of Network value 0 200 400 600 1000 2000 3000 4000 Number of Mass P oints 0 50000 100000 150000 200000 1000 2000 3000 4000 T otal Likelihood Evaluations alpha = 0.5 alpha = 20 alpha = 300 Figure 4: Posterior means for number of mass points, and total likelihood computations, for sub- sampled networks. The extent to which our method can scale for datasets with many mor e individuals (large N ) depends on how both the network density and k change as N increases. Ultimately , these are both empirical questions, the second of which we cannot know up front because not only is k unobserved, but it can be inﬂuenced by the choice of H 0 and α . However , the expected number of mass points can be asymptotically appr oximated as E ( k ) ≈ α log  α + N α  (Antoniak 1974; Escobar 1994). Thus, if α is small, the expected number of mass points is also expected to be small, but it will grow for larger datasets. If α is large, the number of mass points for smaller datasets might be larger , but this number will not grow as quickly for larger datasets. T o test how well this approximation works in practice, we estimated the full model using successively larger subsets of our original network. W e then ﬁxed α at three differ ent values: 0.5, 20 and 300 (instead of placing a weakly informative prior on α , as we did in the main analysis). W e also computed the total number of likelihood computations for each sweep of the Gibbs sampler , which is just ( k 2 ) , plus the number of non-empty dyads in the dataset. Figure 4 plots the posterior mean of k (the number of mass points) and the total number of likelihood evaluations, against the size of the network. For the number of mass points, we observe 21 the expected pattern. For small α , the number of mass points is small, but the incremental number of mass points grows with network size. For large α , the number of mass points is lar ge, but the incremental change goes down with network size. The asymptotic approximation suggests that incremental computational effort would decrease even more for even larger values of N , even though the number of total dyads continues to grow quadratically . In terms of total computation, for low α , the number of computations increases more rapidly with N than for higher values of α , but when α is high, the relationship becomes more linear . Even though the number of dyads grows quadratically with k , larger networks will tend to have lar ger number of nonempty dyads. For low α , computation grows faster than linear , but the number of latent dyads is low to begin with, because of the increased clustering. Collectively , our results suggest that, if using a DP prior is not computationally feasible for a particular dataset, the incremental effort likely comes from the inability to aggr egate the observed data, and not fr om an inability to aggr egate the latent parameters. Of course, this is no differ ent fr om scalability problems faced by Bayesian hierar chical modelers who use MCMC to update model parameter estimates from non-networked data. 4 Interpretation and usefulness of the latent space In Figure 5(a), we plot a single draw fr om the joint posterior distribution of the latent coordinates from the Full model with D = 2 (we chose the draw with the lar gest conditional likelihood). Each person in the network occupies a position in the latent space, and we deﬁne a cluster as all individuals who share the same coordinates in the latent space (this is akin to two observations having the same mass point in a r ealization fr om a mixture of Dirichlet processes, and we counted 593 such clusters in this realization). But since multiple individuals are located at the same coor- dinates, to aid visualization we ”jitter” the individual locations of customers by adding a small amount of random noise (drawn from a uniform(-0.03,0.03) distribution) to each coordinate. The scale labels on the axes are included to help refer ence certain parts of the space, and do not have a concrete interpretation themselves. In Figure 5(a), we see that there is considerable clustering, with distinct “super-clusters” (clusters of clusters, or clusters closer to other clusters), of individu- als on the latent space. Also, ther e are some clusters that contain only a few customers, and which are quite separate from the rest of the network of customers, such as the one at { x , y } coor dinate 22 − 2 − 1 0 1 2 − 2 − 1 0 1 2 (a) Latent space x − coordinate y − coordinate − 0.6 − 0.4 − 0.2 0.0 − 0.8 − 0. 6 − 0.4 − 0.2 (b) Segmentation and T argeting x − coordinate ! A ! B ! C ! D ! E Figure 5: Illustration of the full latent space, and connections among some selected customers. In both panels the black dots represent jittered locations for individual customer on { x , y } coordi- nates. Lines repr esent observed connections among customers. In panel (b), we labeled several customers who have hypothetically adopted a new product. Cir cles around customers ar e used to identify other customers who may be similar to the tar geted customers and ther efore have similar adoption likelihoods. (-0.9,-2.2). Note that the latent space is a random variable, so this ﬁgure repr esents just one pos- sible conﬁguration of the individuals into clusters. Figure5(b) “zooms in” on a small partition of the latent space. Having provided and discussed a graphical depiction of latent space, the next question becomes: what use is this to marketers? W e examine this in the context of segmentation and targeting. Segmentation and targeting using interactions data Segmentation and targeting is central to the development of ef fective marketing strategy . The fundamental idea behind segmentation is to ﬁnd people who are similar to one another , with the assumption that they will respond in simi- lar ways, and therefor e can be targeted using similar methods (e.g. the same price discount, the same promotion, or same advertising copy). In our study we reveal two ways marketers can use 23 network based data (our interactions observations) in practice. As is well documented by authors such as Novak et al. (1992) and Hill et al. (2006), following observed interactions to or from cus- tomers who have alr eady adopted a pr oduct or service can help identify other potential customers. Their results show impr ovements in response rates, compared with methods using observed, tra- ditional segmentation and targeting bases. While these network-based methods are powerful tools for eliciting new customers, our graphical r epresentation of the latent space highlights that there are sometimes interactions among customers who are quite differ ent from one another . W e see this in Figure 5(b). In this ﬁgure, we identify ﬁve individuals for whom a marketer might have some speciﬁc information (e.g., an existing customer , or a r espondent to a pr omotion). The lines radiating from these individuals represent observed links in the dataset. W e also placed circles of common radius around these focal individuals. Network marketing tactics that “follow the links,” would use the lines to determine the next potential customers to target. While this is useful in reaching new clusters, there ar e many mar- keting tactics that have nothing to do with following links or word of mouth, and instead de- pend more on understanding which customers can be gr ouped into mor e homogeneous segments. Given the interpretation of the latent space as repr esenting similarity , anyone in close pr oximity (within the circle) to a focal customer should also be a target. Although most observed contacts also occur within a cluster , there are certainly interactions among dissimilar individuals as well. As an example, consider a marketer of trendy casual clothing, tar geting a college student who interacts with two people: a classmate at the same college, who shares similar demographic traits such as age, education, gender , values; and an older relative with whom ther e is a closer personal relationship, but nothing else in common in terms of pur chase patterns. While the college student might have identical observed interaction patterns with his classmate and with the relative, he shares many more common friends with his classmate than with his relative. Our model places the student closer on the latent space to his classmate than to his r elative, and the r elative is closer to her own friends and others in her social cir cle. This is useful for marketers to be aware of, be- cause the classmate and the r elative repr esent quite dif ferent marketing pr ospects. If the marketer were to identify prospects based on the observed interactions alone, however , he could be target- ing the r elative, and her friends, who ar e unlikely to behave in the same way as the focal customer (the student). T argeting these prospects incurs additional costs with little expected return. In ad- 24 dition, there are many individuals within the circle who never talk to our focal customer , but still “travel in the same social circles,” or who might otherwise be exposed to, or susceptible to, similar marketing activities. W e can compare the use of targeting based on proximity in latent space, with targeting based on geodesic distance 8 . In fact, many more customers can be identiﬁed for targeting than if one were to use a geodesic-type distance metric repr esented by observed interactions. W e calculate that if one wer e to follow the ﬁrst degr ee geodesic distance, on average the marketer would expect to r each eight customers (rounded up from 7.56) . If this data wer e available and the marketer also included in the target set the ”second degree,” or friends of friends, the marketer then expects to reach on average 97 customers. Drawing a circle of radius equal to 0.1 around the customer , the marketer may expect to reach, on average, 232 customers 9 . Since homophily implies that similar individuals ar e more likely to interact, then tar geting based on latent space means that the customers identiﬁed for tar geting ar e mor e likely to be similar to the focal customer . The geodesic distance in some cases could be connections which span much of the space and therefore may lead to leads that ar e substantially dif ferent than the original customer . For example, in Figur e 5(b), the customer labelled ”C” has seven interactions, but two of these interactions are to customers who are at substantially different locations in the latent space. Given the desir e for marketers to ﬁnd customers similar to the focal customer , we assert that it is better to target those customers close to the labeled customers in the network. The latent space model pr esents an opportunity to r eﬁne these targeting methods, and using the Dirichlet process to model the latent space makes the approach computationally feasible for marketing data. Extracting information from limited data Another advantage of using our probability model- ing approach is that the interpretation of the posterior latent space is only loosely dependent on the kind of data that one uses to estimate it. Of course, larger , richer datasets, collected over longer periods of time, might lead to better posterior distributions, but the data itself could really be anything, as long as the underlying data-generating process is dyad-speciﬁc, and depends on 8 A third perspective is that perhaps one should ﬁrst follow connections within the clusters, then the similar cus- tomers, and then follow the geodesic links outside of some cluster 9 Of course, this depends on the size of the circle, but since customers in the same cluster occupy identical coordi- nates, even a circle of minimal radius gives us the result that more customers are identiﬁed for targeting, on average, than the ﬁrst-degree geodesic. 25 similarities in a monotonic way (events are more likely if latent distances are small). The data that is available could be limited in terms of time (a censored dataset), or by the fact that ob- served cell phone interactions are only one of many possible means of communication. Interac- tions occur among customers at heterogeneous rates, and since it is not practical for marketers to wait extended amounts of time to see if interactions will occur among customers, it may be that some interactions that exist among customers occur at such a rate that they may not be ob- served in a small observation window . One might be tempted to treat the addition or subtraction of observed interactions as evidence of nonstationarity . The Bayesian approach to data analysis makes it straightforward to update our estimates of the latent space as new data becomes avail- able, even when the space itself is stationary . Therefore, what Kossinets and W atts (2006) refer to as an ”evolving social network” may , in effect, be an artifact of the censoring of the data. All the observed data does is pr ovide some clues fr om which the true underlying similarities must be inferred. Related to censoring, one of the concerns about using data like phone call records is that they do not necessarily r epresent the universe of interactions among the population. Unless customers reveal all of the people with whom they ever interact with, observed data cannot be an authori- tative document of the underlying social network. There may be other modes of communication, such as email or face-to-face contact. For example, Ansari et al. (2008) consider the case of a Swiss music sharing website, where the connection between users could be described alternatively as “friendship,” “communication,” or “download.” So what value does using only a network of cell phone calls have, if it is an incomplete repr esentation of all interactions? W e can think of any “true” observed interaction network as a population of subnetworks, and each observed network (e.g., the cell phone data), as a single draw from that population (Gelman 2007). W e then treat that observed network as a single data point, and use it to update our beliefs about the structur e of the latent space. If we had observed another mode of communication ﬁrst, we might get a dif- ferent posterior latent space but, in any event, the posterior of the latent space after observing one network becomes the prior before observing the next. Focus on Diffusion and WOM W e see two key contributions of the latent space useful to mar- keters managing the diffusion of innovation of information thr ough customer networks. The ﬁrst 26 involves the concept of ”W ord of Mouth” (e.g. Arndt 1967). While contagion could occur via non-explicit advocacy (e.g. fashion can be seen by people that one does not interact with), explicit communication is well regar ded as an important source of information for customers. WOM is at the heart of models of information diffusion in networks (e.g. Goldenberg et al. 2001). As testi- mony to the importance of consumer reviews, there ar e many services and organizations focused on collecting and presenting such information on just about every product or service. Of key interest in the WOM literature is how the network structur e affects diffusion patterns. The con- tribution of our work is in considering that WOM may work via some geodesic distance versus distance measur ed in latent space. From r elations data, only the geodesic distance can be studied, but there is likely considerable value to considering distance in latent space as a channel for WOM. The second ar ea where the latent space model could be useful in practice is in identifying inﬂuential customers. The concept of a market maven, or opinion leader (Katz and Lazarsfeld 1955; Feick and Price 1987; Iyengar et al. 2008; Kratzer and Lettl 2009) has frequently been stud- ied in the context of diffusion research. However , recent resear ch challenges the notion that such inﬂuentials are the primary reason for ”global cascading inﬂuence” (W atts and Dodds 2007), i.e. the contagious diffusion of innovation or information throughout an entire network. These in- sights suggests that it is vital for marketers to understand how inﬂuence can occur among all customers, and that there is more to WOM marketing than focusing attention on just inﬂuentials. Observations from practice certainly seem to support this, from the popularity of services such as BzzAgent, and Procter & Gamble’s V ocalpoint, which are not selective about recruiting only ”opinion leaders,” but rather would prefer mor e people in their network. Further research opportunities The sociological theory of homophily , coupled with the latent space framework, yields a stochastic r epresentation of the r elative latent characteristics underlying interactions data. The latent characteristics are repr esented on a latent space, and we propose a Bayesian nonparametric for the latent space using Dirichlet processes. Latent spaces are well known in social network analysis, and Dirichlet processes and probability models are known in marketing. However , due to the computational obstacle involved in estimating lar ger networks, the concepts have not yet fully integrated across disciplines. Our resear ch lowers this obstacle, and makes probability modeling more accessible to marketing r esearchers who possess data on 27 customer interactions. This approach maintains the properties of interdependence, heterogeneity and interpr etability , a goal that is har der to accomplish with extant classical or machine learning approaches. W e readily admit that our interpretation of the latent space, and our suggestions on how to use it, depend on an acceptance of two premises. First, we need to believe that similarities drive interactions, and so one can use interactions to infer similarities. W e use the volumes of resear ch of homophily to support this contention, but we have not tried to test this directly . Second, our recommendation that managers consider using latent distance, rather than observed geodesic dis- tance, to segment and tar get customers assumes that similar individuals have correlated pur chase prefer ences or behavior . This is a premise that could be tested, and we hope that both researchers and practitioners will undertake that challenge. Unfortunately , the data that we have at our dis- posal does not allow us to follow that path, but we would like to describe brieﬂy how we think this might work. The output of the latent space model is a posterior distribution of conﬁgurations on a latent space. Figure 5 is one such conﬁguration. Although the distances among individuals in a sin- gle conﬁguration do not have a physical interpretation, we can still treat them as distances in a statistical sense. So we could draw upon the methods of hierarchical spatial modeling to infer a correlation structure among individuals, similar to those described in Banerjee et al. (2004). An ex- ample of this kind of tr eating a non-physical distance as a physical one is Y ang and Allenby (2003), who computed a demographic distance between peopl e based on pr oﬁles of personal characteris- tics. So just as they modeled correlations in prefer ences as functions of observed geographic and demographic distance, we propose modeling these correlations as functions of latent distances. W e hypothesize that since observed interactions repr esent only one possible path for the sharing of information, it is latent distance, rather than geodesic, demographic or geographic distance, that would best predict these correlations. In or der to conduct a test like this, one would need two types of data for the same set of people: dyad-level interactions data to infer the latent space, and individual-level purchase data to see if latent distance explains corr elations in purchase behavior . As more and more business is conducted through mobile communications devices, we anticipate that data like this will become more available. A corollary to this resear ch stream would be to incorporate individual-level demographics or covariates into the latent space model, and to better 28 understand how that information might complement interactions data in unde rstanding pur chase behavior . Appendices A Choosing dimensionality of the latent space There are a number of approaches one could take to selecting D , and ﬁnding a general method for choosing among dif ferent speciﬁcations of Bayesian hierar chical models, especially those that incorporate nonparametric priors, remains an area of active resear ch among statisticians. W e be- lieve that because of the abstract nature of the latent space, there is no “correct” value of D that one needs to infer fr om the data. Hoff (2005) notes that one should choose the smallest value of D that offers a r easonable model ﬁt, erring on the side of parsimony . Adding more dimensions impr oves model ﬂexibility , but can also lead to over-ﬁtting. As far as objective measur es go, he suggests ex- amining the log mar ginal likelihoods (LML) (we use the holdout LML for the Full model), as well as the posterior predictive checks (PPC) for test statistics that capture important characteristics of the data (we discuss PPCs in Section 2.2). In T able 3 we pr esent relative estimated LML of the HMCR and Full models, for different val- ues of D , from both the calibration and holdout datasets. Our estimates were generated using cumulant approximations, and adjusting for the discrepancy between posterior and prior sup- port, using the methods proposed in Lenk (2009). Results are normalized such that the reported estimate for the HMCR model, with D = 2 is 0, for each of the calibration and holdout datasets, and then scaled by 1,000 for readability . calibration holdout D HMCR Full HMCR Full 2 0 -40 0 -337 3 221 -310 -378 -690 4 -396 -1,609 -190 -1,792 5 -1,550 -1,130 -3,183 -1,560 6 -4,729 -829 -5,069 -1,573 T able 3: Estimates of log marginal likelihoods (LML) of models, by dimensionality of latent space and model variant. Estimates are normalized with respect to the D = 2 HCMR model for each dataset. 29 In three of the four cases, D = 2 is preferr ed, and in the remaining one, D = 3 is preferr ed. Also, we found essentially no differ ence among values of D in posterior predictive checks. So, following Hoff ’s advice, we use the D = 2 models for subsequent analysis. B Model speciﬁcation for China Mobile example In this appendix, we derive the data likelihood and hyperprior speciﬁcations for our Chongqing Mobile application. Let p i j be the probability that a dyad between i and j is open, and let λ i j be the rate of contacts between i and j (assuming exponentially-distributed inter contact times), if the dyad is open. W e also deﬁne an auxilliary latent Bernoulli variable s i j that indicates whether a dyad is open ( s i j = 1) or closed ( s i j = 0). T o remain consistent with our general model speciﬁcation in the text, we use y i j to denote the vector of observed intercontact times, and y ∗ i j to denote the count of observed contacts. Also, let T be the duration of the observation period. The data likelihood is similar to an “exponential never triers” model (Fader et al. 2003). There are two ways a dyad could be empty (i.e., y ∗ = 0): the dyad could be closed ( s i j = 0), or it could be open, but the contact rate λ i j is suf ﬁciently low that ther e just happened to be no contacts during the observation period. If we observe any contacts at all, we know the dyad must be open. Therefor e, the data likelihood is f  y i j | λ i j , p i j  =  1 − p i j  I h y ∗ i j = 0 i + p i j λ y ∗ i j i j exp  − λ i j T  (4) W e incorporate dyad-wise unobserved heterogeneity in λ i j by using a gamma distribution with dyad-speciﬁc shape parameter r i j and dyad-speciﬁc scale parameter a i j f  λ i j | r i j , a i j  = a r i j i j Γ  r i j  λ r i j − 1 i j exp  − a i j λ i j  (5) After integrating over λ i j , f  y i j | p i j , r i j , a i j  =  1 − p i j  I h y ∗ i j = 0 i + p i j Γ  r i j + y ∗ i j  Γ  r i j   a i j a i j + T  r i j  1 a i j + T  y ∗ i j (6) W e will also use the r eparameterizations r i j = µ 2 i j / v and a i j = µ i j / v , wher e µ i j and v are the dyad- 30 speciﬁc mean and common variance of the gamma distribution, respectively . Linking back to our general model formulation in Section 1.1, φ i j =  p i j , µ i j , v  . The deﬁnitions of these parameters ar e described in Section 2.1. Hyperprior speciﬁcs For the choice of H 0 , we decompose each latent coordinate into two components: the distance from the origin (a “radius”) and the location on the surface of a hyperspher e that has that radius. W e then choose a H 0 that factors into a prior on these two components. In other words, we think of the elements of z i in terms of their spherical, rather than Cartesian, coor dinates. If the Cartesian coordinates of z i (herein suppressing the i subscript) ar e z 1 , z 2 , ..., z D , then its polar coor dinates ar e ( ϑ 1 , ..., ϑ D − 1 , ρ ) , where ρ is a distance from the origin and the ϑ 0 s are angles, expressed in radians, such that 0 < ϑ 1 < 2 π , and 0 < ϑ j < π for 2 ≤ j ≤ D − 1. W e can then factor H 0 as g 0 ( z ) = f ( ϑ 1 , ..., ϑ D − 1 , ρ ) = f ( ϑ 1 , ..., ϑ D − 1 | ρ ) f ( ρ ) (7) Conditioning on ρ , we want to place a distribution on ϑ = ( ϑ 1 , ..., ϑ D − 1 ) , such that there is a uniform pr obability of being at any location on a D − dimensional hyperspher e with radius ρ . This is achieved by letting f ( ϑ | ρ ) be a multivariate power sine distribution (Johnson 1987; Nachtsheim and Johnson 1988), where f ( ϑ ) ∝ D − 1 ∏ j = 1 sin j − 1 ϑ j (8) Thus, ϑ 1 has a uniform distribution, f ( ϑ 2 ) ∝ sin ( ϑ 2 ) , f ( ϑ 3 ) ∝ sin 2 ( ϑ 3 ) , and so forth. Johnson (1987, ch 7) pr oposes some algorithms for simulating fr om a multivariate power sine distribution. For f ( ρ ) , recall that ρ is deﬁned on the positive real line, with E ( ρ ) = 1. W e also need the ability to trade off tail weight (probability of draws of z being far away from the origin) against kurtosis (likelihood of draws of z being clustered around the origin). Beginning with the gen- eralized Laplace distribution (Kotz et al. 2001, sec. 4.4.2), we center , and then fold, at zer o, to get f ( ρ ) =  κ 1 κ σ Γ  1 + 1 κ   − 1 exp h −  ρ κ σ  κ i , ρ > 0 (9) 31 where E ( ρ ) = κ 1 κ σ Γ  2 κ  Γ  1 κ  (10) Setting E ( ρ ) = 1, σ = Γ  1 κ  Γ  2 κ  κ 1 κ (11) so the density of ρ , constrained so E ( ρ ) = 1, is f ( ρ | κ ) = κ Γ  2 κ  Γ  1 κ  2 exp " − ρ Γ  2 κ  Γ  1 κ  ! κ # (12) The parameter κ controls the tradeof f between tail weight and kurtosis. If κ = 1, f ( ρ | κ ) reduces to an exponential distribution, and if κ = 2, f ( ρ | κ ) is a half-normal distribution. As κ becomes large, the mode of ρ becomes less and less peaked, and f ( ρ | κ ) converges to a uniform ( 0, 2 ) distri- bution. The “correct” value for κ is inferred through the estimation process, letting the data drive the tradeoff between placing a mode on ρ and bounding the locations such that ρ ≤ 2. Note that this prior using the multivariate power sine distribution and our r estricted half-Laplace distribu- tion adds only one additional parameter to the model, compared to an independent multivariate normal hyperprior , which adds many more. It turns out that f ( ρ | κ ) is a special case of a power gamma distribution. T o see this, perform a change of variables so v = ρ κ , ρ = v 1 k and d ρ = 1 κ v 1 κ − 1 . Then, f ( v | κ ) = Γ  2 κ  Γ  1 κ  v 1 κ − 1 exp " − Γ  2 κ  Γ  1 κ  ! κ v # (13) which is a gamma distribution with shape parameter 1 κ and rate parameter  Γ ( 2 κ ) Γ ( 1 κ )  κ . This result makes it easy to simulate values of ρ ; just draw v from this gamma distribution, and transform ρ = v 1 κ . After simulating values of ϑ and ρ , it is often convenient to convert z i j back to its Cartesian coordinates. The elements of z can be expressed as: z 1 = ρ cos ϑ 1 z j = ρ cos ϑ j j − 1 ∏ l = 1 sin ϑ l , for 2 ≤ j < D z D = ρ D − 1 ∏ l = 1 sin ϑ l (14) 32 (Johnson 1987, ch 7). W e selected the other hyperpriors to balance weak information content against numerical sta- bility . Following Escobar and W est (1995), we place a weakly informative gamma prior on α , with mean of 4 and variance of 80. Since β and v ar e all population-level parameters that appear only in the deﬁnitions of φ i j , we can combine them all into a single parameter vector Ξ (log-transforming parameters when necessary), with a multivariate normal prior Ξ 0 , centered at the origin, with covariance matrix A = 10 I . Note that if κ = 2, then H 0 is a multivariate normal distribution. Since we were concerned about a mode of H 0 introducing too much prior information, we used a gamma prior with a mean of 3 and a variance of about 5. Experimenting with alternative values led to no substantive effect. C Estimation algorithm In this section we present the complete MCMC sampling algorithm for the general latent space model. The parameters to be estimated ar e α , κ , β , and z i , i = 1... N . Recall that since H is discr ete, at each iteration there ar e only k possible values that any z i can take. C.1 Simulate α | · Let r α and a α be the parameters of the gamma hyperprior on α . Using the algorithm proposed by Escobar and W est (1995), do the following. 1. Starting with the current value of α , draw a temporary variable η fr om a Beta ( α + 1, N ) distribution. 2. Draw τ from a Bernoulli trial with probability r α + k − 1 N ( a − log ( η )) + r α + k − 1 3. If τ = 0, draw α fr om a gamma ( r α + k − 1, a α − log ( η ) ) distribution. If τ = 1, draw α from a gamma ( r α + 1, a α − log ( η ) ) distribution. C.2 Simulate β , v |· T o simplify notation, we combine β and v into a single parameter vector Ξ . The conditional pos- terior distribution of Ξ depends on the data likelihood and the prior . The data likelihood we care 33 about here is the mar ginal likelihood in Section 1.2, after integrating over θ i , multiplied acr oss all dyads (using our assumption of conditional independence across dyads). Note that φ i j is a function of Ξ , z i and z j . The prior on Ξ is a multivariate normal with mean Ξ 0 and covariance A . Thus, the log conditional posterior (without normalizing constant) for Ξ is log f ( Ξ |· ) = N ∑ i = 1 N ∑ j = i + 1 log f ( y i j | Ξ , z i , z j ) − 1 2 ( Ξ − Ξ 0 ) 0 A − 1 ( Ξ − Ξ 0 ) (15) W e simulate Ξ using a random walk Metr opolis sampler (Rossi et al. 2005, ch. 3). C.3 Simulate κ | · W e place a gamma ( r κ , a κ ) prior on κ . Let ρ 1: k be the radii of the k distinct values of z . Combining the likelihood of the radii in 12 with the prior , the conditional posterior distribution for κ is f ( κ | · ) ∝ " κ Γ  2 κ  Γ  1 κ  2 # k exp " − k ∑ j = 1 ρ j Γ  2 κ  Γ  1 κ  ! κ − a κ κ # κ r κ − 1 (16) There are many dif ferent ways to simulate from this univariate density . W e chose to use sampling- importance resampling (SIR) (Smith and Gelfand 1992), but one might choose Metropolis, grid- based inverse CDF , or slice sampling methods instead. C.4 Simulate z | · This step, in which we draw each of the z i vectors from the mixtur e of Dirichlet processes (MDP), is an adaptation Algorithm 8 in Neal (2000). W e direct the reader ther e for an explanation of how and why the algorithm works, but we present a summary here, using our terminology and notation. The distribution of the z i ’s is discrete, so at each iteration of the estimation algorithm, there are only k possible values that z i can take. Let z ∗ =  z ∗ 1 . . . z ∗ k  deﬁne these k distinct latent coordinates, let z ∗ − i be the distinct mass points when not including person i , and let k − i be the number of distinct mass points in z ∗ − i when not including person i ( z ∗ and z ∗ − i , and k and k − i ,will differ only if i is a “singleton” who is the only person located at z i ). At the current state of the sampler , each person is “assigned” to one of the z ∗ j , in the sense that there is exactly one j for which z i = z ∗ j . Let N j be the number of people assigned to z ∗ j , and let N − i , j be the number of 34 people assigned to z ∗ j , when not counting person i . The algorithm involves choosing some number of proposal values (determined by a pr espec- iﬁed control parameter m ) for each z i . Deﬁne f i ( y i | z i , z − i ) as the likelihood contribution for all dyads that involve person i , given the current values of z i assigned to i , and of z − i assigned to everyone else. There are two cases that we need to consider . The ﬁrst is if there is some other person i 0 for which z i = z i 0 (i.e., i is not a singleton). In this case, draw m pr oposal draws fr om H 0 , call them z ∗ k + 1 , ..., z ∗ k + m , and let ˜ z =  z ∗ 1 , ..., z ∗ k , z ∗ k + 1 , ..., z ∗ k + m  . Intuitively , ˜ z is the union of the set of all latent vectors that are already assigned to someone in the population, with the set of m new proposal vectors. Next, compute f i  y i | ˜ z j , z − i  , for all j = 1 . . . ( k + m ) . These are the likelihood contributions for all dyads involving i , if z i were set to each of the values in ˜ z . These “proposal likelihoods” form a set of weights that we use to draw a new z i for each i . Thus, draw a new value for z i from ˜ z using the following probabilities: Pr  z i = ˜ z j  =  ω n − i , j N − 1 + α F i  y i | ˜ z j , z − i  for 1 ≤ j ≤ k ω ( α / m ) N − 1 + α F i  y i | ˜ z j , z − i  for k + 1 ≤ j ≤ k + m where ω is a normalizing constant (and does not need to be known for the purposes of random sampling). Thus, z i can take on the value of any of the existing elements of z ∗ , or one of the m new candidate values. Which value is selected depends on thr ee values: the likelihood of the data for each ˜ z j (values of ˜ z j that yield a high likelihood ar e more likely to be chosen), the number of other people who also are assigned to ˜ z j (coordinates where the prior distribution has more mass are more likely to be chosen), and the DP contr ol parameter α , which governs how close the DP prior on z is to H 0 . If i is a singleton, then there are only k − i = k − 1 elements in z ∗ . In this case, draw m + 1 candidate values from H 0 , re-index them so ˜ z =   z ∗ − i  , z ∗ k , ..., z ∗ k + m  , and select according to the probabilities Pr  z i = ˜ z j  =  ω N − i , j N − 1 + α F i  y i | ˜ z j , z − i  for 1 ≤ j ≤ k − 1 ω ( α / m ) N − 1 + α F i  y i | ˜ z j , z − i  for k ≤ j ≤ k + m References George A. Akerlof. Social distance and social decisions. Econometrica , 65(5):1005–1027, Sept. 1997. 35 Asim Ansari and Carl F . Mela. E-customization. Journal of Marketing Research , 40:131–145, May 2003. Asim Ansari, Oded Koenigsberg, and Florian Stahl. Modeling multiple relationships in social networks. working paper , Dec. 2008. C.E. Antoniak. Mixtures of Dirichlet pr ocesses with applications to nonparametric problems. An- nals of Statistics , 2(4):1152–1174, 1974. Johan Arndt. Role of product-related conversations in the dif fusion of a new product. Journal of Marketing Research , 4(3):291–295, August 1967. Sudipto Banerjee, Bradley P . Carlin, and Alan E. Gelfand. Hierarchical Modeling and Analysis for Spatial Data . CRC Press, 2004. Frank M. Bass. A new pr oduct gr owth model for consumer durables. Management Science , 15(5): 215–227, Jan. 1969. W illiam O. Bearden and Michael J. Etzel. Reference group inﬂuence on product and brand pur- chase decisions. Journal of Consumer Research , 9, Sept. 1982. David R. Bell and Sangyoung Song. Neighborhood effects and trial on the Internet: Evidence fr om online grocery r etailing. Quantitative Marketing and Economics , 5:361–400, 2007. David Blackwell and James B. MacQueen. Fer guson distributions via polya urn schemes. The Annals of Statistics , 1(2):353–355, Mar . 1973. Peter M. Blau. Inequality and Heterogeneity: A Primitive Theory of Social Structure . Free Pr ess, New Y ork, 1977. Eric T . Bradlow and David C. Schmittlein. The little engines that could: Modeling the performance of world wide web search engines. Marketing Science , 19(1):43–62, W inter 2000. Michael Braun, Peter S. Fader , Eric T . Bradlow , and Howard Kunreuther . Modeling the “Pseudo- deductible” in homeowners’ insurance. Management Science , 52(8):1258–1272, August 2006. Jacqueline Johnson Brown and Peter H. Reingen. Social ties and wor d-of-mouth referral behavior . Journal of Consumer Research , 14(3):350–362, Dec. 1987. Jeonghye Choi, Sam K. Hui, and David R. Bell. Spatiotemporal analysis of imitation behavior across new buyers at an online grocery retailer . Journal of Marketing Resear ch , 47(1):75–89, Feb. 2010. Koustuv Dasgupta, Rahul Singh, Balaji V iswanathan, Dipanjan Chakraborty , Sougata Mukherjea, Amit A. Nanavati, and Anupam Joshi. Social ties and their relevance to churn in mobile telecom networks. In Proceedings of the 11th International Conference on Extending Database T echnology: Advances in Database T echnology , volume 261 of ACM International Conference Proceeding Series , pages 668–677, New Y ork, 2008. ACM. Nathan Eagle, Alex Pentland, and David Lazer . Inferring friendship network structure by using mobile phone data. Proceedings of the National Academy of Sciences , 106(36):15274–15278, Septem- ber 2009. 36 Michael D. Escobar . Estimating normal means with a dirichlet process prior . Journal of the American Statistical Association , 89(425):268–277, March 1994. Michael D. Escobar and Michael W est. Computing nonparametric hierar chical models. In D. Day , P . M ¨ uller , and D. Sinha, editors, Practical Nonparametric and Semiparametric Bayesian Statistics , chapter 1, pages 1–22. Springer-V erlag, 1998. Michael D. Escobar and Mike W est. Bayesian density estimation and inference using mixtures. Journal of the American Statistical Association , 90(430):577–588, 1995. Peter S. Fader , Bruce G. S. Hardie, and Robert Zeithammer . Forecasting new product trial in a controlled test market envir onment. Journal of For ecasting , 22:391–410, 2003. Lawrence F . Feick and Linda L. Price. The market maven: A diffuser of marketplace information. Journal of Marketing , 51(1):83–97, Jan. 1987. Thomas S. Ferguson. A Bayesian analysis of some nonparametric pr oblems. The Annals of Statistics , 1(2):209–230, 1973. Jeffr ey D. Ford and Elwood A. Ellis. A reexamination of group inﬂuence on member brand pref- erence. Journal of Marketing Research , 17(1):125–132, Feb. 1980. Hubert Gatignon and Thomas S. Robertson. A pr opositional inventory for new diffusion research. Journal of Consumer Research , 11(4):849–867, Mar . 1985. Andrew Gelman. Comment on Handcock, et. al., ”Model-based clustering for social networks”. Journal of the Royal Statistical Society , Series B , 170(2):337, 2007. Andrew Gelman, Xiao-Li Meng, and Hal Stern. Posterior predictive assessment of model ﬁtness via realized discr epancies. Statistica Sinica , 6(4):733–807, 1996. Lisa Getoor and Christopher P . Diehl. Link mining: A survey . ACM SIGKDD Explorations Newslet- ter , 7(2):3–12, 2005. David Godes and Dina Mayzlin. Firm-cr eated word-of-mouth communication: Evidence fr om a ﬁeld test. Marketing Science , 28(4):721–739, Jul.-Aug. 2009. Jacob Goldenberg, Barak Libai, and Eitan Muller . T alk of the network: A complex systems look at the underlying process of wor d-of-mouth. Marketing Letters , 12(3):211–223, 2001. Mark S. Handcock, Adrian E. Raftery , and Jer emy M. T antrum. Model-based clustering for social networks. Journal of the Royal Statistical Society . Series A , 170(2):301–354, 2007. Shawndra Hill, Foster Provost, and Chris V olinsky . Network-based marketing: Identifying likely adopters via consumer networks. Statistical Science , 21(2):256–276, 2006. Peter D. Hoff. Bilinear mixed-effects models for dyadic data. Journal of the American Statistical Association , 100(469):286–295, March 2005. Peter D. Hoff, Adrian E. Raftery , and Mark S. Handcock. Latent space approaches to social net- work analysis. Journal of the American Statistical Association , 97(460):1090–1098, Dec. 2002. David R. Hunter , Steven M. Goodreau, and Mark S. Handcock. Goodness of ﬁt of social network models. Journal of the American Statistical Association , 103(481):248–258, March 2008. 37 Hemant Ishwaran and Lancelot F . James. Gibbs sampling methods for stick-br eaking priors. Jour- nal of the American Statistical Association , 96(453):161–173, Mar . 2001. Raghuram Iyengar , Christophe V an den Bulte, and Thomas W . V alente. Opinion leadership and social contagion in new product diffusion. T echnical Report 08-120, Marketing Science Institute, 2008. working paper , University of Pennsylvania. Glen Jeh and Jennifer W idom. Simrank: A measure of structural-context similarity . In Proceedings of the ACM SIGKDD International Confer ence on Knowledge Discovery and Data Mining , pages 271– 279, New Y ork, 2003. ACM Press. Mark E. Johnson. Multivariate Statistical Simulation . W iley Series in Probability and Mathematical Statistics. W iley , 1987. Elihu Katz and Paul F . Lazarsfeld. Personal Inﬂuence . Fr ee Press, New Y ork, 1955. Leo Katz. A new status index derived fr om sociometric analysis. Psychometrika , 18(1):39–43, March 1953. Jin Gyo Kim, Ulrich Menzefricke, and Fred M. Feinberg. Assessing heterogeneity in discrete choice models using a dirichlet process prior . Review of Marketing Science , 2(1):1–39, 2004. Gueorgi Kossinets and Duncan J. W atts. Empirical analysis of an evolving social network. Science , 311:88–90, Jan. 2006. Samuel Kotz, T omasz J. Kozubowski, and Krzysztof Podgorski. The Laplace Distribution and Gen- eralizations: A Revisit with Applications to Communications, Economics, Engineering and Finance . Birkhauser , Boston, 2001. Jan Kratzer and Christopher Lettl. Distinctive roles of lead users and opinion leaders in the social networks of schoolchildren. Journal of Consumer Research , 36(4):646–659, Dec. 2009. Paul Lazarsfeld and Robert K. Merton. Friendship as a social pr ocess: a substantitive and method- ological analysis. In M. Berger , editor , Freedom and Control in Modern Society , chapter 2, pages 18–66. V an Nostrand, New Y ork, 1954. Peter Lenk. Simulation pseudo-bias correction to the harmonic mean estimator of integrated like- lihoods. Journal of Computational and Graphical Statistics , 18(4):941–960, 2009. David Liben-Nowell and Jon Kleinberg. The link-prediction pr oblem for social networks. Journal of the American Society for Information Science and T echnology , 58(7):1019–1031, 2007. Miller McPherson, L ynn Smith-Lovin, and James M. Cook. Birds of a feather: Homophily in social networks. Annual Review of Sociology , 27:415–444, 2001. Donald G. Morrison and David C. Schmittlein. Predicting future random events based on past performance. Management Science , 27(9):1006–1023, 1981. Donald G. Morrison and David C. Schmittlein. Generalizing the NBD Model for Customer Pur- chases: What ar e the Implications and is it Worth the Effort? Journal of Business and Economic Statistics , 6(2):145–159, 1988. 38 Christopher J. Nachtsheim and Mark E. Johnson. A new family of multivariate distributions with applications to Monte Carlo studies. Journal of the American Statistical Association , 83(404):984– 989, Dec. 1988. Sungjoon Nam, Puneet Manchanda, and Pradeep K. Chintagunta. The effects of service quality and word of mouth on customer acquisition, r etention and usage. working paper , March 2007. Radford M. Neal. Markov chain sampling methods for Dirichlet process mixtur e models. Journal of Computational and Graphical Statistics , 9(2):249–265, June 2000. Thomas P . Novak, Jan de Leeuw , and Bruce MacEvoy . Richness curves for evaluating market segmentation. Journal of Marketing Research , 29(2):254–267, May 1992. Anthony O’Hagan and Jonathan Forster . Kendall’ s Advanced Theory of Statistics , volume 2B: Bayesian Inference. Arnold, London, 2nd edition, 2004. C. Whan Park and V . Parker Lessig. Students and housewives: Differences in susceptibility to refer ence group infuence. Journal of Consumer Research , 4(2):102–110, Sept. 1977. Peter H. Reingen and Jerome B. Kernan. Referral networks in marketing: Methods and illustration. Journal of Marketing Research , 23(4):370–378, Nov . 1986. Peter H. Reingen, Brian L. Foster , Jacqueline Johnson Br own, and Stephen B. Seidman. Brand congruence in interpersonal relations: A social network analysis. Journal of Consumer Research , 11(3):771–783, Dec. 1984. Peter E. Rossi, Greg M. Allenby , and Robert McCulloch. Bayesian Statistics and Marketing . W iley Series in Probability and Statistics. John W iley and Sons, Chichester , UK, 2005. Donald B. Rubin. Bayesianly justiﬁable and relevant fr equency calculations for the applied statis- tician. Annals of Statistics , 12(4):1151–1172, 1984. J. Sethuraman. A constructive deﬁnition of dirichlet priors. Statistica Sinica , 4(2):639–650, 1994. A.F .M. Smith and A.E. Gelfand. Bayesian statistics without tears: A sampling-resampling per- spective. The American Statistician , 46(2):84–88, 1992. Riitta T oivonen, Lauri Kovanen, Mikko Kivel ¨ a, Jukka-Pekka Onnela, Jari Saram ¨ aki, and Kimmo Kaski. A comparative study of social network models: Network evolution models and nodal attribute models. Social Networks , 31:240–254, 2009. Marshall van Alstyne and Erik Brynjolfsson. Global village or cyber -balkans? modeling and measuring the integration of electronic communities. Management Science , 51(6):851–868, June 2005. Duncan J. W atts. Networks, dynamics and the small-world phenomenon. American Journal of Sociology , 105(2):493–527, Sept. 1999. Duncan J. W atts and Peter Sheridan Dodds. Inﬂuentials, networks and public opinion formation. Journal of Consumer Research , 34:441–458, December 2007. Duncan J. W atts and Steven H. Strogatz. Collective dynamics of ”small-world” networks. Natur e , 393:440–42, June 1998. 39 Michel W edel and Jie Zhang. Analyzing brand competition across subcategories. Journal of Mar- keting Research , 41(4):448–456, Nov . 2004. Robert E. W itt and Grady D. Bruce. Group inﬂuence and brand choice congruence. Journal of Marketing Research , 9(4):440–443, Nov . 1972. Sha Y ang and Greg M. Allenby . Modeling inter dependent consumer prefer ences. Journal of Mar- keting Research , 40(3):282–294, 2003. 40

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