Scalable Inference of Customer Similarities from Interactions Data using Dirichlet Processes
Abstract
Under the sociological theory of homophily, people who are similar to one another are more likely to interact with one another. Marketers often have access to data on interactions among customers from which, with homophily as a guiding principle, inferences 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 probability 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 present a novel Bayesian nonparametric approach, using Dirichlet processes, to moderate the scalability problems that marketing researchers encounter when working with networked data. We find that this framework is a powerful way to draw insights into latent similarities of customers, and we discuss how marketers can apply these insights to segmentation and targeting activities.
Introduction
Marketers have long been interested in the notion that interactions among customers will affect behavior. For example, knowledge of how customers relate to one another improves our understanding on how preferences are formed (Reingen et al. 1984), how preferences are correlated within groups (Witt and Bruce 1972; Park and Lessig 1977; Ford and Ellis 1980; Bearden and Etzel 1982), or how useful referrals are for marketers when developing new markets (Reingen and Kernan 1986). Connections among customers are opportunities for preference influence (e.g. contagion 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 are more likely to interact with one another, so given the need for marketers to find efficient ways to attract and cultivate customers, there exists vast opportunity in leveraging interactions data to infer similarity and connect this to marketing behavior (e.g., Yang 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 customers (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 more likely to interact with one another, and share information and influence, than customers who are not. There is a substantial 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. We consider interactions to be the “data” that records some observable action between two individuals, while similarities form a latent, unobserved construct (though possibly correlated with other observed measurements) that determines which individuals are more likely to interact with others. In this paper, we present an illustrative yet parsimonious model, grounded in the theory of homophily, that allows marketers to infer latent similarities from 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 better understand why customers interact with whom they do, or why they behave the way they do, in terms that are useful to marketers.
We build on a class of probability 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 multidimensional space. When estimated on relationship data (e.g. a list of selfreported 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 relationships.^{1}^{1}1Latent space methods are, of course, not limited to examining social network data, and could be used to model similarities between units in two distinct groups (Bradlow and Schmittlein 2000), or to model the difference in knowledge by individuals (van Alstyne and Brynjolfsson 2005). Further applications are discussed in Toivonen et al. (2009). What is becoming 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 are and what they talk about), or the nature of the relationship 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 theoretical 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 revenue 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 “word of mouth.” Knowing how similar customers are to one another is of direct relevance to marketing practitioners 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 distributions of latent distances), we can segment and target customers accordingly. Ordinarily, this segmentation is done based on observed characteristics of individuals. Very little attention has been paid to how marketers might be able exploit the information contained in interactions data for traditional, nonnetworked marketing tactics, like deciding in which publications (online or otherwise) to advertise. Indeed, the company that uses interactions data for segmentation and targeting (e.g. an online retailer) 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 individuals 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 probability models for larger observational network datasets. A dataset with individuals involves dyads (the binomial coefficient is defined as ). For the exemplar dataset that we use in this paper, there are 11,426,590 sets of dyadspecific 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 restrictive, we need to compute all of these dyadspecific likelihoods, and the same number of dyadspecific parameters, at each iteration of our estimation algorithm. The problem with scale makes probability models of social interactions computationally intractable for all but the smallest datasets.
The modeling challenge is therefore to reveal similarities in heterogeneous characteristics from customers’ interaction data, in a scalable and interpretable way. We 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 are clustered on common locations on the latent space. So if this discrete distribution has mass points, there are only distinct distances on the latent space (the comes from the zero distance between two individuals at the same latent coordinate). Since must be smaller than , there are substantially fewer distinct likelihoods to compute and parameters to estimate.
In this research, we show how marketers could use latent space models to segment customers based on posterior inferences of latent similarities, using this more efficient Bayesian nonparametric approach. An output of our algorithm is a posterior estimate of the latent space that is inferred 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 circles. Also, we recognize that while interactions typically occur among similar customers, there is also the possibility that dissimilar customers (who may have different purchase patterns and preferences) may interact at some time. To demonstrate the power and utility of this approach to modeling interactions data, we apply it to a dataset of observed interactions from a cellular communication network. We propose a probability specification for this particular dataset, in which the incidence and rates of interactions are functions of distances in latent space. We validate the approach in two ways: by showing that adding the latent space structure 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 are 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 heterogeneous interdependencies among individuals leads to a model that simply does not represent the observed patterns in interactions.
We then assess the computational improvements and scalability issues surrounding our Bayesian nonparametric approach, 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 are available to us do not let us offer hard evidence of a correlation between similarities and purchase preferences but, given the findings in the marketing literature that show the importance of similarities and interactions in customer behavior, it is reasonable to expect that marketing mix efforts benefit from being able to distinguish interactions among similar customers from interactions among dissimilar customers. The computational improvements from using a DP prior for the latent space make these inferences attainable for the datasets that marketers typically encounter.
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 represent a wide variety of behavior. Examples include binary indicators of relationships, counts of transactions (among customers), times between interactions, or combinations thereof. However simple or complicated the data are, they should be treated as some output of a stochastic process that is governed by dyadspecific parameters (and possibly some additional populationlevel parameters). Data that is generated from a network of customers differs from individuallygenerated data, such as household purchase data, in that we can no longer assume that the datagenerating processes 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 dyadlevel 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.
We determine dyadlevel parameters so that similar individuals will have higher incidence of interaction than dissimilar individuals. The characteristics upon which this similarity is based are likely unobservable by the researcher. Therefore, we represent unobserved, exogenous characteristics of the individual (and thus, the individual himself), as a dimensional vector on some latent space (Hoff et al. 2002; Handcock et al. 2007; Bradlow and Schmittlein 2000; van Alstyne and Brynjolfsson 2005). Similarity between two individuals is measured by the distance between their latent coordinates across this latent space, and we can express the rates or probabilities of interaction between two people as a decreasing function of the latent distance between them. Note that these distances and locations do not directly represent physical or geographic locations in any way (although they may, of course, be incidentally correlated with them). Instead, they are individuallevel 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 nonstationary observed phenomenon that Kossinets and Watts (2006) describe as evolving), the underlying rates and probabilities of these incidences remain the same. Thus, our stationary model can still capture nonstationary behavior in the observed data. Also, we want to emphasize that a latent space model is an abstraction of reality, and we caution researchers not to place too much concrete meaning 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 be a vector of observed data that is attributable to the dyad of person and person , and let be the likelihood of observing , given the dyadspecific parameter vector . Next, let be heterogeneous across dyads, with each drawn randomly from a dyadspecific prior distribution . A model in which is common across all dyads, or itself distributed independently (drawn from its own mixing distribution), would imply crossdyad independence of , which may not make sense in a network setting. To incorporate some networkbased dependence in the distribution of , we instill a pattern of heterogeneity of that allows for a useful, intuitive interpretation of the similarities. Thus, there are two sources of heterogeneity that generate : independent dyadlevel variation from , and networkinduced interdependence in the distribution of .
Before explaining how we model heterogeneity in , 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 be a dimensional vector that is associated with person , and let be the collection of all of these vectors. Since each is unobserved, we call it a “latent coordinate,” and the dimensional space on which it lies a “latent space,” as in Hoff et al. (2002) and Handcock et al. (2007). Even if the vectors in are distributed independently on the latent space, the distances between every pair of (the “latent distances”) are not. By expressing as a monotonic function of the distance between and , we induce dependency among all the and, in turn, all the . As an example, suppose that represents a rate of contact between and , and the distribution of depends positively on (for example, the mean of increases with ). We determine by evaluating a monotonically decreasing function of the latent distance, so as and are less similar (the distance between and goes up), the rate of interaction between and goes down. But we never need to estimate or directly. We need only to estimate the locations of for all people to get the values of for all dyads.
1.3 Mixtures of Dirichlet processes: what they are, and how to use them to model the latent space
Even if we model as a function of the latent distance among the ’s, we still have the issue that there are many ’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 values of , and corresponding data likelihoods. When is small, scalability becomes less of a problem, and one could use the original parametric formulation of the latent space model. But as becomes even moderately large, estimating the latent coordinates becomes computationally infeasible. We reduce the number of distinct values of by using a discrete distribution, for the distribution of on the latent space. If this discrete distribution has mass points, then there are only distinct latent distances. For a given network size, a larger difference between and leads to a greater computational savings by having fewer distinct values of to consider. To avoid having to estimate each directly, we choose and such that we can integrate over analytically, and express the marginal distribution in closed form. However, we do not want to prespecify the functional form of , because we don’t know for certain what it is, nor do we want to prespecify , since we do not know what the “correct” number of mass points for is.
Our approach is to use a mixture of Dirichlet processes as a Bayesian nonparametric prior distribution 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 filtering), Kim et al. (2004) (identifying clusters of customers in discrete choice models), Wedel and Zhang (2004) (analyzing brand competition across subcategories) and Braun et al. (2006) (estimating 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 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 realization of a DP is a discrete distribution, with its support having a finite number of mass points ( in the previous paragraph), so the DP can be thought of as a prior distribution on discrete distributions.^{2}^{2}2The formal definition of what makes a stochastic distribution a DP is a more technical issue. Essentially, the probabilities of certain events occurring must follow a Dirichlet distribution with parameters that depend on and ). There is an accessible and readable explanation in O’Hagan and Forster (2004, ch. 13).
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 representation, with straightforward sampling properties, and does not require a prespecification of the number of mass points. In our latent space framework, we let be a realization from , and then have each be a draw from . The first parameter, , is itself a probability distribution, and is the “mean” of the distributions that the DP generates. A scalar controls the variance of the realizations of the DP around . This variance is low when is high, so for high , realizations from will look a lot like the distribution function of . This concentration of the DP towards results from a DP that generates a discrete distribution with a lot of mass points (a high ). When is low, realizations from look much less like (high variance), because this DP generates discrete distributions with fewer mass points. So plays an important role in determining just how discrete (i.e. value of ), or clustered, a DPgenerated distribution really is. Reasonable choices for are those distributions for that one might use in a purely parametric model (note that 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 or set it directly.
Given and we need to know how to simulate from and then each from . Since is nonparametric, even though we know it was generated by the , the posterior distribution of any new depends on all the other . Consequently, there is no obvious way to draw a from directly. The “trick” is to integrate out analytically, and treat as if it were drawn from this marginal distribution, a mixture of Dirichlet processes (MDP, see Antoniak 1974). The probability of any one , given the empirical distribution of all the other , is
(1) 
(Blackwell and MacQueen 1973; Escobar 1994). So in the estimation algorithm, determines how likely it is that any new draw of 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 , as a new value.
To illustrate how this works, Figure 1 shows simulations from an MDP when is a univariate standard normal distribution, for different values of . In the figure, the heavy black line is the standard normal cdf, and each colored line is a single realization from the MDP. We see that when is low, there are fewer mass points in each realization, and when is high, the higher number of mass points allows the realizations to approximate the normal cdf. In Figure (b)b, for each we present histograms from draws of a single realization of the MDP (so these are draws from a distribution that the MDP generated). Again, we see fewer distinct clusters (low ) when is low, more clusters when is high. In our network model, we are dealing with more dimensions and a richer specification of , but the basic idea remains the same.
How we select and , and the priors we place on them, is described in more detail in Appendix B. Selection of an appropriate distribution for requires that we introduce some identifying restrictions on the location vectors (). 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 . To handle this problem, we constrain the prior distribution of so the mean distance of any from the origin is one. However, we need to do this without introducing too much “incorrect” prior information. For example, a simple choice for could be a standard multivariate normal distribution; setting the mean at the origin and the variance at one addresses the translation and scale identification issues. The problem with defining as a multivariate normal is that it implies that our prior on the distribution of 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 posterior. An alternative specification for could be a bounded uniform distribution (so the mean distance from the origin remains one), but that would constrain all to be inside a hypersphere, effectively placing an upper bound on the latent distance between customers. This, too, seems like an unreasonable expression of prior information. Our solution involves using spherical coordinates for , consisting of two components: a radius representing the distance from the origin, and the location on the surface of a hypersphere that has that radius. We show in Appendix B that can be factored into priors for these two components from which it is straightforward to draw samples.
This prior on , combined with the data likelihood, leads to conditional posterior distributions that are easily incorporated into Gibbs samplers. Escobar (1994) and Escobar and West (1998) describe some of the theory and derivations behind this, while Neal (2000) details stepbystep instructions on how to add MDPs to Gibbs samplers for both conjugate and nonconjugate models.^{3}^{3}3This way of expressing the MDP (and the approach we took in our estimation is known as the “Polya urn” representation. There is another, equally useful approach known as the “stickbreaking” representation (Sethuraman 1994) that one can also use to build conditional posterior distributions for Gibbs samplers Ishwaran and James (2001). Thus, MDPs allow marketing modelers to relax many of their distributional assumptions by adding only one additional step to the parametric Gibbs sampling algorithm. We give details of our estimation algorithm in Appendix C. Our exploitation of the discreteness property of Dirichlet processes also lets us reduce the computational burden substantially, as we demonstrate in Section 3.
2 Example: telephone calls
We now turn to a specific application of our model, using a dataset provided by Chongqing Mobile, a subsidiary of China Mobile, the largest cellular phone operator in China. Cellular phone networks have been reported to be highly representative of selfreported friendships (Eagle et al. 2009), making such data ideal for studies of network based interdependencies among customers. The data consist of contact record information (for phone calls and SMS messages) for a panel of residents of Chongqing who are members of the “silver tier”, “gold tier” or “diamond tier” of the company’s preferred customer program. Each record contains the identifiers for both parties in the contact, and the date of when the contact takes place. For the purposes of this example, we ignore contacts with people outside this person network.^{4}^{4}4Our 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. Therefore, 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 predictions to a greater population, ignoring outofnetwork calls could influence specific parameter estimates. There are an additional 209 silver, gold or diamond customers in the panel for whom there were no observed calls to other silver, gold or diamond customers during the observation period. 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). We divide the observation period into a sixmonth calibration period and a sixmonth holdout period. Descriptive statistics for this dataset are summarized in Table 1. Of the nonempty dyads in the dataset, only appear in both the calibration and holdout samples. dyads are nonempty in calibration, but empty in holdout, and are empty in calibration, but nonempty in holdout.
Calibration  Holdout  Full  

Weeks  26  26  52 
Customers  4,781  4,781  4,781 
Nonempty 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 nonempty dyad  7.5 (18.7)  7.6 (18.7)  15.1 (36.0) 
Mean (s.d.) shared friends in nonempty dyad  3.0 (2.7)  3.3 (2.8)  5.7 (2.8) 
One way to describe the structure of the observed network is to compare it to the “small world” networks described in Watts and Strogatz (1998) and Watts (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). We can assess the extent to which a network is “small world” comparing the mean geodesic distances and clustering coefficients to those that we would expect to see from a network in which connections are determined at random for the same number of people (4,781) and average number of “friends” per person (7.6). Using the asymptotic approximations in Watts (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 (illustrated in Figure 4) will bear this out. We also note that our network would not qualify as a “scale free” network, in that the degree distribution clearly does not follow a power lawtype distribution (we show the observed degree distribution in Figure 2).
2.1 Model specifics
Using the notation introduced in Section 1, 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 are no observed contacts in the dyad, is the length of the observation period, and we call that dyad “empty.” If there are observed calls, the dyad is “nonempty.” The definition of follows the logic of the “exponential nevertriers” model in (Fader et al. 2003), which in turn draws from the “hardcore neverbuyers” model in Morrison and Schmittlein (1981) and Morrison and Schmittlein (1988). First, there is a probability that a dyad will remain forever empty, no matter how long we wait. We call dyads like this “closed”. Next, for dyads that are “open,” (with probability ), intercontact times follow an exponential distribution with rate . To link these specifics with our general model, . Note that there are 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 appropriate for this dataset is ultimately an empirical question, but we choose it for four reasons. First, we do not need to make special provisions for leftcensoring because of the memorylessness property. Second, the number of contacts is a sufficient statistic for the individual elements in . We were able to exploit these two features of the exponential distribution to gain computational savings without compromising the fundamental purpose of the research. Third, we did run the model on a much smaller dataset where is governed by a “Weibull nevertriers” model, in order to allow for duration dependence, and found that since the shape parameter of the Weibull was close to , it reduced to the exponential distribution anyway. Finally, we chose the exponential distribution because it forms a conjugate pair with our choice of , a gamma distribution for with dyadspecific mean and common variance , and a degenerate distribution over , so that at this level of the hierarchy, is homogeneous for all dyads (we will add heterogeneity to later through the latent space). The exponentialgamma pair lets us integrate over analytically, further easing computational effort. The vector therefore contains three elements, ( is contained in both and .)
To evaluate whether latent space is worth adding to a model of interactions data, we estimated the model with three different definitions of the elements of . For a “Baseline” model, we let , a common value for all dyads (note that we still maintain dyadlevel heterogeneity in , but it does not appear explicitly in the data likelihood). For a second model, HMCR (for “Homogeneous Mean Contact Rate”), and remain homogeneous across dyads, but is now determined by the distribution on the latent space. Specifically, we define
(2) 
where the ’s are coefficients to be estimated, and is the latent distance between and . For a third model, named “Full,” retains the same definition as in Equation 2, except that is now heterogeneous across dyads, defined as
(3) 
Equations (2) and (3) allow the respective relationships to latent distance to be concave, linear or convex. The parameters are constrained to be nonnegative, because as latent distance increases, the probability of contact, and the rate of contact, should decrease. We selected Euclidean distance as our distance measure, after experimenting with others that did not perform as well (van Alstyne and Brynjolfsson 2005).^{5}^{5}5Here, 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. Another candidate for this distance metric 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 nonparametric nature of the estimated latent space means the dimensions are already differentially scaled. Also, the Euclidean distance is computationally more efficient. As with the parametric specification, the functions in Equations 2 and 3, and the distance measure, 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 interdependence is an important characteristic of a model of customer interactions. However, one could falsify this claim by showing that models in which dyadlevel parameters are independent fit no worse than models that incorporate a latent space. We ran our algorithm with latent spaces of different dimensionality and, based on estimates of log marginal likelihoods, we decided that the parsimonious choice of 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 represents the datagenerating 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 goodnessoffit for social networking data: the degree distribution, the dyadwise shared partner distribution, and the distribution of geodesic distances. We also examine the histogram of the number of calls made within nonempty dyads, the density of the network, and the clustering coefficient for the network. All of our PPCs in this article are with respect to the 26week holdout sample. Figure 2 shows the results for the distributional PPCs, and Figure 3 shows the PPCs for the density and clustering coefficients. The xaxis in each panel is the count of individuals or dyads, and the yaxis is the log proportion of those individuals with each count. The dark dots represent the log probabilities generated from the actual dataset, and the boxandwhisker plot represents the distribution of log probabilities across 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 nonempty provides a lot of information about the likelihood of future emptiness, because nonempty dyads must be open. However, closer examination reveals 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 structure fit better on these test statistics. However, we do not see much difference 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 performs at the dyad level. Our approach here is to predict which of the dyads that are 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. We 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. To assess the predictive ability of a model, we first identify, individual by individual, the top lift (or the top lift, where ) most likely, previously uncontacted, individuals observed to contact during the holdout period^{6}^{6}6For any individual , the top q lift requires rank ordering all potential customers based on the predicted probability of interaction. For a holdout sample, the top lift of this rank ordered list is equal to the proportion of the top customers for whom we observe interactions with , divided by the proportion of total interactions made by customer .. For a completely random or naive model, the percentage of the top most likely empty dyads to become nonempty should be equal to . For any other model, if the value for the top metric is greater than , then the model provides some ”betterthanchance” predictive value. The use of the lift metric ensures that the maximum of this value is always equal to one.
Table 2 presents these lift metrics for different models and values of , and different calibration/holdout samples. Results are presented for all three model variants, with for the latent space models. In addition, we present results for a “Condition on Observed” prediction rule, under which empty dyads are to remain empty in holdout, and nonempty dyads remain nonempty in holdout. The “Geodesic Distance” model ranks dyads according to their geodesic distances, as in Kossinets and Watts (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 Baseline 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, 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 dyadlevel prediction dramatically. We do not, however, see any substantive differences between the HMCR and Full models, suggesting that, in this application, all of the action is on the open/closed probability and not on the contact rates. Nevertheless, our results indicate that the use of the latent space structure for networked data is a better model than assuming independence across dyads.^{7}^{7}7There are of course many different 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 Widom 2003). Getoor and Diehl (2005) provides a detailed review of link mining methods, and LibenNowell and Kleinberg (2007) compare the performance of some of them. We 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 ), the other models “caught up” when was increased. However, our model offers behavioral intuition (see Section 4) that machine learning algorithms cannot provide, and we are willing trade off some predictive power for managerial interpretability. Nevertheless, our objective in predicting future link formation is only to demonstrate the value of accounting for latent network structure when modeling interactions data.
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 
3 Scalability and computation
Having demonstrated the contribution of latent space models, we now turn to the issue of scalability and computation. The amount of computational improvement one can expect from 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 different 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 discrete representation of the latent space will have little effect. However, the density of many (if not most) social networks tends to be very low. Even if nonempty 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 observed data is discrete, then even more aggregation is possible. Of course, aggregation according to observed data is standard practice when a model is homogeneous, or marginal likelihoods are 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 pair, and the same value of , 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 pair, we can compute the log likelihood for that pair once for each , and multiply by the number of dyads with that pair and that . Among all the data zeros, there are only possible likelihood values. If is less than , there is a computational saving, even if all of the nonzero values of are different (as happens when is continuous). If is discrete (so is the number of distinct values of ), there are at most possible likelihoods. For a continuous , but with a large number of zeros, the number of possible likelihoods is , plus the number of nonzero ’s. Clearly, the more distinct observed data patterns there are, the less one can take advantage of the discretization of the latent space that is generated by the DP. But in the social networking applications that are common in marketing, networks are often very sparse, so we have at least one very large group of dyads with the same data.
To assess just how much computational savings there is, consider the Full model in the telephone call example. The calibration dataset has nonempty dyads; likelihoods for each of these dyads must be computed individually. The mean of is , so there are distinct distances between mass points on the latent space. Instead of computing separate likelihoods for each of the empty dyads, we only need to compute of them. Thus, the number of likelihoods to compute at each MCMC iteration is . This represents a reduction in computational requirements.
The extent to which our method can scale for datasets with many more individuals (large ) depends on how both the network density and change as increases. Ultimately, these are both empirical questions, the second of which we cannot know up front because not only is unobserved, but it can be influenced by the choice of and . However, the expected number of mass points can be asymptotically approximated as (Antoniak 1974; Escobar 1994). Thus, if is small, the expected number of mass points is also expected to be small, but it will grow for larger datasets. If is large, the number of mass points for smaller datasets might be larger, but this number will not grow as quickly for larger datasets. To test how well this approximation works in practice, we estimated the full model using successively larger subsets of our original network. We then fixed at three different values: 0.5, 20 and 300 (instead of placing a weakly informative prior on , as we did in the main analysis). We also computed the total number of likelihood computations for each sweep of the Gibbs sampler, which is just , plus the number of nonempty dyads in the dataset.
Figure 4 plots the posterior mean of (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 the expected pattern. For small , the number of mass points is small, but the incremental number of mass points grows with network size. For large , the number of mass points is large, 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 , even though the number of total dyads continues to grow quadratically. In terms of total computation, for low , the number of computations increases more rapidly with than for higher values of , but when is high, the relationship becomes more linear. Even though the number of dyads grows quadratically with , larger networks will tend to have larger number of nonempty dyads. For low , 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 aggregate the observed data, and not from an inability to aggregate the latent parameters. Of course, this is no different from scalability problems faced by Bayesian hierarchical modelers who use MCMC to update model parameter estimates from nonnetworked data.
4 Interpretation and usefulness of the latent space
In Figure 5(a), we plot a single draw from the joint posterior distribution of the latent coordinates from the Full model with (we chose the draw with the largest 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 realization from a mixture of Dirichlet processes, and we counted 593 such clusters in this realization). But since multiple individuals are located at the same coordinates, 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 reference 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 “superclusters” (clusters of clusters, or clusters closer to other clusters), of individuals on the latent space. Also, there 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 coordinate (0.9,2.2). Note that the latent space is a random variable, so this figure represents just one possible 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? We examine this in the context of segmentation and targeting.
Segmentation and targeting using interactions data
Segmentation and targeting is central to the development of effective 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 similar ways, and therefore 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 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 customers who have already adopted a product or service can help identify other potential customers. Their results show improvements in response rates, compared with methods using observed, traditional segmentation and targeting bases. While these networkbased methods are powerful tools for eliciting new customers, our graphical representation of the latent space highlights that there are sometimes interactions among customers who are quite different from one another. We 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 respondent to a promotion). The lines radiating from these individuals represent observed links in the dataset. We 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 are many marketing tactics that have nothing to do with following links or word of mouth, and instead depend more on understanding which customers can be grouped into more homogeneous segments. Given the interpretation of the latent space as representing similarity, anyone in close proximity (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, targeting 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 there is a closer personal relationship, but nothing else in common in terms of purchase 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 relative, and the relative is closer to her own friends and others in her social circle. This is useful for marketers to be aware of, because the classmate and the relative represent quite different marketing prospects. If the marketer were to identify prospects based on the observed interactions alone, however, he could be targeting the relative, and her friends, who are unlikely to behave in the same way as the focal customer (the student). Targeting these prospects incurs additional costs with little expected return. In addition, 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.
We can compare the use of targeting based on proximity in latent space, with targeting based on geodesic distance^{8}^{8}8A third perspective is that perhaps one should first follow connections within the clusters, then the similar customers, and then follow the geodesic links outside of some cluster. In fact, many more customers can be identified for targeting than if one were to use a geodesictype distance metric represented by observed interactions. We calculate that if one were to follow the first degree geodesic distance, on average the marketer would expect to reach eight customers (rounded up from 7.56) . If this data were 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}^{9}9Of course, this depends on the size of the circle, but since customers in the same cluster occupy identical coordinates, even a circle of minimal radius gives us the result that more customers are identified for targeting, on average, than the firstdegree geodesic.. Since homophily implies that similar individuals are more likely to interact, then targeting based on latent space means that the customers identified for targeting are more 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 are substantially different than the original customer. For example, in Figure 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 desire 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 presents an opportunity to refine 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 modeling 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 datagenerating process is dyadspecific, and depends on 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 observed cell phone interactions are only one of many possible means of communication. Interactions 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 observed 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 available, even when the space itself is stationary. Therefore, what Kossinets and Watts (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 provide some clues from 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 represent 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 authoritative document of the underlying social network. There may be other modes of communication, such as email or facetoface 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 representation of all interactions? We 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). We then treat that observed network as a single data point, and use it to update our beliefs about the structure of the latent space. If we had observed another mode of communication first, we might get a different 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
We see two key contributions of the latent space useful to marketers managing the diffusion of innovation of information through customer networks. The first involves the concept of ”Word of Mouth” (e.g. Arndt 1967). While contagion could occur via nonexplicit advocacy (e.g. fashion can be seen by people that one does not interact with), explicit communication is well regarded 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 testimony to the importance of consumer reviews, there are 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 structure affects diffusion patterns. The contribution of our work is in considering that WOM may work via some geodesic distance versus distance measured in latent space. From relations 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 area 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 studied in the context of diffusion research. However, recent research challenges the notion that such influentials are the primary reason for ”global cascading influence” (Watts and Dodds 2007), i.e. the contagious diffusion of innovation or information throughout an entire network. These insights 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 Vocalpoint, which are not selective about recruiting only ”opinion leaders,” but rather would prefer more people in their network.
Further research opportunities
The sociological theory of homophily, coupled with the latent space framework, yields a stochastic representation of the relative latent characteristics underlying interactions data. The latent characteristics are represented 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 larger networks, the concepts have not yet fully integrated across disciplines. Our research lowers this obstacle, and makes probability modeling more accessible to marketing researchers who possess data on customer interactions. This approach maintains the properties of interdependence, heterogeneity and interpretability, a goal that is harder to accomplish with extant classical or machine learning approaches.
We 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. We use the volumes of research 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 distance, to segment and target customers assumes that similar individuals have correlated purchase preferences 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 disposal 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 single 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 example of this kind of treating a nonphysical distance as a physical one is Yang and Allenby (2003), who computed a demographic distance between people based on profiles of personal characteristics. So just as they modeled correlations in preferences as functions of observed geographic and demographic distance, we propose modeling these correlations as functions of latent distances. We hypothesize that since observed interactions represent 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 order to conduct a test like this, one would need two types of data for the same set of people: dyadlevel interactions data to infer the latent space, and individuallevel purchase data to see if latent distance explains correlations 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 research stream would be to incorporate individuallevel demographics or covariates into the latent space model, and to better understand how that information might complement interactions data in understanding purchase behavior.
Appendices
Appendix A Choosing dimensionality of the latent space
There are a number of approaches one could take to selecting , and finding a general method for choosing among different specifications of Bayesian hierarchical models, especially those that incorporate nonparametric priors, remains an area of active research among statisticians. We believe that because of the abstract nature of the latent space, there is no “correct” value of that one needs to infer from the data. Hoff (2005) notes that one should choose the smallest value of that offers a reasonable model fit, erring on the side of parsimony. Adding more dimensions improves model flexibility, but can also lead to overfitting. As far as objective measures go, he suggests examining the log marginal 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 Table 3 we present relative estimated LML of the HMCR and Full models, for different values of , from both the calibration and holdout datasets. Our estimates were generated using cumulant approximations, and adjusting for the discrepancy between posterior and prior support, using the methods proposed in Lenk (2009). Results are normalized such that the reported estimate for the HMCR model, with 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 
In three of the four cases, is preferred, and in the remaining one, is preferred. Also, we found essentially no difference among values of in posterior predictive checks. So, following Hoff’s advice, we use the models for subsequent analysis.
Appendix B Model specification for China Mobile example
In this appendix, we derive the data likelihood and hyperprior specifications for our Chongqing Mobile application. Let be the probability that a dyad between and is open, and let be the rate of contacts between and (assuming exponentiallydistributed intercontact times), if the dyad is open. We also define an auxilliary latent Bernoulli variable that indicates whether a dyad is open () or closed (). To remain consistent with our general model specification in the text, we use to denote the vector of observed intercontact times, and to denote the count of observed contacts. Also, let 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., ): the dyad could be closed (), or it could be open, but the contact rate is sufficiently low that there just happened to be no contacts during the observation period. If we observe any contacts at all, we know the dyad must be open. Therefore, the data likelihood is
(4) 
We incorporate dyadwise unobserved heterogeneity in by using a gamma distribution with dyadspecific shape parameter and dyadspecific scale parameter
(5) 
After integrating over ,
(6) 
We will also use the reparameterizations and , where and are the dyadspecific mean and common variance of the gamma distribution, respectively. Linking back to our general model formulation in Section 1.1, . The definitions of these parameters are described in Section 2.1.
Hyperprior specifics
For the choice of , we decompose each latent coordinate into two components: the distance from the origin (a “radius”) and the location on the surface of a hypersphere that has that radius. We then choose a that factors into a prior on these two components. In other words, we think of the elements of in terms of their spherical, rather than Cartesian, coordinates. If the Cartesian coordinates of (herein suppressing the subscript) are , then its polar coordinates are , where is a distance from the origin and the s are angles, expressed in radians, such that , and for . We can then factor as
(7) 
Conditioning on , we want to place a distribution on , such that there is a uniform probability of being at any location on a dimensional hypersphere with radius . This is achieved by letting be a multivariate power sine distribution (Johnson 1987; Nachtsheim and Johnson 1988), where
(8) 
Thus, has a uniform distribution, , and so forth. Johnson (1987, ch 7) proposes some algorithms for simulating from a multivariate power sine distribution.
For , recall that is defined on the positive real line, with . We also need the ability to trade off tail weight (probability of draws of being far away from the origin) against kurtosis (likelihood of draws of being clustered around the origin). Beginning with the generalized Laplace distribution (Kotz et al. 2001, sec. 4.4.2), we center, and then fold, at zero, to get
(9) 
where
(10) 
Setting
(11) 
so the density of , constrained so , is
(12) 
The parameter controls the tradeoff between tail weight and kurtosis. If , reduces to an exponential distribution, and if , is a halfnormal distribution. As becomes large, the mode of becomes less and less peaked, and converges to a uniform distribution. The “correct” value for is inferred through the estimation process, letting the data drive the tradeoff between placing a mode on and bounding the locations such that . Note that this prior using the multivariate power sine distribution and our restricted halfLaplace distribution adds only one additional parameter to the model, compared to an independent multivariate normal hyperprior, which adds many more.
It turns out that is a special case of a power gamma distribution. To see this, perform a change of variables so , and . Then,
(13) 
which is a gamma distribution with shape parameter and rate parameter
(14) 
(Johnson 1987, ch 7).
We selected the other hyperpriors to balance weak information content against numerical stability. Following Escobar and West (1995), we place a weakly informative gamma prior on , with mean of 4 and variance of 80. Since are all populationlevel parameters that appear only in the definitions of , we can combine them all into a single parameter vector (logtransforming parameters when necessary), with a multivariate normal prior , centered at the origin, with covariance matrix . Note that if , then is a multivariate normal distribution. Since we were concerned about a mode of 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.
Appendix C Estimation algorithm
In this section we present the complete MCMC sampling algorithm for the general latent space model. The parameters to be estimated are , , , and , . Recall that since is discrete, at each iteration there are only possible values that any can take.
c.1 Simulate
Let and be the parameters of the gamma hyperprior on Using the algorithm proposed by Escobar and West (1995), do the following.

Starting with the current value of , draw a temporary variable from a distribution.

Draw from a Bernoulli trial with probability

If , draw from a distribution. If , draw from a distribution.
c.2 Simulate
To simplify notation, we combine into a single parameter vector . The conditional posterior distribution of depends on the data likelihood and the prior. The data likelihood we care about here is the marginal likelihood in Section 1.2, after integrating over , multiplied across all dyads (using our assumption of conditional independence across dyads). Note that is a function of . The prior on is a multivariate normal with mean and covariance . Thus, the log conditional posterior (without normalizing constant) for is
(15) 
We simulate using a random walk Metropolis sampler (Rossi et al. 2005, ch. 3).
c.3 Simulate
We place a prior on . Let be the radii of the distinct values of . Combining the likelihood of the radii in 12 with the prior, the conditional posterior distribution for is
(16) 
There are many different ways to simulate from this univariate density. We chose to use samplingimportance resampling (SIR) (Smith and Gelfand 1992), but one might choose Metropolis, gridbased inverse CDF, or slice sampling methods instead.
c.4 Simulate
This step, in which we draw each of the vectors from the mixture of Dirichlet processes (MDP), is an adaptation Algorithm 8 in Neal (2000). We direct the reader there 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 ’s is discrete, so at each iteration of the estimation algorithm, there are only possible values that can take. Let define these distinct latent coordinates, let be the distinct mass points when not including person , and let be the number of distinct mass points in when not including person ( and , and and ,will differ only if is a “singleton” who is the only person located at ). At the current state of the sampler, each person is “assigned” to one of the , in the sense that there is exactly one for which . Let be the number of people assigned to , and let be the number of people assigned to , when not counting person .
The algorithm involves choosing some number of proposal values (determined by a prespecified control parameter ) for each . Define as the likelihood contribution for all dyads that involve person , given the current values of assigned to , and of assigned to everyone else. There are two cases that we need to consider. The first is if there is some other person for which (i.e., is not a singleton). In this case, draw proposal draws from , call them , and let . Intuitively, is the union of the set of all latent vectors that are already assigned to someone in the population, with the set of new proposal vectors. Next, compute , for all . These are the likelihood contributions for all dyads involving , if were set to each of the values in . These “proposal likelihoods” form a set of weights that we use to draw a new for each . Thus, draw a new value for from using the following probabilities:
where is a normalizing constant (and does not need to be known for the purposes of random sampling). Thus, can take on the value of any of the existing elements of , or one of the new candidate values. Which value is selected depends on three values: the likelihood of the data for each (values of that yield a high likelihood are more likely to be chosen), the number of other people who also are assigned to (coordinates where the prior distribution has more mass are more likely to be chosen), and the DP control parameter , which governs how close the DP prior on is to . If is a singleton, then there are only elements in . In this case, draw candidate values from , reindex them so , and select according to the probabilities
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