Appl. Statist. (2013) 62, Part 5, pp. 705–722
Hierarchical longitudinal models of relationships in social networks Sudeshna Paul and A. James O’Malley Harvard Medical School, Boston, USA [Received September 2011. Final revision December 2012] Summary. Motivated by the need to understand the dynamics of relationship formation and dissolution over time in real world social networks we develop a new longitudinal model for transitions in the relationship status of pairs of individuals (‘dyads’). We first specify a model for the relationship status of a single dyad and then extend it to account for important interdyad dependences (e.g. transitivity—‘a friend of a friend is a friend’) and heterogeneity. Model parameters are estimated by using Bayesian analysis implemented via Markov chain Monte Carlo sampling. We use the model to perform novel analyses of two diverse longitudinal friendship networks: an excerpt of the Teenage Friends and Lifestyle Study (a moderately sized network) and the Framingham Heart Study (a large network). Keywords: Bayesian; Dyadic independence; Latent variables; Longitudinal model; Social networks and health; Transitivity
1.
Introduction
Relationship formation and dissolution in social networks are a complex process. Relationships (ties) develop between individuals (actors) over time and strengthen or decay depending on individual habits or behaviours, external factors and interaction with other actors. It is often of interest to determine whether changes in the network can be attributed to dyadic characteristics (e.g. similarity or spatial proximity of the actors) or to network structure near the dyad (e.g. density or average number of ties, reciprocation of ties or a third actor in common). Definitions of density and the other network terms that are used in this paper can be found in Kolaczyk (2009), Wasserman and Faust (1994) and the on-line supplemental material A. Dynamic modelling of social networks has not been studied extensively in the past owing to the lack of adequate data sets and complexity of the models required. However, with increasing availability of longitudinal network data and advances in computational methods, tractable solutions to a wide array of real world problems involving evolution of social networks, e.g. sexual networks (Morris and Kretzschmar, 1997), interaction networks between professionals or researchers (Kapferer, 1972) and classroom friendship networks (Udry, 2003), are now possible. The methodological research in this paper is motivated by two longitudinally resolved real world networks of 48 high school children (observed over three time points) and 831 individuals from the offspring cohort of the Framingham Heart Study (FHS) (observed over eight time points). Address for correspondence: A. James O’Malley, Department of Health Care Policy, Harvard Medical School, 180 Longwood Avenue, Boston, MA 02115-5899, USA. E-mail:
[email protected] Reuse of this article is permitted in accordance with the terms and conditions set out at http://wileyonline library.com/onlineopen#OnlineOpen Terms. © 2013 Royal Statistical Society
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Models treating dyads as independent or conditionally independent bivariate random variables (Fienberg et al., 1985; Holland and Leinhardt, 1981; Yang and Yong, 1987) have, until recently, been the traditional approach to modelling data on the relationships between individuals. For example, the p2 -model assumes dyadic independence conditional on random effects for the actors in a single network (Duijn et al., 2004). The p2 -model accounts for the prevalence of ties (density), clustering of ties due to reciprocation (reciprocity) and heterogeneity in the number of ties to and from actors (in- and out-degree distributions). The conditional independence structure ensures that a model for a single dyad induces the model for a network of individuals. However, the primary limitation of the p2 - and other dyadic independent models is that they do not allow for clustering induced by groups of three or more actors. Several generalizations of p2 -type models have been proposed to account for dependence between dyads. These utilize latent factors (Hoff, 2005; Hoff et al., 2002), interpreted as positions in a ‘social space’, or bilinear effects (Hoff, 2005). However, this approach has its own challenges: determining the dimension of the factor space is not straightforward and estimation of the latent factors encounters the identifiability and computational problems akin to exploratory factor analysis. Rather than treating the dyad as the unit of analysis, the network may be treated as a single observation and modelled by using exponential random graphs (also known as pÅ -models) (Frank and Strauss, 1986; Wasserman and Pattison, 1996). These models directly accommodate predictors representing a wide array of important sociological constructs, enabling (for example) transitivity and cyclical effects to be tested (Besag, 2000; Handcock, 2000; Snijders, 2002). But a major limitation of the pÅ -models is that estimators with desirable statistical properties such as consistency can be elusive as regular asymptotic results do not apply. Because of irregularly shaped likelihood surfaces, parameter estimates may lie on or close to the edge of the parameter space. Furthermore, degenerate networks (the networks containing no ties or all possible ties) are often generated under estimated (and actual) values of model parameters; the phenomenon known as ‘degeneracy’ (Handcock, 2000). However, the extent to which dynamic pÅ -models are susceptible to degeneracy has not yet been extensively studied. In this paper, we analyse two real life longitudinal networks by developing a network model that combines features of the p2 - and pÅ -models developed for cross-sectional data to accommodate higher order network effects while retaining conditional independence across dyads. By insisting on a conditionally independent model, we ensure that the appealing properties of likelihood-based estimators under standard regularity conditions apply. Assuming regularly spaced observations across time, we develop a Markov model for discrete time dyadic transitions. A generalized logistic regression model links the multinomial transition probabilities to density and reciprocity effects and modifications thereof by current and previous tie and dyadic covariates. Conditional on its previous state, the state probabilities of the dyad resemble those of a p2 -model. Specifically, we extend the p2 -model by incorporating clustering due to transitivity and other terms through judiciously defined lagged predictors. The critical feature is that the current state of a dyad can depend on the preceding state of all other dyads without violating the (conditional) dyadic independence assumption. Our approach differs from existing models for longitudinal network data. Continuous time Markov processes were among the first dynamic models for networks (Holland and Leinhardt, 1977; Wasserman, 1977). Although avoiding discretization of time is appealing, closed form expressions for the probabilities that a dyad is in a given state at a given time are available for only the simplest models. Hence, estimation of any realistic model requires numerical methods and is often impractical (Leenders, 1995). Existing discrete time longitudinal models for network data are based on pÅ -models (which
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are also known as exponential random-graph models). A temporal extension in the discrete Markov domain was proposed by Hanneke and Xing (2007) and more recently by Desmarais and Cranmer (2011) where the conditional probabilities of tie changes are described by a pÅ model with lagged and contemporaneous network statistics and node attributes as predictors (Cranmer and Desmarais, 2011; Desmarais and Cranmer, 2011; Hanneke and Xing, 2007). Krivitsky and Handcock (2010) extended this approach to develop a separable discrete time dynamic model for tie duration and tie formation. Snijders and colleagues have developed ‘edge-oriented’ and ‘actor-oriented’ models that combine a continuous time process controlling the ‘opportunity’ of change with a ‘propensity’ of change based on a utility function (Snijders, 2005, 2006). Because they allow a wide array of network hypotheses to be tested, variants of dynamic pÅ -models are appealing but like their cross-sectional counterparts they suffer from degeneracy and require substantial computational resources (Goldenberg et al., 2010; O’Malley and Marsden, 2008). Our hierarchical modelling approach avoids these concerns. The remainder of the paper is arranged as follows. In Section 2 we describe our proposed model and discuss its potential benefits and limitations. Section 3 describes an efficient algorithm for implementing Bayesian model estimation by using Markov chain Monte Carlo (MCMC) sampling to estimate the model. In Section 4, we analyse the Teenage Friends and Lifestyle (TFL) Study data set from the ‘Simulation investigation for empirical network analysis’ (www.stats.ox.ac.uk/∼snijders/siena/) and the FHS friendship network data set. The paper concludes in Section 5. The programs that were used to analyse the data can be obtained from http://www.blackwellpublishing.com/rss 2.
Transition model
We assume that (a) the length of time between observations is ignorable (regular longitudinal spacing), (b) given the current state, the future status of the dyad is independent of all past states (Markov dependence) and (c) both ties within a dyad can change status within a time interval (e.g. a ‘null’ dyad (0 ties) at t may be ‘mutual’ at t + 1 and vice versa). Compared with continuous time Markov chain models, assumption (a) is a restriction and (c) is a relaxation. .t/ Let Yij indicate a tie from actor i to j and Y .t/ be the n × n adjacency matrix containing .t/ Yij as its ijth element at time t. The dyad comprising actors i and j .i < j/ at time t, denoted .t/ .t/ (Yij , Yji /, is either null (‘00’), asymmetric (‘01’ or ‘10’) or mutual (‘11’). Therefore, under the Markov assumption, the dependence of a dyad’s transition probabilities on its past states is represented by a 4 × 4 matrix. Let pij|kl denote the probability of transition from state kl to state ij. To gain insight into the dyadic transition probabilities, consider the simple case where actors are homogeneous. Because the ordering of individuals within dyads is arbitrary, transition probabilities involving state ‘10’ must have the same probability as the analogous transition involving state ‘01’. Therefore, for model identifiability p10|00 = p01|00 , p10|11 = p01|11 , p00|10 = p00|01 , p11|10 = p11|01 , p10|10 = p01|01 and p10|01 = p01|10 (Table 1). Because the probabilities along each row sum to 1, there are 42 − 9 = 7 free probabilities. However, unless p10|01 = p01|10 = 0, the state space of dyadic transitions cannot be reduced to a 3 × 3 transition matrix even when the actors are homogeneous. Actor level heterogeneity in individual-specific covariates and unmeasured (latent) variables
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S. Paul and A. J. O’Malley Table 1. Dyadic transition matrix when the actors are homogeneous† Wave
State
t −1
00 01 10 11
t 00
01
10
11
p00|00 p00|01 p00|01 p00|11
p01|00 p01|01 p10|01 p01|11
p01|00 p10|01 p01|01 p01|11
p11|00 p11|01 p11|01 p11|11
†pij|kl denotes the probability of transition from state kl at time t − 1 to state ij at time t. Because five probability equalities are implied by homogeneity and each row sums to 1, there are only seven free probabilities.
leads to asymmetry in transition probabilities involving asymmetric states. For example, the probability of a null to asymmetric transition is the sum of the probabilities of each actor naming the other; unlike the homogeneous case, these probabilities may not be equal because of covariates defined from the perspective of the sender or receiver of the tie. Therefore, it is easier to model the data by using a four-component as opposed to a three-component multinomial dis.t/ .t/ tribution. Assuming a generalized logit link function in which .Yij = 0, Yji = 0/ is the reference category, the model is ln
.t/
.t/
.t/
.t/
.t−1/
Pr.Yij = yij , Yji = yji |yij
.t−1/
, yji
, αt , βt /
.t/ .t/ .t−1/ .t−1/ Pr.Yij = 0, Yji = 0|yij , yji , αt , βt / .t−1/
.t/
.t/
.t/ .t/
= θijt yij + θjit yji + ρijt yij yji :
.1/
.t−1/
In model (1), θijt and ρijt may depend on .yij , yji /, the past state of the dyad, and actor, tie or dyadic observed and latent variables. The quantity ρijt contains predictors and associated .t/ .t/ coefficients affecting the dependence within .Yij , Yji /; a non-zero value implies that the status of ties within a dyad must be jointly modelled. For a fixed t, model (1) reduces to a p2 -model that includes lagged covariates. Additivity is imposed further through the decomposition θijt = μijt + αit + βjt , where μijt is the tie level mean and αit and βjt are actor-specific variables at t, and the incorporation of covariates and latent variables: ⎫ .t−1/ .t−1/ .t−1/ .t−1/ μijt = μ + γ11 yij + γ12 yji + γ13 yij yji + η1 z1ijt , ⎪ ⎪ ⎪ ⎪ ⎬ αit = ait + x1kit ξ1k , .2/ ⎪ βjt = bjt + x2kjt ξ2k , ⎪ ⎪ ⎪ .t−1/ .t−1/ .t−1/ .t−1/ ⎭ ρ = ρ + δ .y +y /+δ y y +η z ijt
1
ij
ji
2 ij
ji
2 2ijt
where γ11 , γ12 and γ13 modify density whereas δ1 and δ2 modify reciprocity according to the past state of the dyad. The covariates Z1ijt , Z2ijt , X1it and X2jt in system (2) are tie, dyad or actor specific and may be time varying. Asymmetry arises when Z1ijt = Z1jit , ait = bit or X1it = X2it (Z2ijt is symmetric by construction). The transition probabilities in the heterogeneous case follow from the substitution of equations (2) into model (1) and application of the homogeneity constraints that are implied by Table 1. Each row of the resulting transition matrix is a p2 -model (Table 2).
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Table 2. Transition matrix for heterogeneous dyads under the model defined by equations (1) and (2)† Wave
State
t 00
t −1
01
10
00
−1 k00
−1 k00 exp.μ + αjt + βit /
−1 exp.μ + αit + βjt / k00
01
−1 k01
−1 k01 exp.μ + γ11 + αjt + βit /
−1 k01 exp.μ + γ12 + αit + βjt /
10
−1 k10
−1 k10 exp.μ + γ12 + αjt + βit /
−1 k10 exp.μ + γ11 + αit + βjt /
11
−1 k11
−1 k11 exp.μ + γ11 + γ12 + γ13 + αjt + βit /
−1 k11 exp.μ + γ11 + γ12 + γ13 + αit + βjt /
11 −1 exp.2μ + αit + βjt + αjt k00 +βit + ρ/ −1 k01 exp.2μ + γ11 + γ12 + αit + βjt + αjt + βit + ρ + δ1 / −1 k10 exp.2μ + γ11 + γ12 + αit + βjt + αjt + βit + ρ + δ1 / −1 k11 exp{2.μ + γ11 + γ12 + γ13 / + αit + βjt + αjt + βit + ρ + 2δ1 + δ2 }
†For ease of presentation covariates are hidden.
Interactions between the dyadic covariates Z1ijt and past dyad status are incorporated by substituting η1 Z1ijt with .t−1/
.η10 + η11 yij
.t−1/
+ η12 yji
.t−1/ .t−1/ yji /z1ijt :
+ η13 yij
.3/
In expression (3), η10 is the effect of z1ijt on tie formation .i → j/ when originating from the ‘null’ state and η11 , η12 and η11 + η12 + η13 are the modifying effects of the 10, 01 and 11 states respectively. The term η2 Z2ijt can also be augmented with interactions to allow covariate effects affecting reciprocity to depend on current dyad status. The actor-specific effects αit and βit depend on latent variables ait and bit representing the propensity for actors i and j to send (‘expansiveness’) and receive (‘popularity’) ties respectively. In general .ai2 , : : : , aiT / and .bi2 , : : : , biT / can take any form of dependence across time. However, here we consider only the stationary case ait = ai and bit = bi for t = 2, : : : , T . To accommodate correlation between .ai , bi / we suppose that they obtain from a bivariate normal distribution with mean 0 and covariance matrix σa2 αab σa σb Σ= : αab σa σb σb2 The correlation coefficient αab represents the extent that expansive individuals (who nominate many friends) are also popular (who are nominated by many). In practice, one or more cells in the contingency table may have close to 0 or exactly 0 frequencies. For example, because the relationship must cease in one direction and form in the other between observation times, transitions .01 → 10/ and .10 → 01/ may be rare or even nonexistent in social networks. A reduced model that does not allow such transitions is obtained by fixing p01|10 = p10|01 = 0 and rederiving the model. The interpretation of some of the other model coefficients may change under structural zero constraints (see the on-line supplemental material B). 2.1. Third-order dependence Transitivity is the most researched form of dependence involving groups of three or more actors. In a binary network, transitivity occurs if yij = 1 and yjk = 1 implies yik = 1 or in a friendship network ‘a friend of a friend is a friend’ (Snijders et al., 2006; Wasserman and Faust, 1994).
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Another form of third-order dependence is third-order cyclicity, which is the tendency to form cycles of length 3, defined as yij = 1 and yjk = 1 implies yki = 1. A major limitation of crosssectional dyadic independence models is that they do not accommodate transitivity or other forms of interdyad clustering. However, the availability of longitudinal data enables lagged values of predictors depending on the status of other dyads to enter the model while maintaining dyadic independence at the current time. For example, to allow for the possibility that a third actor having both i and j as friends increases the likelihood of a tie between i and j, we include
.t−1/ .t−1/ two-start−1 = I y y > 0 ij ki kj k=i,j
as a predictor. Here, I.·/ is an indicator function defined as I.event/ = 1, if the event occurs, and I.event/ = 0 otherwise. Likewise, because a two-path between individuals i and j may make a direct tie from i to j more likely, we also include
.t−1/ .t−1/ two-patht−1 = I y y > 0 ij ik kj k=i,j
as a predictor. Similarly, one can test for a tendency towards cyclicity by including twot−1 patht−1 ji = 1 if there exists k such that yjk = yki = 1 and two-pathji = 0 otherwise to predict transitions from actor i to j. Since our focus is primarily on the role of transitivity in the network, intransitive triadic closure is of less interest herein. If directionality has no effect or the network is non-directional, one might use
.t−1/ .t−1/ .t−1/ .t−1/ triadt−1 = I .y +y /.y + y / > 0 : ij ik ki jk kj k=i,j
Alternatively, counts or other non-binary functions may be used as triadic covariates. These terms depict forms of triadic closure and have positive coefficients if triad closure induces or reinforces ties. The above predictors (or variants) may be included in system (2) as main effects or multiplied t−1 t−1 t−1 t−1 by yij , yji or yij yji to allow effect heterogeneity by lagged dyad status. For example, .t−1/ .t−1/ × two-patht−1 .1 − yij / × two-patht−1 ij and yij ij allow the effect of a two-path from actor i to actor j on a new tie forming between them to differ from the effect on reinforcement of an existing tie. Triadic covariates that are asymmetric with respect to i and j can only be included in μijt whereas those that are symmetric may be included in μijt or ρijt . 2.2. Likelihood function Under Markov dependence, the joint likelihood function satisfies L= =
.1/
.i
