Bayesian inference for dynamic social network data

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Abstract

We consider a continuous-time model for the evolution of social networks. A social network is here conceived as a (di-) graph on a set of vertices, representing actors, and the changes of interest are creation and disappearance over time of (arcs) edges in the graph. Hence we model a collection of random edge indicators that are not, in general, independent. We explicitly model the interdependencies between edge indicators that arise from interaction between social entities. A Markov chain is defined in terms of an embedded chain with holding times and transition probabilities. Data are observed at fixed points in time and hence we are not able to observe the embedded chain directly. Introducing a prior distribution for the parameters we may implement an MCMC algorithm for exploring the posterior distribution of the parameters by simulating the evolution of the embedded process between observations.

Introduction

The methods of social network analysis (e.g., Wasserman and Faust, 1994, Carrington et al., 2005) have long been employed in studying social phenomena ranging from friendship patterns of children in school classes (Moreno, 1934) to the spread of HIV (Klovdahl, 1985). The importance of studying the dynamics of social networks is generally stressed and there are some obvious shortcomings of cross-sectional data. Snijders (2001), proposed a statistical model for the analysis of longitudinal social network data (for empirical applications see e.g. van de Bunt et al., 1999, and van Duijn et al., 2003). Inference has so far been limited to an MCMC implementation of the method of moments, the properties of which (i.e. the estimators) are not fully understood. Here we propose a Bayesian inference scheme for these models as well as a model formulation that makes provisions for relaxing some of the assumptions and that allows for a wider scope for the type of data that may be analysed.

Section snippets

Preliminaries

In this paper, we consider social networks construed as graphs or directed graphs, where the vertices represent actors in a network and edges and arcs represent how the actors are relationally tied to each other. The set of actors is assumed to be fixed and we are interested in how the set of edges or arcs changes over time.

Here follows a brief characterization of the general class of models. We restrict attention to binary social network data. Denote by V={1,,n} a fixed set of actors and

Model specification

We now proceed to express the transition intensities as the product of two more easily interpretable factors. We define the class of models considered in this paper from the point of view of the embedded chain of a continuous-time process on X. The transition probabilities in the embedded chain areη(θ,x,y),and the time spent in xX exponentially distributed with rateλ(θ,x).It is assumed that η(θ,x,y)>0 for x,yX such that |x-y|=1 and 0 otherwise. The p×1 parameter vector θΘ includes all

Parameter inference

Assuming that we have a model in the form specified in the previous section, and that we have specified a prior distribution π(θ), the main objective of the proposed inference scheme is to obtain the posterior distributionπ(θ|x(t0),,x(tM))π(θ)=0M-1LD(θ;x(t+1),x(t)).In accordance with the account of the model formulation above, since the likelihood factors, we may treat each pair of consecutive observations separately.

Under the assumption that the augmented likelihood L(θ;w,x(t0)) is cheap

Example: the electronic information exchange system

In this section we analyse the electronic information exchange system (EIES) data collected by Freeman and Freeman (1979), using one of the models fitted with the method of moments in Snijders and van Duijn (1997). The relation (originally valued) is directed and was measured at two points in time for a network consisting of n=32 actors (N=992). The data set is described in Wasserman and Faust (1994). We consider the relation defined by the respondent clamning to have met the other person or

Example: university freshmen

We now turn to look at friendship nominations for a group of 32 university freshmen in a Dutch university. There are observations for three points in time, t0, t1, and t2, with three weeks between each observation (the observations used here were labelled t2, t3, and t4, respectively, in the original study, van de Bunt et al., 1999). We fit a reciprocity model (as described in Section 2) as well as an extended model to x(t0) and x(t1). The extended model is similar to the model in the preceding

Discussion

In this paper we have proposed an alternative estimation procedure to the method of moments estimator for longitudinal social network data. In place of the approximate standard errors from the MM-analysis as a measure of uncertainty, we are provided with an entire sample from the posterior distribution of the parameters. We may explore the posterior distribution thus obtained to investigate marginal as well as joint distributions of the parameters and in addition the distribution of functions

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The research was in part supported by the Swedish Institute.

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