Essays on Bayesian Inference for Social Networks

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In paper number two a method for modelling reports given by actors (or other informants) on their social ... List of included papers. I. Bayesian ... from a number of people. Regrettably, I cannot name you all. First of all I would like to thank .... in various forms dates back to Erdös (1947)(see Janson et al., 2000, on random.
ESSAYS ON BAYESIAN INFERENCE FOR SOCIAL NETWORKS JOHAN KOSKINEN

Department of Statistics, Stockholm University, 2004

Doctoral Dissertation Department of Statistics Stockholm University S-106 91 Stockholm Abstract: This thesis presents Bayesian solutions to inference problems for three types of social network data structures: a single observation of a social network, repeated observations on the same social network, and repeated observations on a social network developing through time. A social network is conceived as being a structure consisting of actors and their social interaction with each other. A common conceptualisation of social networks is to let the actors be represented by nodes in a graph with edges between pairs of nodes that are relationally tied to each other according to some definition. Statistical analysis of social networks is to a large extent concerned with modelling of these relational ties, which lends itself to empirical evaluation. The first paper deals with a family of statistical models for social networks called exponential random graphs that takes various structural features of the network into account. In general, the likelihood functions of exponential random graphs are only known up to a constant of proportionality. A procedure for performing Bayesian inference using Markov chain Monte Carlo (MCMC) methods is presented. The algorithm consists of two basic steps, one in which an ordinary Metropolis-Hastings up-dating step is used, and another in which an importance sampling scheme is used to calculate the acceptance probability of the Metropolis-Hastings step. In paper number two a method for modelling reports given by actors (or other informants) on their social interaction with others is investigated in a Bayesian framework. The model contains two basic ingredients: the unknown network structure and functions that link this unknown network structure to the reports given by the actors. These functions take the form of probit link functions. An intrinsic problem is that the model is not identified, meaning that there are combinations of values on the unknown structure and the parameters in the probit link functions that are observationally equivalent. Instead of using restrictions for achieving identification, it is proposed that the different observationally equivalent combinations of parameters and unknown structure be investigated a posteriori. Estimation of parameters is carried out using Gibbs sampling with a switching devise that enables transitions between posterior modal regions. The main goal of the procedures is to provide tools for comparisons of different model specifications. Papers 3 and 4, propose Bayesian methods for longitudinal social networks. The premise of the models investigated is that overall change in social networks occurs as a consequence of sequences of incremental changes. Models for the evolution of social networks using continuos-time Markov chains are meant to capture these dynamics. Paper 3 presents an MCMC algorithm for exploring the posteriors of parameters for such Markov chains. More specifically, the unobserved evolution of the network in-between observations is explicitly modelled thereby avoiding the need to deal with explicit formulas for the transition probabilities. This enables likelihood based parameter inference in a wider class of network evolution models than has been available before. Paper 4 builds on the proposed inference procedure of Paper 3 and demonstrates how to perform model selection for a class of network evolution models. Keywords:Bayesian inference; social network analysis; Markov chain Monte Carlo; exponential random graphs; cognitive social structures; longitudinal social networks. ISBN 91-7265-888-6 c Johan Koskinen

Jannes Snabbtryck Kuvertproffset HB, Stockholm 2004





List of included papers I

Bayesian analysis of exponential random graphs – estimation of parameters and model selection. Research Report 2004:2, Department of Statistics, Stockholm University.

II

Model selection for cognitive social structures. Research Report 2004:3, Department of Statistics, Stockholm University.

III

Bayesian inference for longitudinal social networks. Research Report 2004:4, Department of Statistics, Stockholm University.

IV

Model selection for longitudinal social networks. Research Report 2004:5, Department of Statistics, Stockholm University.

Acknowledgments During the work with my thesis I have received valuable support from a number of people. Regrettably, I cannot name you all. First of all I would like to thank Ove Frank for valuable comments and suggestions. I am also deeply grateful for the comments and suggestion that I have received from Tom Snijders, who also arranged for me to visit Groningen, a stay which proved to be very important and inspiring. Mattias Villani has been a great inspiration. Jan Hagberg has never refused me help and, quite frankly, I don’t think he can resist helping people. As regards the more substantial contributions to the thesis I thank Alan Gibson, for thoroughly researching various translations of Daniil Kharms, and Silke Burestam for choosing the right colour red. I thank all those who have stoically allowed me access to their computers, notably Hakan, Bruce, Mattias, and Jelenor; my fellow doctoral students, especially Carlson and Tallberg, for guiding me through; friends and family of varying description and, of course, Milusheva. Stockholm, April 2004

Johan Koskinen

Contents 1. Introduction ix 1.1. Social network analysis ix 1.2. Bayesian inference x 2. Summary of papers xi 2.1. Paper I: Bayesian analysis of exponential random graphs – estimation of parameters and model selection xiii 2.2. Paper II: Model selection for cognitive social structures xiv 2.3. Paper III: Bayesian inference for longitudinal social networks xiv 2.4. Paper IV: Model selection for longitudinal social networks xv References xv



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1. Introduction Broadly defined, social network analysis (SNA) is concerned with the relation between social structure and individual entities. Social structure both constrains and enables the actions of individual units. The social structure itself is however made up of individuals and their actions and interactions. What is commonly called social network analysis is a highly formalised analytical tool for studying social structure (for an exhaustive introduction to SNA see Wasserman and Faust, 1994). This thesis deals exclusively with social networks conceived as patterns of interaction among individuals, interactions which are assumed to be measurable. More specifically, we deal only with social networks consisting of fixed sets of individuals and the relationships between these, measured in a way such that the social network can be represented by a graph. By graph we mean a set of vertices that may have edges or arcs between them representing the status of the relationships between pairs of vertices. The potential of this representation as an analytical tool is well testified (as, allegedly, first discovered by Moreno, 1934; classic works include Coleman et al., 1957; Milgram, 1967; Granovetter, 1973; for a review see e.g. Wasserman and Galaskiewicz, 1994) but it clearly raises several epistemological and methodological issues. To deal with whether social interaction can be measured in a meaningful way that lends itself to such a simple representation, or whether the ideosyncracies of every unique network cancel whatever general tendencies we would like to uncover, is beyond the scope of this thesis. Social network analysis as used here is a theoretical model of ”reality”, a theoretical construct upon which we can for example build statistical models. 1.1. Social network analysis. Typically a fixed set of social entities is considered. The elements of this set are commonly referred to as actors. For these actors a collection of relations are defined that specifies how these actors are relationally connected pairwise to each other. For a non-directed (e.g. ”friendship”) relation the actors are represented by vertices in a graph in which there are edges between pairs of actors that are relationally tied together. These relational ties may also be directed (e.g. ”give advice to”), in which case we differentiate between when an actor i has a tie to another actor j from when j has a tie to i. Directed ties can be represented by directed edges, arcs, between pairs of vertices of the network. For a single relation the make-up of the overall pattern of social interaction can be studied by drawing its graph. Structural properties of the graph that may caracterize the social structure of the network include the level of clustering of ties, the presence of actors who serve as cut-points between otherwise disconnected groups, etc. The social networks literature is rich in examples of measures meant to capture different aspects of the social networks. A basic structural feature of a directed relation is for example the number of ties in the network and the variation in the number of ties each vertex has. The adjacency matrix of a graph on vertices 1, . . . , n, is matrix x that has elements xij = 1 if i has a tie to j, and xij = 0 otherwise. The statistical task is hence to model these binary adjacency matrices. Probability modeling of random graphs

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in various forms dates back to Erd¨os (1947)(see Janson et al., 2000, on random graphs with a focus asymptotic theory). Statistical inference for social networks is notoriously complicated by the dependencies that arise as a natural consequence of the research context (Frank, 1971;Frank, 1997; see also the forthcoming book Carrington et al., 2004; for data from samples or otherwise incompletely observed networks see Frank, 1988). As argued in Robins and Pattison (2004; or Wasserman and Robins, 2004), whereas in many familiar statistical models observations on different sample units can be assumed to be independent, this is not reasonable to assume for social networks. In fact, the dependencies between variables is exactly what we are interested in modeling. 1.2. Bayesian inference. Bayesian inference is a well established school within statistics and there is an ever growing literature covering the basic concepts. For a comprehensive treatment of Bayesian inference see for example Bernardo and Smith (1994), Lindley (1965), and Box and Tiao (1973). Here follows a simplified description of some of the concepts that appear in the papers1. Assume that there is a parametric model which describes an observable phenomenon with a distribution p(·|θ). Under the assumption that this model is true and the parameter θ an unknown constant, different possible values on θ represent different statements about the observable phenomena. Since we have preconceptions about the observable phenomenon, some observations might seem highly implausible – such as for example getting all heads when flipping a coin a 100 times – we should have some idea about the likely values of θ. If this information is quantified into a probability distribution on θ, a prior p(θ), Bayes theorem tells us how we can (should) up-date our belief regarding the likely values of θ. When we obtain data x generated by the model, the posterior distribution of θ given data is given by (1.1)

p(θ|x) = R

p(x|θ)p(θ) . p(x|θ)p(θ)dθ

This is Bayes theorem which follows from standard probability calculus, and in which p(x|θ) is called the likelihood function. It commonly emphasised that posterior ∝ likelihood × prior, i.e. that what we believe when we have made an observation is proportional to what we believed beforehand scaled by the ”plausibility” of what we have observed. The term reference prior is used to signify a prior distribution that reflects ”little” prior information. What ”little” information is and how this should be represented has been widely discussed (Jeffreys, 1961; Bernardo, 1979, 1997). When employed in the papers to follow, reference priors are used merely as points of reference for communicating and interpreting results. Ideally, a reference prior should be clear enough for other researcher to easily take the influence of the prior 1We

refer to the sources mentioned above for a more stringent and rigorous presentation of the theory as well as an account of the more subtle aspects; note in particular the simplified description of model selection as being a choice between a limited number of models, one of which is assumed to be true.

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used into consideration when interpreting the results. One might be tempted to set the reference prior proportional to a constant, so that the posterior only depends on the likelihood function, in which case we will refer to it as a vague prior. In some cases this constant prior will be a proper distribution and in other cases it is improper. For improper distributions there are cases when there is no proper posterior distribution. Once we have obtained a posterior distribution this contains all information regarding the uncertainty of the parameters. The Bayes point estimate (under square loss) for a parameter is for example given by the posterior expected value of the parameter given data. We also obtain posterior distributions of transformations and functions of the parameters. Assume that we have two models for data, represented by distributions p1 (·|θ) and p2 (·|φ), that we wish to compare and that our prior belief in the first model is given by π and by (1 − π) for the second model. With prior distributions p1 (θ) and p2 (φ), the marginal likelihood of each model is given by Z p1 (x) = p1 (x|θ)p1 (θ)dθ, and p2 (x) =

Z

p2 (x|φ)p2 (φ)dφ.

From Bayes theorem we then have that the posterior probability of model 1 given data is given by p1 (x)π . p1 (x)π + p2 (x)(1 − π) The ratio of posterior probabilities, the posterior odds, of the models p1 (x) π , p2 (x) (1 − π) where the first fraction is called the Bayes factor of model 1 against model 2, and the second fraction is called the prior odds. Model selection is hence being done in terms of probabilistic statements in the Bayesian framework. As a consequence the marginal likelihood (which is proportional to the posterior probability of a model for equal prior odds) is the only quantity relevant for model selection (no penalties for model complexity are needed). 2. Summary of papers Bayesian analysis gives us a rich picture of uncertainty, something which is essential when we have complex models and relatively few observations. This combination limits the use of approximations and asymptotics for performing nonBayesian inference in the absence of closed form expressions for, e.g. maximum likelihood estimates and their standard errors. We do not intend to explore this issue further here, merely state our view that Bayesian statistics, for want of an alternative, is well suited for SNA. However, to perform a fully subjective analysis would require of the researcher not only a thorough knowledge about the models employed but also to some extent the effects of different prior specifications. The

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complexity of this task, i.e. quantifying the prior information, increases with the complexity of the models. An aspect of this problem is addressed in Paper II. A few of the pitfalls and issues that can be encountered in these circumstances are discussed. Although no definite solution to these problems is given, it illustrates the ambiguity of the concepts objectivity and non-informative. This theme also appears in Paper IV. As mentioned above, the network structure of data complicates statistical modeling a great deal. The papers presented here are instances of three different data structures that might appear in SNA: a single observation on a social network, repeated measures on the same social network and, repeated measures on a social network developing through time. The difficulties mentioned are perhaps best illustrated by the numerous efforts in the literature to create models to accommodate data consisting of a single observation on a network. The usual trade-off between realism and parsimony in statistical models is somewhat complicated in SNA. One reason is that in order to create workable models, one has had to omit exactly those features of the network structure in which we are most interested. The second type of data structure gives us more information on the network. In the setting of Cognitive Social Structures (CSS), the focus is mainly on data where different reporters have supplied information regarding the same network. Naturally, there are some discrepancies, the reports on the network are not identical. The way one deals with these discrepancies also signifies a statement about the very core of empirical investigations on social networks. One can choose to regard these reports as data laden with noise or bias and the analysis simplifies to finding out what reporters are wrong, lie or are untrustworthy. Taking another stance, saying that in principle there is no social network other than that perceived by the actors, compromises the quantitative methodology in SNA altogether - how can we model data when the data we observe represent nothing more substantial than the inner worlds of different actors? (Naturally, not all social network data consists of self reported ties. The study of affiliation networks, such as in study of interlocking directorates, does not rely on data reported by the actors, to take one example.) This is an important issues in SNA and there is (to the best of knowledge) no known non-Bayesian procedure that can handle the ambiguities and the dynamics of these two perspectives. However, making the analysis fully subjective is not straightforward either and this is the subject matter of Paper II. The motivation behind studying the third type of structure, social networks observed through time, is derived in a sense from the problems arising when modeling single observations. The social network as an outcome of social actions and processes is central to the analysis in Pattison and Robins (2002). Studying the actual processes can however only be done when the notion of change and action is incorporated explicitly in the modeling. Longitudinal analysis of social networks revives the notion that social networks are instances of social interaction and that the ties between actors are not entities independent of the actors. Since change in the network gives rise to endogenous feed-back loops, to subject these models to

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empirical evaluation becomes a fairly complex task. Posterior distributions provide a precise measure of uncertainty both regarding parameter estimates as well as model specifications. These issues are considered in the final Papers III and IV. Most computations in these essays are done by means of Markov chain Monte Carlo (MCMC) techniques. We refer to Gilks et al. (1996) and Fishman (1996), for introductions and Tierney (1994), for technical results.

2.1. Paper I: Bayesian analysis of exponential random graphs – estimation of parameters and model selection. As summarized in Wasserman and Pattison (1996), quantitative research on social networks during the period from the 1930’s to the seventies, was largely dominated by the investigation of distributions of various network statistics under a variety of null models. That is, if the studies were not merely concerned with descriptive statistics. Although probabilistic models for graphs has been around for a long time (e.g. Bernoulli graphs, Erd¨os, 1947), one might say that that the first effort of realistic modeling of social networks was given by Holland and Leinhardt (1981) and their p1 model. Apart from some minor remarks regarding the maximum likelihood estimators (Wong, 1987), the main weakness of the p1 model is the assumption of independent dyads. The nature of the dependencies in social networks were thoroughly investigated by Frank and Strauss (1986) and resulted in a proposed model for social networks called Markov graphs. The study of dependence structure for social networks has later been further studied by Frank and Nowicki (1993), Robins (1998), Pattison and Robins (2002), and Robins and Pattison (2004). Wasserman and Pattison (1996) proposed a generalization of the Markov graph to include arbitrary functions of the graph and actor attributes called the p∗ -model or exponential random graph (a shift of focus has however been made from seeing the p∗ -model as an approximate auto-logistic regression model back to how the p∗ -model follows from different assumptions regarding the dependence graph, cf Wasserman and Robins, 2004). This model is the focus of Paper I. Many authors have reported difficulties in handling the exponential random graph model (Frank and Strauss, 1986; Dahmstr¨om and Dahmstr¨om, 1993; Corander et al., 1998; Besag, 2000; Hancock, 2000; Corander et al., 2002; Snijders, 2002; Snijders and van Duijn, 2002; Snijders et al., 2004). The problems encountered can roughly be divided into problems to do with estimation and pure model deficiencies. In this paper we briefly discuss how Bayesian inference to a certain extent might alleviate inference problems. The key idea is that parameter combinations known to lead to model instability should be penalized by the prior distributions. The main topic is however to provide an inference procedure for conducting Bayesian inference for exponential random graphs. A simple MCMC algorithm is proposed for exploring the posterior distribution of the parameters. The construction of the MCMC algorithm enables Bayesian model selection to be performed by using the method presented by Chib and Jeliazkov (2001). A working procedure for conduction Bayesian model selection means that we may compare quite diverse models and that we are not limited to comparing nested models.

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2.2. Paper II: Model selection for cognitive social structures. The study of how the actors perceive their network has a long tradition in SNA (see e.g. Moreno, 1934,and Newcomb, 1961; a review and analysis of social cognition is given by Pattison, 1994). A series of studies (e.g. Bernard et al., 1980; Killworth and Bernard, 1979) concluded that interaction as reported by the actors was not to be trusted. Krackhardt (1987) proposed a methodology for studying and comparing actor reports that involves collection information on the whole network from each actor. He called the resulting three-way array a Cognitive Social Structure (CSS). Many studies have been done in that tradition, (Bondonio, 1998; Casciaro, 1998; Casciaro et al., 1999, , to name a few), and the aim has been to correlate discrepancies in the actor reports with structural features and actor attributes. A statistical model for studying CSS was proposed by Batchelder et al. (1997) that specified the model conditional on an assumed true network structure. The reports given by the actors were seen as Bernoulli trials with probabilities of success (false positives and ”true” positives, respectively) dependent upon the true structure. How to analyse their model with a Bayesian approach was shown in Koskinen (2002b). An extension of this model was proposed in Koskinen (2002a), which modeled the Bernoulli probabilities with probit link functions. This is further elaborated here with a special focus on finding standard reference priors that enables model selection. The main obstacle is that the model is not fully identified, something which can not be solved in any obvious way through restrictions or highly informative priors. The proposed solution is to asses, a posteriori, which are the main determinants of identifying conditions. We present a procedure for choosing prior distributions and provide the necessary adjustments to the original MCMC sampling scheme of Koskinen (2002a).

2.3. Paper III: Bayesian inference for longitudinal social networks. To study the dynamics of social networks many authors have suggested using continuoustime Markov chains (Holland and Leinhardt, 1977a,b; Wasserman, 1977, 1980b,a; Leenders, 1995b,a). Snijders (1996) proposed a class of models that allow for greater flexibility in defining the dynamic components, relaxing the restrictions on the type of dependence structures that could be modeled (for some recent applications see e.g van de Bunt, 1999; van de Bunt et al., 1999; van Duijn et al., 2003; Snijders and Baerveldt, 2003). Previously, estimation of the parameters in such models has been based on a Markov chain Monte Carlo (MCMC) implementation of the method of moments (Snijders, 2001; Snijders and van Duijn, 1997; Snijders, 2004). In Paper III we generalize the class of stochastic actor-oriented models, and propose an MCMC algorithm for exploring the posterior distribution of the parameters. The generalized class of stochastic actor oriented models can handle un-directed, bipartite and valued social networks in addition to the dichotomous directed networks of the stochastic actor oriented models. The proposed solution for performing likelihood based inference is that, instead of dealing directly with the transition probabilities, the un-observed evolution of the network in between observations is treated as a latent variable. The observed data is then

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augmented (Tanner and Wong, 1987) by the latent variables in a way that enables parameter inference to be performed by MCMC techniques. 2.4. Paper IV: Model selection for longitudinal social networks. Paper IV continues the line of investigation of Paper III. Arguably, the main theoretical motivation behind models for longitudinal social networks is to infer what components are important in the dynamics of social interaction. This calls for statistical procedures for testing hypothesis, something which in the absence of procedures for conducting model selection, is limited to inspection of posterior credibility regions. The Bayesian paradigm is well suited for model selection but the relative complexity of this class of models prevents the use of any standard techniques for calculating the relevant quantities. Although an analytically tractable form for the likelihood function is not strictly necessary for performing parameter inference as shown in Paper III, most model selection techniques rely heavily on the assumption that the likelihood function is easy to evaluate. We define a family of models, with the property that they include the reciprocity model (Wasserman, 1977) as special case. Since there is an analytically tractable form for the likelihood function in the case of the reciprocity model (as shown by e.g. Wasserman, 1977; Leenders, 1995b,a), the scheme of Chib and Jeliazkov (2001) for estimating the marginal likelihood can be adapted, thus providing the posterior distribution over a set of models in this family of models. If the analysis is restricted to comparisons between nested models, the likelihood function does not have to be evaluated and model selection need not be restricted to models with the reciprocity model as a special case. References Batchelder, W., Kumbasar, E., and Boyd, J. (1997). “Consensus analysis of threeway social network data.” Journal of Mathematical Sociology, 22, 29–58. Bernard, H., Killworth, P., and Sailer, L. (1980). “Information accuracy in social network data IV: A comparison of clique-level structure in behavioral and cognitive network data.” Social Networks, 2, 191–218. Bernardo, J. M. (1979). “Reference posterior distributions for Bayesian inference.” Journal of the Royal Statistical Society, 41, 113–147. With discussion. — (1997). “Non–informative priors do not exist.” Journal of Statistical Planning and Inference, 65, 159–189. With discussion. Bernardo, J. M. and Smith, A. F. M. (1994). Bayesian Theory. New York: Wiley. Besag, J. (2000). “Markov chain Monte Carlo for statistical inference.” working paper, University of Washington, Center for Statistics and the Social Sciences. Bondonio, D. (1998). “Predictors of accuracy in perceiving informal social networks.” Social Networks, 20, 301–330. Box, G. P. and Tiao, G. C. (1973). Bayesian Inference in Statistical Analysis. Reading, Massachusetts: Addison–Wesley. Carrington, P. J., Scott, J., and Wasserman, S., eds. (2004). Models and Methods in in Social Network Analysis. New York: Cambridge University Press.

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Casciaro, T. (1998). “Seeing things clearly: social structure, personality, and accuracy in social network perception.” Social Networks, 20, 331–351. Casciaro, T., Carley, K., and Krackhardt, D. (1999). “Positive affectivity and accuracy in social network perception.” Motivation and Emotion, 23, 285–306. Chib, S. and Jeliazkov, I. (2001). “Marginal likelihood from the MetropolisHastings output.” J. Amer. Statist. Assoc., 96, 453, 270–281. Coleman, J., Katz, E., and Mentzel, H. (1957). “The diffusion of an innovation among physicians.” Sociometry, 20, 253–270. Corander, J., Dahmstr¨om, K., and Dahmstr¨om, P. (1998). “Maximum likelihood estimation for Markov graphs.” Research report 1998:8, Stockholm University, Department of Statistics. — (2002). “Maximum likelihood estimation for exponential random graph model.” In Contributions to Social Network Analysis, Information Theory, and Other Topics in Statistics; A Festschrift in honour of Ove Frank , ed. J. Hagberg, 1–17. Stockholm: Dept. of Statistics, Stockholm University. Dahmstr¨om, K. and Dahmstr¨om, P. (1993). “ML-estimation of the clustering parameter in a Markov Graph model.” Research report 1993:4, Stockholm University, Department of Statistics. Erd¨os, P. (1947). “Some remarks on the theory of graphs.” Bull. Amer. Math. Soc., 53, 292–294. Fishman, G. S. (1996). Monte Carlo – Concepts, Algorithms, and Applications. New York: Springer–Verlag. Corrected third printing, 1999. Frank, O. (1971). “Statistical Inference in Graphs.” Ph.D. thesis, Stockholm. — (1988). “Random sampling and social networks. A survey of various approaches.” Math. Sci. Humaines, 104, 19–33. — (1997). “Composition and structure of social networks.” Mathematique Informatique et Science Humaines, 137, 11–23. Frank, O. and Nowicki, K. (1993). “Exploratory statistical analysis of networks.” In Quo vadis, graph theory? , vol. 55 of Ann. Discrete Math., 349–365. Amsterdam: North-Holland. Frank, O. and Strauss, D. (1986). “Markov Graphs.” Journal of the American Statistical Association, 81, 832–842. Gilks, W., Richardson, S., and Spiegelhalter, D. J., eds. (1996). Markov Chain Monte Carlo in Practice. London: Chapman & Hall. Granovetter, M. (1973). “The strength of weak ties.” American Journal of Sociology, 81, 1287–1303. Hancock, M. (2000). “Progress in statistical modeling of drug user and sexual networks.” Unpublished manuscript, University of Washington, Center for Statistics and the Social Sciences. Holland, P. and Leinhardt, S. (1977a). “A dynamic model for social networks.” Journal of Mathematical Sociology, 5, 5–20. — (1977b). “Social structure as a network process.” Zeitschrift f¨ ur Soziologie, 6, 386–402. — (1981). “An exponential family of probability distributions for directed graphs (with discussion).” Journal of the American Statistical Association, 76, 33–65.

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Janson, S., Luczak, T., and Rucinsnki, A. (2000). Random graphs. New York: Wiley. Jeffreys, H. (1961). Theory of Probability. Oxford: University Press. Killworth, P. and Bernard, H. (1979). “Information accuracy in social network data III, or: A comparison of triadic structure in behavioural and cognitive data.” Social Networks, 2, 19–46. Koskinen, J. (2002a). “Bayesian analysis of cognitive social structures with covariates.” Research Report 2002:3, Department of Statistics, Stockholm University. — (2002b). “Bayesian analysis of perceived social networks.” Research Report 2002:2, Department of Statistics, Stockholm University. Krackhardt, D. (1987). “Cognitive social structures.” Social Networks, 9, 109–134. Leenders, R. (1995a). “Models for network dynamics: a Markovian framework.” Journal of Mathematical Sociology, 20, 1–21. — (1995b). “Structure and Influence. Statistical Models for the Dynamics of Actor Attributes, Network Structure and their Interdependence.” Ph.D. thesis, Amsterdam. Lindley, D. V. (1965). Introduction to Probability and Statistics from a Bayesian Viewpoint. Cambridge: Cambridge University Press. Vol. 1 and 2. Milgram, S. (1967). “The small world problem.” Psychology Today, 22, 61–67. Moreno, J. (1934). Who Shall Survive? Foundations of Sociometry, Group Psychotherapy and Sociodrama. Washington, D.C.: Nervous and Mental Disease Publishing Co. Reprinted in 1978 (Third Edition) by Becon House, inc., Beacon, NY. Newcomb, T. (1961). The Acquaintance Process. New York: Holt, Rinehart and Winston. Pattison, P. (1994). “Social cognition in context – some applications of social network analysis.” In Advances in Social Network Analysis: Research in the Social and Behavioral Sciences, eds. S. Wasserman and J. Galaskiewicz, 79– 109. Thousand Oaks: Sage. Pattison, P. and Robins, G. (2002). “Neighbourhood-based models for social networks.” Sociological Methodology, 32, 301–337. Robins, G. (1998). “Personal Attributes in Interpersonal Contexts: Statistical Models for Individual Characteristics and Social Relationships.” Ph.D. thesis, University of Melbourne, Department of Psychology. Robins, G. and Pattison, P. (2004). “Interdependencies and social processes: dependence graphs and generalized dependence structures.” Forthcoming in Models and Methods in Social Network Analysis, P. J. Carrington and J. Scott and S. Wasserman (eds), New York: Cambridge University Press. Snijders, T. and Baerveldt, C. (2003). “A multilevel network study of the effects of delinquent behavior on friendship Evolution.” Journal of Mathematical Sociology, 27, 123–151. Snijders, T. A. B. (1996). “Stochastic actor-oriented models for network change.” Journal of Mathematical Sociology, 21, 149–172. Also published in Doreian and Stockman (1997).

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— (2001). “The statistical evaluation of social network dynamics.” Sociological Methodology, 30, 361–395. — (2002). “Markov chain Monte Carlo estimation of exponential random graph models.” Journal of Social Structure, 3, 2. — (2004). “Models for Longitudinal Network Data.” To appear as Chapter 11 in P. Carrington, J. Scott, and S. Wasserman (Eds.), Models and methods in social network analysis. New York: Cambridge University Press. Snijders, T. A. B., Pattison, P. E., and Robins, G. (2004). “New specifications for exponential random graph models.” In preparation. Snijders, T. A. B. and van Duijn, M. A. J. (1997). “Simulation for statistical inference in dynamic network models.” In Simulating social phenomena, eds. R. Conte, R. Hegselmann, and P. Terna, 493–512. Berlin: Springer. — (2002). “Conditional maximum likelihood estimation under various specifications of exponential random graph models.” In Contributions to Social Network Analysis, Information Theory, and Other Topics in Statistics; A Festschrift in honour of Ove Frank , ed. J. Hagberg, 117–134. Stockholm: Dept. of Statistics, Stockholm University. Tanner, M. A. and Wong, W. (1987). “The calculation of posterior distributions by data augmentation (with discussion).” Journal of the American Statistical Association, 82, 528–550. Tierney, L. (1994). “Markov chains for exploring posterior distributions.” Ann. Statist., 22, 4, 1701–1762. With discussion and a rejoinder by the author. van de Bunt, G. (1999). “Friends by Choice. An Actor-Oriented Statistical Network Model for Friendship Networks through Time.” Ph.D. thesis, Amsterdam. van de Bunt, G., Van Duijn, M., and Snijders, T. (1999). “Friendship networks through time: An actor-oriented statistical network model.” Computational and Mathematical Organization Theory, 5, 167–192. van Duijn, M. A., Zeggelink, E. P. H., Huisman, M., and Stokman, F. M. (2003). “Evolution of sociology freshmen into a friendship network.” Journal of Mathematical Sociology, 27, 153–191. Wasserman, S. (1977). “Stochastic Models for Directed Graphs.” Ph.D. thesis, University of Harvard, Department of Statistics. — (1980a). “Analyzing social networks as stochastic processes.” Journal of the American Statistical Association, 75, 280–294. — (1980b). “A stochastic model for directed graphs with transition rates determined by reciprocity.” Sociological Methodology, 11, 392–412. Wasserman, S. and Faust, K. (1994). Social Network analysis: Methods and Applications. New York and Cambridge: Cambridge University Press. Wasserman, S. and Galaskiewicz, J., eds. (1994). Advances in Social Network Analysis: Research in the Social and Behavioral Sciences. Thousand Oaks: Sage. Wasserman, S. and Pattison, P. (1996). “Logit models and logistic regression for social networks: I. An introduction to Markov graphs and p∗ .” Psychometrika, 61, 401–425.

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Wasserman, S. and Robins, G. (2004). “An introduction to random graphs, dependence graphs, and p∗ .” Forthcoming in Models and Methods in Social Network Analysis, , P. J. Carrington and J. Scott and S. Wasserman (eds), New York: Cambridge University Press. Wong, G. Y. (1987). “Bayesian models for directed graphs.” Journal of the American Statistical Association, 82, 140–148.