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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.