A deeper understanding of the relation between individual behavior and individual actions on one hand and the embeddedness of individuals in social structures on the other hand can be obtained by empirically studying the dynamics of individual outcomes and network structure, and how these mutually affect each other. In methodological terms, this means that behavior of individuals -indicators of performance and success, attitudes and other cognitions, behavioral tendencies -and the ties between them are studied as a social process evolving over time, where behavior and network ties mutually influence each other. We propose a statistical methodology for this type of investigation and illustrate it by an example. * We thank the Chief Scientists Office of the Scottish Home and Health Department for permission to use the data of the Teenage Friends and Lifestyle Study.

In applications to dependent data, first and foremost relational data, a number of discrete exponential family models has turned out to be near-degenerate and problematic in terms of Markov chain Monte Carlo simulation and statistical inference. We introduce the notion of instability with an eye to characterize, detect, and penalize discrete exponential family models that are near-degenerate and problematic in terms of Markov chain Monte Carlo simulation and statistical inference. We show that unstable discrete exponential family models are characterized by excessive sensitivity and near-degeneracy. In special cases, the subset of the natural parameter space corresponding to non-degenerate distributions and mean-value parameters far from the boundary of the mean-value parameter space turns out to be a lower-dimensional subspace of the natural parameter space. These characteristics of unstable discrete exponential family models tend to obstruct Markov chain Monte Carlo simulation and statistical inference. In applications to relational data, we show that discrete exponential family models with Markov dependence tend to be unstable and that the parameter space of some curved exponential families contains unstable subsets.

A model for network panel data is discussed, based on the assumption that the observed data are discrete observations of a continuous-time Markov process on the space of all directed graphs on a given node set, in which changes in tie variables are independent conditional on the current graph. The model for tie changes is parametric and designed for applications to social network analysis, where the network dynamics can be interpreted as being generated by choices made by the social actors represented by the nodes of the graph. An algorithm for calculating the Maximum Likelihood estimator is presented, based on data augmentation and stochastic approximation. An application to an evolving friendship network is given and a small simulation study is presented which suggests that for small data sets the Maximum Likelihood estimator is more efficient than the earlier proposed Method of Moments estimator.

Summary Dependent phenomena, such as relational, spatial and temporal phenomena, tend to be characterized by local dependence in the sense that units which are close in a well-defined sense are dependent. In contrast with spatial and temporal phenomena, though, relational phenomena tend to lack a natural neighbourhood structure in the sense that it is unknown which units are close and thus dependent. Owing to the challenge of characterizing local dependence and constructing random graph models with local dependence, many conventional exponential family random graph models induce strong dependence and are not amenable to statistical inference. We take first steps to characterize local dependence in random graph models, inspired by the notion of finite neighbourhoods in spatial statistics and M-dependence in time series, and we show that local dependence endows random graph models with desirable properties which make them amenable to statistical inference. We show that random graph models with local dependence satisfy a natural domain consistency condition which every model should satisfy, but conventional exponential family random graph models do not satisfy. In addition, we establish a central limit theorem for random graph models with local dependence, which suggests that random graph models with local dependence are amenable to statistical inference. We discuss how random graph models with local dependence can be constructed by exploiting either observed or unobserved neighbourhood structure. In the absence of observed neighbourhood structure, we take a Bayesian view and express the uncertainty about the neighbourhood structure by specifying a prior on a set of suitable neighbourhood structures. We present simulation results and applications to two real world networks with ‘ground truth’.