exponential family theory. It’s been worked out a long time ago and the best place to start is a seminal book by Barndorff-Nielsen (1978). The last relevant literature is that coming from the graphical modeling community. They work on related ideas that are actually closer than it would seem by just judging the commonality in terms used. A good pointer is Lauritzen and Spiegelhalter (1988).
Networks are very complex things, and how we choose to model them will reflects what we have tried to represent well and what we are essentially not trying to represent well. That choice is going to be driven much by the objectives we have in mind. This is an obvious point, a point that drives through much of scientific modeling, but I have found that it’s very important for the networks area because of the complexity of the underlying models and the complexity of the phenomena we are interested in.
What we are probably interested in first would be the nature of the relationships themselves, questions such as how the behavior of individuals depends on their location in a social network, or how the qualities of the individuals influence the social structure. Then we might be interested in how network structure influences processes that develop over a network. Dynamic or otherwise, the classic examples would include the spread of infection, the diffusion of innovations, or the spread of computer viruses, all of which are affected by the network structure.
Lastly—and I think this is of primary importance and why many people actually study networks, even though they will look at it only after understanding the relationships and the network structure—is that we are interested in the effect of interventions. If we change the network structure and/or the processes that develop over that network, how will the network structure play itself out, and how will the process itself be actually changed? If you make changes to the network almost certainly the consequences of those changes will not be obvious.
Another important point is that our objectives define our perspectives. There is a difference between a so-called network-specific viewpoint and a population process. By network-specific I mean we look at a given network. It might be sampled or have missing data in it, but our scientific objective is that particular network. That is in contrast with a population viewpoint, where we view the data we have, whether it’s complete or sampled, as a realization from some underlying social phenomena typically represented through a stochastic process, and we wish to understand the properties of that stochastic process. In that case, the observed network is conceptualized as a realization of a stochastic process.
I think a lot of the different approaches in the literature look very different because they have these different perspectives and objectives in that mind. For example, repeating a point