  Consider longitudinal networks whose edges turn on and off according to a
discrete-time Markov chain with exponential-family transition probabilities. We
characterize when their joint distributions are also exponential families with
the same parameter, improving data reduction. Further we show that the
permutation-uniform subclass of these chains permit interpretation as an
independent, identically distributed sequence on the same state space. We then
apply these ideas to temporal exponential random graph models, for which
permutation uniformity is well suited, and discuss mean-parameter convergence,
dyadic independence, and exchangeability. Our framework facilitates our
introducing a new network model; simplifies analysis of some network and
autoregressive models from the literature, including by permitting closed-form
expressions for maximum likelihood estimates for some models; and facilitates
applying standard tools to longitudinal-network Markov chains from either
asymptotics or single-observation exponential random graph models.
