Longitudinal network models and permutation-uniform Markov chains

We offer a general approach to modeling longitudinal network data, including exponential random graph models (ERGMs), that vary according to certain discrete-time Markov chains. We connect conditional and Markovian exponential families, permutation-uniform Markov chains, various (temporal) ERGMs, and statistical considerations such as dyadic independence and exchangeability. By removing models' temporal dependence but not interpretability, our approach simplifies analysis of some network and autoregressive models from the literature, including closed-form expressions for maximum likelihood estimators. We also introduce "exponential random $t$-multigraph models", motivated by our result on replacing $t$ observations of permutation-uniform Markov chains of graphs with single observations of corresponding multigraphs.