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.

翻译：考虑边根据具有指数族转移概率的离散时间马尔可夫链开关的纵向网络。我们刻画了当这些网络的联合分布也是具有相同参数的指数族时的条件，从而改进了数据约简。进一步，我们证明这些链的置换一致子类可解释为同一状态空间上的独立同分布序列。随后，我们将这些思想应用于时间指数随机图模型——该类模型特别适合置换一致性，并讨论了均值参数收敛性、二元独立性及可交换性。我们的框架有助于引入新网络模型；简化了文献中部分网络模型和自回归模型的分析，包括使某些模型的最大似然估计具有闭式表达式；并促进了从渐近理论或单观测指数随机图模型出发将标准工具应用于纵向网络马尔可夫链。