Longitudinal networks are becoming increasingly relevant in the study of dynamic processes characterised by known or inferred community structure. Generalised Network Autoregressive (GNAR) models provide a parsimonious framework for exploiting the underlying network and multivariate time series. We introduce the community-$\alpha$ GNAR model with interactions that exploits prior knowledge or exogenous variables for analysing interactions within and between communities, and can describe serial correlation in longitudinal networks. We derive new explicit finite-sample error bounds that validate analysing high-dimensional longitudinal network data with GNAR models, and provide insights into their attractive properties. We further illustrate our approach by analysing the dynamics of $\textit{Red, Blue}$ and $\textit{Swing}$ states throughout presidential elections in the USA from 1976 to 2020, that is, a time series of length twelve on 51 time series (US states and Washington DC). Our analysis connects network autocorrelation to eight-year long terms, highlights a possible change in the system after the 2016 election, and a difference in behaviour between $\textit{Red}$ and $\textit{Blue}$ states.

翻译：纵向网络在具有已知或推断社区结构的动态过程研究中日益重要。广义网络自回归（GNAR）模型为利用底层网络与多元时间序列提供了简约的框架。本文提出社区-α GNAR交互模型，该模型利用先验知识或外生变量分析社区内及社区间的交互作用，并能描述纵向网络中的序列相关性。我们推导了新的显式有限样本误差界，验证了使用GNAR模型分析高维纵向网络数据的有效性，并揭示了其优良特性。通过分析1976年至2020年美国总统选举期间红州、蓝州和摇摆州的动态变化（即51个时间序列单元上长度为12的时间序列），我们进一步展示了该方法的应用价值。分析结果表明：网络自相关性与八年选举周期存在关联，2016年大选后系统可能出现结构性变化，且红州与蓝州的行为模式存在显著差异。