  Network time series are becoming increasingly important across many areas in
science and medicine and are often characterised by a known or inferred
underlying network structure, which can be exploited to make sense of dynamic
phenomena that are often high-dimensional. For example, the Generalised Network
Autoregressive (GNAR) models exploit such structure parsimoniously. We use the
GNAR framework to introduce two association measures: the network and partial
network autocorrelation functions, and introduce Corbit (correlation-orbit)
plots for visualisation. As with regular autocorrelation plots, Corbit plots
permit interpretation of underlying correlation structures and, crucially, aid
model selection more rapidly than using other tools such as AIC or BIC. We
additionally interpret GNAR processes as generalised graphical models, which
constrain the processes' autoregressive structure and exhibit interesting
theoretical connections to graphical models via utilization of higher-order
interactions. We demonstrate how incorporation of prior information is related
to performing variable selection and shrinkage in the GNAR context. We
illustrate the usefulness of the GNAR formulation, network autocorrelations and
Corbit plots by modelling a COVID-19 network time series of the number of
admissions to mechanical ventilation beds at 140 NHS Trusts in England & Wales.
We introduce the Wagner plot that can analyse correlations over different time
periods or with respect to external covariates. In addition, we introduce plots
that quantify the relevance and influence of individual nodes. Our modelling
provides insight on the underlying dynamics of the COVID-19 series, highlights
two groups of geographically co-located `influential' NHS Trusts and
demonstrates superior prediction abilities when compared to existing
techniques.
