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Network time series are becoming increasingly important across many areas inscience and medicine and are often characterised by a known or inferredunderlying network structure, which can be exploited to make sense of dynamicphenomena that are often high-dimensional. For example, the Generalised NetworkAutoregressive (GNAR) models exploit such structure parsimoniously. We use theGNAR framework to introduce two association measures: the network and partialnetwork autocorrelation functions, and introduce Corbit (correlation-orbit)plots for visualisation. As with regular autocorrelation plots, Corbit plotspermit interpretation of underlying correlation structures and, crucially, aidmodel selection more rapidly than using other tools such as AIC or BIC. Weadditionally interpret GNAR processes as generalised graphical models, whichconstrain the processes' autoregressive structure and exhibit interestingtheoretical connections to graphical models via utilization of higher-orderinteractions. We demonstrate how incorporation of prior information is relatedto performing variable selection and shrinkage in the GNAR context. Weillustrate the usefulness of the GNAR formulation, network autocorrelations andCorbit plots by modelling a COVID-19 network time series of the number ofadmissions to mechanical ventilation beds at 140 NHS Trusts in England & Wales.We introduce the Wagner plot that can analyse correlations over different timeperiods or with respect to external covariates. In addition, we introduce plotsthat quantify the relevance and influence of individual nodes. Our modellingprovides insight on the underlying dynamics of the COVID-19 series, highlightstwo groups of geographically co-located `influential' NHS Trusts anddemonstrates superior prediction abilities when compared to existingtechniques.

New tools for network time series with an application to COVID-19 hospitalisations