\section{Global node relevance} \label{app: node relevance}
We propose a diagnostic statistic for quantifying the relevance each node has on the network time series correlation structure. After selecting model order $(\hat{p}, [\hat{s_k}] )$ based on some data-dependent criterion, e.g., quick visual inspection of Corbit plots, we can find the largest active $r$-stage order term $\hat{r} := \max \{ [\hat{s_k}] \}$, which estimates $r^*$. Using the data-dependent $\hat{r}$, we compute the active weights matrix, i.e., $\mathbf{W}$ constrained to nodes at a maximum distance of $\hat{r}$ (no path is longer between active nodes), this matrix is $\bmat{W} \odot \bmat{S}_{(\hat{r})}$, where $\bmat{S}_{(\hat{r})} = \sum_{\ell = 1}^{\hat{r}} \bmat{S}_\ell$. Combining model order selection and the resulting autocovariance bound, as given in Definition \ref{def: nacf}, motivates the global relevance index given below
\begin{equation*}
    \globindex (X_{i, t} | \, \hat{r} ) := \left (\sum_{j = 1}^{d} [\bmat{W} \odot \bmat{S}_{(\hat{r})}]_{j i} \right )
   \left \{ \underset{l \in \mathcal{K}}{\max} \left ( \sum_{j = 1}^{d} [\bmat{W} \odot \bmat{S}_{(\hat{r})}]_{j l} \right ) \right \}^{-1},
\end{equation*}
which computes the ratio between the largest column sum for active nodes and a particular node. We interpret this as the relevance each node has globally on the correlation structure. By \eqref{eq: nacf inequality}, this node upper bounds the covariance structure in the sense that removing it would change the autocovariance bound, the resulting (P)NACF values and possibly our choice of model.

\begin{figure}[H]
    \centering
    \includegraphics[scale=0.6]{images/local_relevance_REF.pdf}
    \caption{Neighbouring node relevance for Trust REF imposed on the network. Values are from zero to one, and equal one for the local node, i.e., REF. See Section~\ref{sec: node influence} for a more detailed discussion. }
    \label{fig: REF local relevance}
\end{figure}

\begin{figure}[H]
    \centering
    \includegraphics[width=0.95\linewidth]{images/top_10_nodes.pdf}
    \caption{Top 10 relevant nodes as given by~\eqref{eq: global index} for $r^* = 1$. The two clusters are well separated between the south and north of England.}
    \label{fig: top 10 nodes}
\end{figure}

\begin{figure}[H]
    \centering
    \includegraphics[width=0.95\linewidth]{images/top_60_nodes.pdf}
    \caption{Top 60 relevant nodes as given by~\eqref{eq: global index} for $r^* = 1$. The two clusters in Figure \ref{fig: top 10 nodes} are now connected. Further, points not previously between relevant nodes in the north are now present. Interestingly, the previous most relevant nodes in the south are the ones that connect London to Bristol and Southampton, i.e., new relevant nodes are concentrated near London.}
    \label{fig: top 60 nodes}
\end{figure}

\begin{figure}[H]
    \centering
    \resizebox{\textwidth}{!}{\includegraphics{images/node_relevance_network_plot_A4_size.pdf}}
    \caption{Node relevance hierarchy imposed on the network. }
    \label{fig: node relevance network A4}
\end{figure}


