\subsection{Correlation structure remarks}
The ideas in this section show how stronger dependence between closer nodes can be exhibited either by pruning the latent graph $\tilde{\mathcal{G}}$ or by applying the soft-threshold operator given by (\ref{eq: cmplx sft}) to $\bmat{S} (\omega)$ (i.e., forms of reparametrization that are equivalent).

\begin{corollary} \label{cor: graph generalization}
Let $\mathcal{G}^{(1)} = (\mathcal{K}, \mathcal{E}^{(1)})$ be the graph associated to the strongest connections with edge set $\mathcal{E}^{(1)}$ given by (\ref{eq: r sequence equivalence}). Then, the graph $G = (\mathcal{K}, E)$ given by Definition \ref{def: conditional indep definition} is the same graph (i.e., $\mathcal{G}^{(1)} = G$).
\end{corollary}
\begin{proof}
    By Theorem \ref{th: pop_sparsity gnar} $(i, j) \in \mathcal{P}_1$ if and only if $\delta_{\mathcal{G}^{(1)}} (i, j) = 1$, and by definition $(i, j) \notin E$ if and only if $ X_{i, t} \perp X_{j, t + h} | \mathcal{A}^{(ij)} \textit{ at all } h$, thus, $E = \mathcal{E}^{(1)}$. Now assume that $\delta_{\mathcal{G}^{(1)}} (i, j) = s$, then 
    $(i, j) \in \mathcal{P}_s$, hence, the two nodal time series become conditionally correlated when active $\rstage$ regression is at least equal to $s$, which means that $\delta_{G} (i, j) = s$ for all nodal pairs, thus, for all $r \in \{1, \dots, r\}$ we have that 
    \begin{equation*}
        \delta_{\mathcal{G}^{(1)}} (i, j) = \delta_{G} (i, j),
    \end{equation*}
    whence considering that both graphs have the same set of nodes and edges finishes the proof.
\end{proof}
Corollary \ref{cor: graph generalization} shows how GNAR processes generalise graphical multivariate time series by constraining VAR models with the introduction of higher-order interactions, furthermore, it shows that the neighbourhood regression coefficients $\beta_{kr}$ exploit the correlation structure induced by $\mathcal{G}$ since they account for node-wise interactions common across the network and manage inclusion in active $\rstage$ conditional dependence paths.
\begin{remark} \label{rma: simplify to graphical models}
Constraining a stationary $\GNAR(p, [s_1, \dots, s_p])$ process to 
\newline
$1$-stage neighbourhood regression simplifies the correlation hierarchy in
% \newline 
Definition \ref{def: rstage hierarchy} to only distinguishing between $[\bmat{S} (\omega)]_{ij} \equiv 0$ and $[\bmat{S} (\omega)]_{ij} \not\equiv 0$. 
\end{remark}
Therefore, in that case, Theorem \ref{th: pop_sparsity gnar} recovers results of \cite{graph_series} and the generalised partial correlation graph given by Definition \ref{def: conditional indep definition} simplifies to a partial correlation graph given by Definition \ref{def{ partial corr graph}}. 
%\newpage
Remark \ref{rma: simplify to graphical models} shows that $\GNAR(p, [1, \cdots, 1])$ models can be interpreted as parsimonious partial correlation graphical models, which satisfy the Markov properties of graphical models; see \cite{graph_series} and \cite{lauritzen}. 
\begin{corollary} \label{cor: adjacency matrix of generalised corr graph}
     Let $\boldsymbol{X}_t$ be a stationary $\GNAR(p, [s_k])$ process with a static underlying network. Then, the adjacency matrix $\bmat{A}_{G}$ of the generalised partial correlation graph given by Definition \ref{def: conditional indep definition} is given by
        $$\bmat{A}_{G} = \bmat{S}_1 + \bmat{S}_2,$$
     where $\bmat{S}_1$ and $\bmat{S}_2$ are the $1$-stage and $2$-stage adjacency matrices of the underlying network.
\end{corollary}
\begin{proof}
    By Corollary \ref{cor: graph generalization} and Theorem \ref{th: pop_sparsity gnar} the edge set of the generalised partial correlation graph $E$ is equal to $\mathcal{E}^{(1)}$, thus,
    \begin{align*}
        (i, j) \in E &\iff (i, j) \in  \mathcal{E}^{(1)} \\
        &\iff \delta_{\mathcal{G}} (i, j) \leq 2, 
    \end{align*}
    where the distance is computed on the underlying network. Thus, if the distance between two nodes is at most two in the underlying network, their distance in the generalised partial correlation graph is equal to one. Thus,
    $$\delta_{\mathcal{G}} (i, j)\leq 2 \iff \delta_{G} (i, j) = 1,$$ 
    which allows us to compute the adjacency matrix of $G$ by summing the $1$-stage and $2$-stage adjacency matrices of $\mathcal{G}$.
\end{proof}
%\newpage
These results and ideas show the equivalence between using the generalised GNAR correlation structure $\mathcal{G}$ and the notions of different degrees of dependence in Definitions \ref{def: conditional indep definition} and \ref{def: rstage hierarchy} with the notions in \cite{corbit_paper}.