\subsection{Conditioning in the GNAR framework}
We will also use the following proposition.
\begin{proposition} \label{prop: var constrains}
Let $\boldsymbol{X}_t$ be a stationary $\GNAR$ process with full rank spectral matrix $\bmat{f} (\omega)$, static underlying network $\mathcal{G} = (\mathcal{K}, \mathcal{E})$ and maximum active $\rstage$ neighbourhood regression $r^*$. Then, we have that
\begin{equation*}
    (i, j) \notin \mathcal{A}_{r^*} \textit{ if and only if } X_{i, t} \perp X_{j, t + h} | \mathcal{A}^{(ij)} \textit{ at all lags } h.
\end{equation*}
\end{proposition}
\begin{proof}
Recall the VAR representation of a $\GNAR(p, [s_1, \dots, s_p])$ process; see \cite{gnar_paper}.
\begin{equation*}
    X_{i, t} = \sum_{k = 1}^{p} \big( \phi_{k}^{ii} X_{i , t- k} + \underset{j \in \mathcal{N} (i)}{\sum} \phi_{k}^{ij} X_{j , t- k}
    \big )+ u_{i, t},
\end{equation*}
for which the autoregressive matrices are given by 
$$\bmat{\Phi}_k = \{\diag(\alpha_{ik}) + \sum_{r = 1}^{s_k} \beta_{kr} (\bmat{W} \odot \bmat{S}_r ) \},$$
and the entries satisfy
\begin{equation*}
    [\bmat{\Phi}_k]_{ij} = \alpha_{ik} \mathbb{I}(i = j) + \beta_{k r} w_{i j} \mathbb{I}(i \neq j),
\end{equation*}
moreover, since $\phi_{k}^{ij} = \beta_{kr} w_{ij}$ and $\phi_{k}^{ji} = \beta_{kr} w_{ji}$ we have that
\begin{equation} \label{eq: ar coeffss equiv}
    \phi_{k}^{ij} \equiv 0 \iff \beta_{kr} \equiv 0 \iff \phi_{k}^{ji} \equiv 0,
\end{equation}
regardless of the connection weights between nodes, it is active $\rstage$ depth which activates a connection between two nodes. Also, that the characteristic matrix polynomial for $z \in \mathbb{C}$ is given by
$$ \bmat{\Phi}(z) := \bmat{I_d} - \sum_{k = 1}^{p} \bmat{\Phi}_k z^k, $$
and recall that $\bmat{S} (\omega)$ satisfies; see \cite{brockwell_davis},
\begin{equation} \label{eq: inverse spec coeffs}
    \bmat{S} (\omega) = \frac{2 \pi}{\sigma^2} \{\bmat{\Phi}(e^{\imath \omega})\}^T \bmat{\Phi}(e^{-\imath \omega}).
\end{equation}
By (\ref{eq: inverse spec coeffs}) we have that
\begin{equation*}
    [\bmat{S} (\omega)]_{ij} = \frac{2 \pi}{\sigma^2} \sum_{\ell = 1}^{d} [\{\bmat{\Phi}(e^{\imath \omega})\}^T]_{i \ell} [\bmat{\Phi}(e^{-\imath \omega})]_{\ell j},
\end{equation*}
hence, we also have that
\begin{equation} \label{eq: inv spec and ar coefs}
    [\bmat{S} (\omega)]_{ij} \equiv 0 \iff  \sum_{l = 1}^{d} [\bmat{\Phi}(e^{\imath \omega})]_{\ell i} [\bmat{\Phi}(e^{-\imath \omega})]_{\ell j} \equiv 0.
\end{equation}
Assume that $[\bmat{\Phi}(e^{\imath \omega})]_{\ell i} [\bmat{\Phi}(e^{-\imath \omega})]_{\ell j} \equiv 0$ for all $\ell \in \mathcal{K}$. Then, the sum in the right hand side of (\ref{eq: inv spec and ar coefs}) is equal to zero, and by combining Lemma \ref{lma: conditional equivalences} and (\ref{eq: inv spec and ar coefs}) we have that
\begin{align} \label{eq: inactive to condtitional independende}
    [\{\bmat{\Phi}(e^{\imath \omega})\}^T]_{i \ell} [\bmat{\Phi}(e^{-\imath \omega})]_{\ell j} \equiv 0 &\implies  [\bmat{S} (\omega)]_{ij} \equiv 0, \nonumber \\
    &\iff X_{i, t} \perp X_{j, t + h} | \mathcal{A}^{(ij)} \textit{ at all lags } h.
\end{align}
Now, suppose that $(i, j) \notin \mathcal{A}_{r^*}$, then $\delta (i, j) > 2 r^*$, so there does not exist $\ell \in \mathcal{K}$ such that $\delta (i, \ell) \leq r^*$ and 
$\delta (j, \ell) \leq r^*$, where the distances are computed on the underlying network $\mathcal{G}$. Thus, for all $X_{\ell, t} \in \mathcal{A}^{(ij)}$ we have that
\begin{align*} \label{eq: first prop cond independence}
    \delta (i, \ell) \leq r^* &\implies \delta (j, \ell) > r^*, \nonumber \\
    &\implies \phi_{k}^{\ell j} \equiv 0,
\end{align*}
which shows that $\ell$ and $j$ do not influence each other, by (\ref{eq: ar coeffss equiv}) and the above for all $\ell$ we have that
\begin{equation*}
    [\bmat{\Phi}(e^{-\imath \omega})]_{\ell j} \equiv 0 \implies [\{\bmat{\Phi}(e^{\imath \omega})\}^T]_{i \ell} [\bmat{\Phi}(e^{-\imath \omega})]_{\ell j} \equiv 0.
\end{equation*}
\par
Note that the argument above holds similarly if $ \delta (j, \ell) \leq r^*$, and that if $\delta (i, \ell) > r^*$ and $\delta (j, \ell) > r^*$, then we have that 
$\phi_{k}^{\ell j} \equiv 0 \equiv \phi_{k}^{\ell i},$
so by (\ref{eq: inactive to condtitional independende}) and the arguments above we have that
\begin{equation}  \label{eq: first prop cond independence end}
    (i, j) \notin \mathcal{A}_{r^*} \implies X_{i, t} \perp X_{j, t + h} | \mathcal{A}^{(ij)} \textit{ at all lags } h.
\end{equation}
We proceed to prove the other direction, assume that $(i, j) \in \mathcal{A}_{r^*}$, then there exists at least one other nodal time series such that its corresponding node $\ell \in \mathcal{K}$, satisfies $\delta (i, \ell) \leq r^*$ and  $\delta (j, \ell) \leq r^*$. Hence, by \eqref{eq: inv spec and ar coefs}, \eqref{eq: inverse spec coeffs}, and \eqref{eq: ar coeffss equiv} we have that 
\begin{equation*}
    (i, j) \in \mathcal{A}_{r^*} \implies  [\bmat{S} (\omega)]_{ij} \not\equiv 0,
\end{equation*}
which by Lemma \ref{lma: conditional equivalences} gives
\begin{equation} \label{eq: second prop cond independence end}
     (i, j) \in \mathcal{A}_{r^*} \implies X_{i, t} \not\perp X_{j, t + h} | \mathcal{A}^{(ij)} \textit{ at all lags } h.
\end{equation}
Finally, combining \eqref{eq: first prop cond independence end} and \eqref{eq: second prop cond independence end} finishes the proof.
\end{proof}