\subsection{Preliminary propositions}
\label{sec: prep propositions}
\begin{proposition} \label{prop: distance for conditional independence}
    Assume that $\boldsymbol{X}_t$ is a stationary $\GNAR(p, [s_1, \dots, s_p])$ process for which the underlying network structure $\mathcal{G} = (\mathcal{K}, \mathcal{E})$ is static with the same notation as above. Then, 
    $(i, j) \notin \mathcal{A}_{r^*}$ if and only if $\mathcal{N} (i) \cap  \mathcal{N} (j) = \varnothing$, and $(i, j) \in \mathcal{P}_{r^*}$ if and only if $\delta (i, j) \in \{2 r^* - 1, 2 r^* \}$.
\end{proposition}
\begin{proof}
    Let $(i, j) \notin \mathcal{A}_{r^*}$. Note that if there is at least one $\ell \in  \mathcal{N} (i) \cap  \mathcal{N} (j)$, then such $\ell$ satisfies $\delta (i, \ell) \leq r^*$ and $\delta (j, \ell) \leq r^*$, which by Proposition \ref{prop: rstage one} gives $\delta (i, j) \leq 2 r^*$, and we have that $(i, j) \in \mathcal{A}_{r^*}$, a contradiction. If $\mathcal{N} (i) \cap  \mathcal{N} (j) = \varnothing$, then there does not exist $\ell \in \mathcal{K}$ such that
    \begin{equation} \label{eq: borough active set condition}
        \delta (i, \ell) \leq r^* \textit{ and   } \delta (j, \ell) \leq r^* ,
    \end{equation}
    which by Proposition \ref{prop: rstage one} gives that $\delta (i, j) \geq 2 r^* + 1$, which gives $(i, j) \notin \mathcal{A}_{r^*}$.
    \par
    Now, suppose that $\delta (i, j) > 2 r^*$, then, by Proposition \ref{prop: rstage one} we can find $\ell \in \mathcal{K}$ such that
    \begin{align*}
        \delta (i, j) &=  \delta \{j, \mathcal{N}_{r^*} (i) \} + \delta (i, \ell) \\
        &= \delta (i, \ell) + \delta (j, \ell),
    \end{align*}
    if $\ell \in \mathcal{N} (i)$, then $\delta (i, \ell) \leq r^*$, and we have that
    \begin{equation*}
        r^* + \delta (i, \ell) \geq \delta (i, \ell) + \delta (j, \ell) > 2 r^*,
    \end{equation*}
    thus, 
    \begin{equation*}
        \delta (j, \ell) > 2r^* - \delta (i, \ell) \geq r^*,
    \end{equation*}
    so we have that $\ell \notin \mathcal{N} (j)$, and that $\ell \in \mathcal{N} (i)$ which gives
    \begin{equation*}
        \ell \notin  \mathcal{N} (i) \cap  \mathcal{N} (j),
    \end{equation*}
   hence, if $\delta (i, j) > 2 r^*$, then $ \mathcal{N} (i) \cap  \mathcal{N} (j) = \varnothing$. The argument for $\ell \in \mathcal{N} (j)$ is the same. 
    Now, decrease $r^*$ to $(r^* - 1)$, since $(i, j) \in \mathcal{A}_{r^* - 1}$ if and only if $\delta (i, j) \leq 2 r^* - 2 < 2 r^*$, we have that $\mathcal{A}_{r^* - 1} \subseteq \mathcal{A}_{r^*}$, and that
    $$2 r^* - 1 \leq \delta (i, j) \leq 2 r^* \iff (i, j) \in \mathcal{A}_{r^*} - \mathcal{A}_{r^* - 1},$$
    hence, $(i, j) \in \mathcal{P}_{r^*}$ if and only if $\delta (i, j) \in \{2 r^* - 1, 2 r^* \}$.
\end{proof}

Recall from time series analysis that the best linear predictor up to a certain lag restricted to a subset of active nodes $\mathcal{A} \subseteq \mathcal{K}$ solves for the optimal linear filter with respect to mean squared error. Let $\mathcal{A}^{(ij)}$ be given as in Section \ref{subsec: notation and sets}. Then, the error of the optimal filter conditioning on $\mathcal{A}^{(ij)}$ is given by

\begin{equation} \label{eq: opt filter}
   u_{i, t}^{(ij)} := u_{i, t | \mathcal{A}^{(ij)} } = X_{i, t} - a_{i} - \sum_{k = 1}^{p} \sum_{l \in \mathcal{A}_{r^*}^{(j)}} 
   \gamma_{k}^{il} X_{l, t - k},
\end{equation}
where $\mathcal{A}_{r^*}^{(j)} = \mathcal{A}^{(ij)} \cap \mathcal{N} (i)$  and $a_i$ is the optimal mean constant; see \cite{brockwell_davis}. 
Following \cite{graph_series}, define the conditional dependence structure based on the cross-spectrum between the conditional time series $X_{i, t} | \mathcal{A}^{(ij)}$ and $X_{j, t} | \mathcal{A}^{(ij)}$ by noting that these two time series are uncorrelated at all lags if and only if the conditional cross-spectrum $f_{i, j | \mathcal{A}^{(ij)}}(\omega)$ is identically zero for all frequencies $\omega \in [-\pi, \pi]$. We will make use of the following result proved in \cite{graph_series}.
%\newpage
\begin{theorem}{[\cite{graph_series}]} \label{th: inv spec cross indentity}
Let $\boldsymbol{X}_t$ be a stationary multivariate time series, if the spectral matrix $\bmat{f}(\omega)$ has full rank, then
\begin{align*}
    f_{i, j | \mathcal{A}^{(ij)}}(\omega) &:= f_{u_{i}^{(ij)} u_{j}^{(ij)}} (\omega), \\
    f_{i, j | \mathcal{A}^{(ij)}}(\omega) &= - [\bmat{S} (\omega)]_{ij}.
\end{align*}
\end{theorem}
The relevance of Theorem \ref{th: inv spec cross indentity} is that it enables us to identify conditional independence by looking at the inverse spectral matrix $\bmat{S} (\omega)$, rather than by computing all possible conditional cross-spectrum values. Hence, the inverse spectral matrix is related to multivariate time series models as the concentration (inverse-covariance) matrix is to Gaussian Graphical Models; see \cite{graph_lasso}. 
This allows us to state results from \cite{graph_series} and \cite{brillinger} in the following Lemma. 
%\newpage
\begin{lemma}{[\cite{graph_series}, and \cite{brillinger} ]} \label{lma: conditional equivalences}
Let $\boldsymbol{X}_{t} \in \mathbb{R}^d$ be a stationary multivariate process, with the same notation as above, then 
\begin{align} \label{eq: conditional independence equivalence}
    &X_{i, t} \perp X_{j, t + h} | \mathcal{A}^{(ij)} \textit{ for all } h  \nonumber \\
    &\iff \cov (u_{i, t}^{(ij)}, u_{j, t + h}^{(ij)} ) \equiv 0, \nonumber \\
    &\iff f_{i, j | \mathcal{A}^{(ij)}}(\omega) \equiv 0, \nonumber \\
    &\iff [\bmat{f}(\omega)]_{ij} - \boldsymbol{f}_{i, \mathcal{A}^{(ij)}} (\omega) \{ \bmat{S}_{\mathcal{A}^{(ij)}} (\omega) \}^{-1}  
    \boldsymbol{f}_{\mathcal{A}^{(ij)}, j} (\omega) \equiv 0, \nonumber \\
    &\iff [\bmat{S} (\omega)]_{ij} \equiv 0.
\end{align}
\end{lemma}
%\newpage
Following \cite{graph_series} we define a partial correlation graph for multivariate time series as follows.
\begin{definition}{[\cite{graph_series}]} \label{def{ partial corr graph}}
    Let $\boldsymbol{X}_t$ be a stationary multivariate time series, the graph $G = (K, E)$ such that
    \begin{equation*}
        X_{i, t} \perp X_{j, t + h} | \mathcal{A}^{(ij)} \textit{ at all } h \iff (i, j) \notin E,
    \end{equation*}
    is the undirected partial correlation graph for which the presence of an edge indicates that the two nodal time series are not uncorrelated given all the other ones for at least one lag.
\end{definition}
A partial correlation graph satisfies the Markov properties if we define the absence of an edge as the relation $ X_{i, t} \perp X_{j, t + h} | \mathcal{A}^{(ij)} \textit{ at all } h$, which makes explicit the relationship between nodes $i$ and $j$ (i.e., $i$ and $j$ are uncorrelated at all lags given all the other nodes in the graph); see \cite{graph_series}. 
%\newpage
\par
GNAR generalises this idea by noting that the conditional independence can be extracted from paths in the graph rather than just from existing edges. 
\begin{definition} \label{def: conditional indep definition}
Let $\boldsymbol{X}_t$ be a stationary multivariate process with an underlying network structure $\mathcal{G} = (\mathcal{K}, \mathcal{E})$, the $\GNAR$ generalised partial correlation graph $G = (\mathcal{K}, E)$ is the undirected graph which satisfies $\delta_{G} (i, j) > r^*$ if and only if the nodal time series $X_{i, t}$ and $X_{j, t}$ are uncorrelated at all lags given all the other nodes, which is denoted by
\begin{equation*}
     X_{i, t} \perp X_{j, t + h} | \mathcal{A}^{(ij)} \textit{ at all } h \iff \delta_{G} (i, j) > r^*.
\end{equation*}
\par
Furthermore, the strength of dependence can be recovered from $G$ in the sense that if $\delta_{G} (i, j) = r$ for some $r \in \{1, \dots, r^* \}$, then constraining the $\GNAR$ model to maximum $\rstage$ depth of $(r - 1)$ will uncorrelate the two nodal time series (i.e., $ X_{i, t} \perp X_{j, t + h}$ at all lags $h$).
\end{definition}

Intuitively, Definition \ref{def: conditional indep definition} proposes that two nodes in the underlying network $\mathcal{G}$ are conditionally uncorrelated if and only if the shortest path between their active $\rstage$ neighbours has a length of at least one (i.e., they do not have common active $\rstage$ neighbours). This condition expresses mathematically that no active regression path is long enough to contain both $i$ and $j$. Before continuing we mention the following result of \cite{graph_series}. 
%\newpage
\begin{proposition}{[\cite{graph_series}]} \label{prop: conditional residual}
Let $\boldsymbol{X}_t$ be a multivariate time series with partial correlation graph $\mathcal{G} = (\mathcal{K}, \mathcal{E})$ and spectral matrix $\bmat{f}(\omega)$ which has full rank for all frequencies $\omega \in [-\pi, \pi]$. Then,
    $$    (i, j) \notin \mathcal{E}  \iff [\bmat{S}(\omega)]_{ij} \equiv 0.
    $$
\end{proposition}