\subsection{Proof of the first direction in Theorem \ref{th: pop_sparsity gnar}.b}
Before proving the second part we mention the following notions, and recall that our underlying motivation and assumption is that the larger $r$ is the weaker the dependence between $\rstage$ neighbours is. First, we introduce the complex soft-threshold operator.
\begin{definition} \label{def: complex sft-thrsh}
    The complex soft-threshold operator is given by
    \begin{align} \label{eq: cmplx sft}
        \sft(z; \rho) : \mathbb{C} &\rightarrow \mathbb{C} \nonumber \\
        z &\mapsto (|z| - \rho)_{+} e^{\imath \phi}.
    \end{align} 
\end{definition}
In Definition \ref{def: complex sft-thrsh}, $\phi = \arg(z) \in [0, 2 \pi]$, $\rho \geq 0$, $\imath$ denotes the imaginary unit (i.e., $\imath^2 = -1$), $(\cdot)_{+}$ denotes the positive part operator, and recall that $|z| e^{\imath \phi} = z$.
\par
Note that if the argument $\phi \in \{0, \pi \}$, then $\sft(z; \rho)$ recovers the real-valued soft-threshold operator.
Next, we state a mathematical form of distinguishing between different degrees of relevance between nodes. Motivated by distance as a measure of dissimilarity, assume that there exists a cross-correlation hierarchy and define $\rstage$ neighbourhood inclusion, with maximum active stage $r^*$, as follows.

\begin{definition} \label{def: rstage hierarchy}
    Let $\boldsymbol{X}_t$ be a stationary $\GNAR$ process with static underlying structure $\mathcal{G} = (\mathcal{K}, \mathcal{E})$ and conditional cross-correlation hierarchy 
    $0 < \tau^{(r^*)}_h < \cdots < \tau^{(r)}_h < \tau^{(r - 1)}_h < \cdots < \tau^{(1)}_h$. Then, $(i, j) \in \mathcal{A}_r \textit{ if and only if }$ the following conditions hold
    \begin{align*}
      \sft (| \cov (u_{i, t}^{(ij)}, u_{j, t + h}^{(ij)} )|;  \tau^{(r - 1)}_h) &\equiv 0, \\
      \sft (| \cov (u_{i, t}^{(ij)}, u_{j, t + h}^{(ij)} )|;  \tau^{(r)}_h) &\not\equiv 0.
    \end{align*}
\end{definition}
Definition \ref{def: rstage hierarchy} makes explicit our assumption that the further nodes are in the graph, the less dependent they are. It enables us to reparameterize a GNAR process by punching out weak dependencies just by decreasing the length of maximum $\rstage$ neighbourhood regression (i.e., if we increase the conditional cross-correlation threshold, then nodes further from each other become conditionally independent). 
\par
A possible way of interpreting Definition \ref{def: rstage hierarchy} is that it makes explicit the idea of a correlation index by distinguishing between different thresholds $\tau^{(r)}_h$ of relevance for which the underlying GNAR structure $\mathcal{G}$ captures the different strengths.
%\newpage
We are ready to prove the second part (i.e., $b$) of Theorem \ref{th: pop_sparsity gnar}, and start with the only if ($\Leftarrow$) direction.
\begin{proof}
A useful consequence of Definition \ref{def: rstage hierarchy} is that if the cross-correlation hierarchy exists, then by Theorem \ref{th: inv spec cross indentity} there exists a cross-spectrum hierarchy 
$\infty > \xi^{(1)} >  \xi^{(2)} > \cdots > \xi^{(r^*)} >  \xi^{(r^* + 1)} = 0 $ such that
\begin{equation*}
    \tau^{(r)}_h \leq |\cov (u_{i, t}^{(ij)}, u_{j, t + h}^{(ij)} )| < \tau^{(r - 1)}_h \iff 
     \xi^{(r)} \leq |[\bmat{S}(\omega)]_{ij}| < \xi^{(r - 1)},
\end{equation*}
for all $h$ and at all $\omega \in [-\pi, \pi]$, thus, defining $\rstage$ neighbourhood inclusion as in Definition \ref{def: rstage hierarchy} guarantees the existence of a cross-spectrum hierarchy.
\par
Now, it is possible to define an $\rstage$ sequence of inverse spectral matrices as
\begin{equation} \label{eq: r stage inv matrix}
    \bmat{S}^{(r)}(\omega) := \sft \big( \bmat{S} (\omega) + \xi^{(r)} \diag\{e^{\imath \phi_{ii}}\};  \xi^{(r)} \big),
\end{equation}
where $\phi_{ii}$ is the argument of the diagonal entries in $\bmat{S} (\omega)$ and $\sft(z; \xi)$ is applied entry-wise to $\bmat{S} (\omega)$. Note that the diagonal of $\bmat{S} (\omega)$ does not change for any $r \in \{1, \cdots, r^*\}$.
And, that the sequence satisfies 
$$ \normx{\bmat{S}^{(r)}(\omega) }_0 \leq \normx{\bmat{S}(\omega) }_0,$$ 
for all active $r$, and since $\xi^{(r + 1)} < \xi^{(r)}$, also, for non-empty GNAR frameworks
\begin{equation} \label{eq: nested inverse spectral matrices}
     \normx{\bmat{S}^{(r)}(\omega) }_0 < \normx{\bmat{S}^{(r + 1)}(\omega) }_0,
\end{equation}
where $\normx{ \cdot }_0$ counts the number of non-zero entries in the matrix. The sequence above makes explicit the different strengths of dependence between nodes, as $r$ decreases the $\bmat{S}^{(r)}(\omega)$ become sparser which results in a model with fewer nodes influencing each other; see Section 4 in \cite{corbit_paper}. Assume that $\bmat{f} (\omega)$ has full rank, then each matrix in the sequence $\{\bmat{S}^{(r)} (\omega) \}$ is non-singular because we are removing entries from linearly independent rows/columns, rather than modifying the entries and introducing colinearity, i.e., the sparse matrices resemble the identity more as we remove entries.
\par
Define the sequence of partial correlation graphs 
$\mathcal{G}^{(r)} = (\mathcal{K}, \mathcal{E}^{(r)})$ as the one which satisfies $(i, j) \notin \mathcal{E}^{(r)}$ if and only if $[\bmat{S}^{(r)}(\omega)]_{ij} \equiv 0$.
By Theorem \ref{th: inv spec cross indentity}, Proposition \ref{prop: var constrains} and Proposition \ref{prop: distance for conditional independence}
we have that
\begin{equation} \label{eq: r sequence equivalence}
    (i, j) \notin \mathcal{E}^{(r)}  \iff \delta (i, j) \geq 2r + 1, 
\end{equation}
where $\delta (i, j)$ is computed on the underlying network $\mathcal{G}$. Also, by (\ref{eq: nested inverse spectral matrices}) the sequence of edge sets satisfies
\begin{equation} \label{eq: edge set sequence}
     \mathcal{E}^{(1)} \subset \cdots \subset \mathcal{E}^{(r)} \subset \mathcal{E}^{(r + 1)} \subset \cdots  \subset \tilde{\mathcal{E}} = \mathcal{E}^{(r^*)},
\end{equation}
where $\tilde{\mathcal{E}}$ is defined as in (\ref{eq: end first part}). Note that once a pair of edges become conditionally independent they stay so since the soft-threshold operator maps that entry to zero for all larger $\xi^{(r)}$ in the cross-spectrum hierarchy, this idea allows us to prune the latent graph $\tilde{\mathcal{G}}$ by removing edges based on the strength (i.e., magnitude) of the conditional cross-spectrum $f_{i, j | \mathcal{A}^{(i j)}}(\omega)$; see Lemma \ref{lma: conditional equivalences}. Now consider a pair of nodes $i$ and $j$ which satisfy
\begin{equation} \label{eq: second part heart}
    \xi^{(r)} \leq |[\bmat{S}(\omega)]_{ij}| < \xi^{(r - 1)},
\end{equation}
for all $\omega \in [-\pi, \pi]$, then, applying the soft-threshold given by Definition \ref{def: complex sft-thrsh} to $\bmat{S} (\omega)$ gives
$(i, j) \in \mathcal{E}^{(r)}$ and $(i, j) \notin \mathcal{E}^{(r - 1)}$ in the graph sequence, thus, we have that 
\begin{align} \label{eq: second part inv equivalences}
    (\ref{eq: second part heart}) &\implies (i, j) \in \mathcal{E}^{(r)} - \mathcal{E}^{(r - 1)}, \nonumber \\
    (i, j) \in \mathcal{E}^{(1)} &\implies (i, j) \in \bigcap_{r = r^*}^{1} \mathcal{E}^{(r)}, \nonumber \\
    (i, j) \in \mathcal{E}^{(r)} &\implies (i, j) \in \bigcap_{s = r^*}^{r} \mathcal{E}^{(s)} \textit{ and } 
    (i, j) \notin \bigcap_{s = r - 1}^{1} \mathcal{E}^{(s)}.
\end{align}

Following the ideas above, consider the graph $\mathcal{G}^{(1)} = (\mathcal{K}, \mathcal{E}^{(1)})$ associated to the strongest connections with edge set $\mathcal{E}^{(1)}$, which is the one with the strongest dependence between nodes as proposed in Definition \ref{def: rstage hierarchy}. 
\par
Using \eqref{eq: second part heart} and \eqref{eq: second part inv equivalences} we have that if $(i, j) \in \mathcal{E}^{(1)}$, then $i$ and $j$ are connected throughout the whole sequence given by (\ref{eq: r stage inv matrix}), and that the edge set sequence given by (\ref{eq: edge set sequence}) satisfies
\begin{align} \label{eq: second part inverse end}
    \mathcal{A}_{1} &= \mathcal{E}^{(1)}, \nonumber \\
    \mathcal{A}_{2} - \mathcal{A}_{1} &= \mathcal{E}^{(2)} - \mathcal{E}^{(1)}, \nonumber \\
    &\cdot \nonumber \\
    \mathcal{A}_{r} - \mathcal{A}_{r - 1} &= \mathcal{E}^{(r)} - \mathcal{E}^{(r - 1)}, \nonumber \\
    &\cdot \nonumber \\
    \mathcal{A}_{r^*} - \mathcal{A}_{r^* - 1} &= \mathcal{E}^{(r^*)} - \mathcal{E}^{(r^* - 1)}.
\end{align}
Moreover, computing the shortest path between nodes $i$ and $j$ in the graph $\mathcal{G}^{(1)}$ can be done as follows. 
\begin{itemize}
    \item  If $(i,j) \in \mathcal{E}^{(1)}$, then $\delta (i, j) = 1$,
    \item  else if $(i,j) \in \mathcal{E}^{(2)} - \mathcal{E}^{(1)}$, then since $(i,j)$ are $1$-stage neighbours in $\mathcal{G}^{(2)}$ there exists another node $\ell$ such that $(i, j, \ell)$ form a triangle in $\mathcal{G}^{(2)}$ and $(i, \ell), (\ell, j) \in \mathcal{E}^{(1)}$, hence, $\delta (i, j) = 2$ in $\mathcal{G}^{(1)}$, 
    \item else if $(i,j) \in \mathcal{E}^{(r)} - \mathcal{E}^{(r - 1)}$, $\dots$ gives $\delta (i, j) = r$ in $\mathcal{G}^{(1)}$.
\end{itemize}
Sequentially carrying this one-step computation on the whole sequence $\{ \mathcal{E}^{(r)}\}$ results in 
\begin{equation*}
    \delta (i, j) = r \mathbb{I}\{(i, j) \in \mathcal{E}^{(r)} - \mathcal{E}^{(r - 1)} \}, 
\end{equation*}
where $\mathcal{E}^{(0)} = \varnothing$. Note that these distances correspond to $\mathcal{G}^{(1)}$, and that since all active neighbourhood paths remain active if the active length increases by one (i.e., $\mathcal{A}_{r - 1} \subseteq \mathcal{A}_{r}$), then, by (\ref{eq: second part inverse end}) we have that 
\begin{equation} \label{eq: end first side part b}
    \xi^{(r)} \leq |[\bmat{S}(\omega)]_{ij}| < \xi^{(r - 1)} \implies (i, j) \in \mathcal{A}_r - \mathcal{A}_{r - 1} = \mathcal{P}_{r}.
\end{equation}
\end{proof}
Furthermore, this shows that defining $\rstage$ neighbourhood inclusion based on Definition \ref{def: rstage hierarchy} results in the same $\mathcal{A}_{r}$ as defining inclusion based on the underlying network $\mathcal{G}$ under the assumption that larger distances between nodes imply weaker dependence (i.e., conditional correlation).