\section{Correlation structure induced by GNAR}
\label{sec: corr structure}

\setcounter{theorem}{1}

We prove Theorem~\ref{th: pop_sparsity gnar} from the main paper.

\begin{theorem} \label{th: pop_sparsity gnar}
    Let $\boldsymbol{X}_t$ be a stationary $\GNAR(p, [s_1, \dots, s_p])$ process with a static network structure $\mathcal{G} = (\mathcal{K}, \mathcal{E})$, full rank spectral matrix $\bmat{f} (\omega)$ and maximum active $\rstage$ depth $r^*$. Then, the inverse spectral matrix $\bmat{S}(\omega)$ and the node-wise distances $\delta (i, j)$ computed on the network $\mathcal{G}$ satisfy
        \par
        $\text{a. }$ There exists a partial correlation graph $\tilde{\mathcal{G}} = (\mathcal{K}, \tilde{\mathcal{E}})$ with the same set of nodes as $\mathcal{G}$ such that
        $$(i, j) \notin \tilde{\mathcal{E}} \textit{ if and only if }  \delta (i, j) \geq 2 r^* + 1.$$
        \par
        $\text{b.}$ There exists a cross-spectral hierarchy $\xi^{(1)} > \cdots > \xi^{(r^*)} >  \xi^{(r^* + 1)} = 0$ and an active $\rstage$ neighbourhood regression which satisfy
         $$\delta (i, j) \in \{2r - 1, 2r\} \textit{ if and only if  } \xi^{(r)} \leq |[\bmat{S}(\omega)]_{i j}| < \xi^{(r - 1)},$$
         for all $\omega \in (-\pi, \pi]$ and for all $r \in \{1, \dots, r^*\}$.
\end{theorem}

\subsection{Notation and preliminary definitions}
\label{subsec: notation and sets}
Assume that $\boldsymbol{X}_t \in \mathbb{R}^d$ is a stationary $\GNAR(p, [s_1, \cdots, s_p])$ process for which the underlying network structure $\mathcal{G} = (\mathcal{K}, \mathcal{E})$ is static and with full rank spectral matrix. Throughout this section the following notation and definitions are used.
\begin{itemize}
    \item $\boldsymbol{X}_t$ multivariate time series.
    \item $r^* = \max(s_1, s_2, \cdots, s_p)$, the maximum active $\rstage$ depth.
    \item $\mathcal{N} (i) := \cup_{r = 1}^{r^*} \mathcal{N}_{r} (i)$, the set of active $\rstage$ neighbours with respect to node $i$.
    \item $\bmat{f}(\omega)$ spectral matrix of $\boldsymbol{X}_t$.
    \item $\bmat{S}(\omega) = \{\bmat{f}(\omega)\}^{-1}$ inverse spectral matrix of $\boldsymbol{X}_t$. 
    \item $f(\cdot) \equiv 0$ the function $f$ is identically zero in its domain.
    \item $f_{i, j | \mathcal{A}^{(ij)}}(\omega)$ conditional cross-spectrum between $X_{i, t}$ and $X_{j, t}$ given all the other nodal time series.
    \item $X_{i, t} \perp X_{j, t + h} | \mathcal{A}^{(ij)}$ the two time series are uncorrelated at all lags ($h \in \mathbb{Z}$) given all the other ones in $\mathcal{A}^{(ij)} := \{ X_{l, t} : l \neq i, j \}$.
    \item $\{\bmat{S}_{\mathcal{A}^{(ij)}} (\omega) \}$ inverse spectral matrix constrained to the set $\mathcal{A}^{(ij)}$.
    \item $ \boldsymbol{f}_{i, \mathcal{A}^{(ij)}} (\omega)$ $i$th column of the spectral matrix without the $i$th row entry.
\end{itemize}
\begin{definition} \label{def: active r nodes}
    Let $\boldsymbol{X}_t$ be a stationary $\GNAR$ process with a static underlying network structure, the set of active nodes at $\rstage$ depth $r$, denoted by $\mathcal{A}_r$, is the set of nodal pairs which collide when considering paths with maximum length $r$. Formally,
    \begin{equation*}
        \mathcal{A}_r := \{ (i, j) \in \mathcal{K} \times \mathcal{K} : \delta (i, j) \leq 2 r \}.
    \end{equation*}
    where $\delta (i, j)$ is the length of the shortest path between nodes $i$ and $j$ on the underlying network.
\end{definition}
Essentially, $\mathcal{A}_r$ extends the notion of an edge between nodes to a path between nodes, this allows us to incorporate higher-order interactions into the GNAR framework. Active nodes are those which collide when considering paths of maximum length $r$, i.e., even if $\delta (i, j) > r$, there might be another node that is at a distance equal or shorter than $r$ from both $i$ and $j$. 
\begin{definition} \label{def: activating r nodes}
    The set of nodes which become active at $\rstage$ depth equal to $r$, denoted by $\mathcal{P}_r$, is the set of nodal pairs which activate at length $r$ and satisfy
    $$\mathcal{P}_{r} = \mathcal{A}_{r} - \mathcal{A}_{r - 1} .$$
\end{definition}
From above we have that $(i, j) \in \mathcal{P}_{r}$ if and only if $2r - 1 \leq \delta (i, j) \leq 2r$. Also, note that since $\mathcal{P}_{r}$ is the set of pairs which activate exactly at a distance equal to $r$ (i.e., the interaction between $i$ and $j$ has order $r$).

\input{corr_structure_props}

\input{corr_structure_conditioning}

\input{corr_structure_first_part}

\input{corr_structre_second_part_a}

\input{corr_structure_part_b}

\input{corr_strcuture_remarks}