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\title{Correlation Structure Induced by a GNAR Framework and The Generalised Correlation Graph}
\author{Daniel Salnikov}


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\begin{center}
    \section*{Abstract}
\end{center}
     This work connects the generalised network autoregressive process (GNAR) formulation with partial correlation graphs and introduces hierarchies of cross-correlation which distinguish between different degrees of relevance among nodes. A GNAR framework induces a specific correlation structure which can be interpreted as a generalisation of a partial correlation graph which enables the identification of higher-order interactions from a latent generalised correlation graph for which distances between edges encode the conditional correlation structure of the underlying GNAR process. Finally, the results are illustrated with a simple example.
\newpage
\section*{Correlation Structure Induced by GNAR}
\begin{theorem} \label{th: pop_sparsity gnar}
    Let $\boldsymbol{X}_t$ be a stationary $\GNAR(p, [s_1, \cdots, s_p])$ 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 correlation structure induced by $\mathcal{G}$ 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
        $$(i, j) \notin \tilde{\mathcal{E}} \textit{ if and only if }  d(i, j) \geq 2 r^* + 1.$$
        \par
        $\text{b.}$ There exists a cross-spectrum hierarchy $\xi^{(1)} > \cdots > \xi^{(r^*)} >  \xi^{(r^* + 1)} = 0$ and an active $\rstage$ neighbouxiod regression which satisfy
         $$d(i, j) \in \{2r - 1, 2r\} \textit{ if and only if } \xi^{(r)} \leq |[\bmat{S}(\omega)]_{i j}| < \xi^{(r - 1)}$$
         for all $\omega \in [-\pi, \pi]$ and for all $r \in \{1, \cdots, r^*\}$, where $d(i, j)$ is computed on the network $\mathcal{G}$.
\end{theorem}
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