## 3 Proposed Graph-Grid-Based Clustering for Energy Hole Detection
represented as \(\{\chi_{\ell}\}_{/=0}^{N-1}\) with corresponding nonnegative eigenvalues \(0\leq\lambda_{0}\leq\ldots\leq\lambda_{N-1}\). Laplacian matrix L, is diagonalized by the eigenvector matrix \(\mathcal{X}=[\chi_{0},\ldots,\chi_{N-1}]\) so that \(\ell=\mathcal{X}\Lambda\mathcal{X}^{\mathsf{T}}\), where is the diagonal eigenvalue matrix. \(\mathcal{L}=I_{N}-D^{-1/2}\
ot\in D^{-1/2}\) is a normalised version [1; 1]. Instead of complex exponentials, the eigenvectors \(\{\chi_{\ell}\}_{/=0}^{N-1}\) of the Laplacian matrix L that meet the orthogonality criterion are employed as decomposition bases for graph-structured data. On a graph, the Fourier transform of a given signal _f(n)_ is defined as Equation (15):  
\[\widehat{f}(\lambda_{\ell})=\sum_{n=0}^{N-1}\chi_{\ell}^{\mathsf{T}}(n)f(n)=X ^{\mathsf{T}}f \tag{15}\]  
Inverse Fourier transformation is represented by Equation (16):  
\[f(n)g(n);g(n)\{\theta_{,r}\}_{=0}^{N-1}\mathcal{G}(\Lambda).f \tag{16}\]  
Convolution is turned into a point-wise product in the Fourier domain as well as reconverted into the vertex domain utilizing the graph Fourier transform as well as the convolution theorem, as in Equation (17):  
\[f\times g=X\Big{(}\sum_{k=0}^{K}\theta_{k}\Lambda^{k}\Big{)}\mathcal{T}^{ \mathsf{T}}f=\Big{(}\sum_{k=0}^{K}\theta_{k}\Big{(}X^{\top}\Lambda^{k}X^{ \mathsf{T}}\Big{)}\Big{)}f=\sum_{k=0}^{K}\theta_{k}\mathcal{L}^{k}f \tag{17}\]A convolution kernel is the graph convolution operation of two graph signals, \(f(n)\) and \(g(n)\), and its transform, \(\mathcal{G}(\Lambda)\). A set of free parameters in the Fourier domain, i.e., Laplacian eigenspace, is used to construct this kernel. Convolution is then written as Equation (18):  
\[\text{f*}g=\mathcal{X}\text{diag}(\theta_{0},\dots,\theta_{N-1})\mathcal{X}^{T }f=\mathcal{X}\mathcal{G}(\Lambda)\mathcal{X}^{T}f\mathcal{G}(\Lambda) \tag{18}\]  
\(\mathcal{G}(\Lambda)\) as an eigenvalue polynomial function: As illustrated in Equation (19), a rapid localised convolution based on low-order polynomial approximation was proposed:  
\[B_{cell}=\sum_{1}^{i}Bi \tag{10}\]0  
which \(\{\theta_{k}\}_{k=0}^{K}\) is the polynomial order, and \(Ki\) is a vector of polynomial coefficients. \(K\) is a tiny positive integer, such as 3, for example. Convolution is then rewritten as Equation (20):  
\[B_{cell}=\sum_{1}^{i}Bi \tag{10}\]1  
The convolution is performed by \(K\) multiplications of the sparse matrix \(\mathrm{L}\), which speeds up computation by avoiding the composition procedure. The following is the updated version of Equation (21) for layer \(l\):  
\[ \begin{array}{c}\widehat{h}_{i}^{l+1}=O_{h}^{l}H_{k=1}\Big{(}\sum_{j\in N_{i }}w_{i,j}^{k,l}\big{)}^{\
u k,l}h_{j}^{l}\Big{)}\\ \hat{c}_{i}^{l+1}=O_{e}^{l}H_{k=1}\Big{(}\widehat{w}_{i,j}^{k,l}\Big{)}\end{array} \]  
where  
\[w_{i,j}^{k,l}=\text{softmax}_{j}\Big{(}\widehat{w}_{i,j}^{k,l}\Big{)} \tag{21}\]  
\[\widehat{w}_{i,j}^{k,l}=\bigg{(}\frac{Q^{k,l}h_{i}^{l}\cdot K^{k,l}h_{j}^{l}}{ \sqrt{d_{k}}}\bigg{)}\cdot E^{k,l}\hat{c}_{i,j}^{l}\]  
with \(Q^{k,l},K^{k,l},V^{k,l},E^{k,l}\in\mathbb{R}d_{k},O_{h}^{l},O_{e}^{l}\in \mathbb{R}^{k\times d},k\in\{1,2,\dots,H\}\) relates the number of attention heads, and where \(O_{h}^{l}\in\mathbb{R}^{d\times d},V^{k,l}\in\mathbb{R}^{d_{k}\times d},d_{k} \mathrm{H}\) denotes the number of heads. Note that \(h\)\(l\)\(i\) is \(i\)-th node's feature at \(l\)-th layer in Equation (22). \[\text{cut}(S_{k},S_{k})=\sum_{v_{i}\in S_{k},v_{j}\in S_{j}}e\big{(}v_{i},v_{ j}\big{)}, \tag{22}\]  
where \(S_{k}\) is \(k\)-th set of a given graph, \(\widehat{S}_{k}\) indicates the remaining sets, except \(S_{k}\), and \(e\big{(}v_{i},v_{j}\big{)}\) is the edge between vertices \(v_{i}\) and \(v_{j}\).