## 3 Materials and Methods
### Preliminaries  
Notations used in the paper are shown in Table 1. Functions are shown in Table 2.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Notation** & **Description** \\ \hline \(N\) & Number of nodes \\ \(F\) & Dimension of node attributes \\ \(H_{i}\)\(P\) & Window size of historical and future \\ \(\chi\in\mathbb{R}^{B\times N\times F\times H}\) & Node attributes that record historical water demand data \\ \(y\), \(\
ot\)\(\
ot\)\(\in\mathbb{R}^{N\times F\times P}\) & Real and predicted water demand data \\ \(G=(V,E,A)\) & A graph defined by nodes, edges, and adjacency matrix \\ \hline \hline \end{tabular}
\end{table}
Table 1: Notations and descriptions.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Function** & **Description** \\ \hline \(softmax(\cdot)\) & The softmax activation function \\ \(\sigma(\cdot)\) & The sigmoid activation function \\ \(permute(\cdot)\) & The dimension conversion function \\ \(squeeze(\cdot)\), \(unsqueeze(\cdot)\) & The dimension compression and expansion functions \\ \hline \hline \end{tabular}
\end{table}
Table 2: Functions and descriptions.  
Problem definition: Given the historical water demand monitoring data for \(N\) residential neighborhoods in a region, our task is to predict the future water demand for each residential neighborhood. Inspired by relevant work, we define the relationship between \(N\) residential neighborhoods belonging to a region as an undirected weighted graph \(G=(V,E,A)\). Where \(V\) is a set of nodes, i.e., \(|V|=N\), \(E\) is a set of edges and \(A\in\mathbb{R}^{N\times N}\) is the weighted adjacency matrix. We denote the historical water demand record on G at time \(t\) as a graph signal \(X_{(t)}\in\mathbb{R}^{N\times F}\). Our task is to predict \(P\) future graph signals using the \(H\) historical graph signals, which are expressed formally as shown in Equation (1).  
\[\left[X_{(t-H+1)},\ X_{(t-H+2)},\ldots,X_{(t)}\right]\overset{f(\cdot)}{ \Rightarrow}\left[\hat{X}_{(t+1)},\hat{X}_{(t+2)},\ldots,\hat{X}_{(t+P)}\right] \tag{1}\]  
Graph convolution network: The correlation analysis in spatial dimension relies on graph convolution operations, which can be categorized into spatial-based and spectral-based methods [23]. [PERSON] et al. [24] introduced spectral graph theory on graph convolution to provide faster processing efficiency and wider versatility compared to spatial-based methods. Subsequently, the graph convolution network is represented as the product of the graph signal \(X_{(t)}\) and the graph convolution kernel \(g_{\theta}\), as shown in Equation (2).  
\[g_{\theta}*X_{(t)}=Ug_{\theta}(\Lambda)U^{T}X_{(t)} \tag{2}\]  
where \(U\) is the matrix constructed from the eigenvectors of the Laplacian matrix corresponding to the graph \(G\), \(\Lambda\) is the diagonal matrix formed by the eigenvalues of this Laplacian matrix, and \(g_{\theta}(\Lambda)\) is the convolution kernel. To reduce the complexity of graph convolution, Chebyshev polynomials are used as convolution kernels [25], as shown in Equation (3).  
\[g_{\theta}(\Lambda)=\sum\
olimits_{k=0}^{K-1}\alpha_{k}T_{k}\left(\widetilde{ \Lambda}\right) \tag{3}\]  
where \(T_{k}(\cdot)\) is a Chebyshev polynomial, \(\alpha_{k}\) is the coefficient of the polynomials of order, and \(\widetilde{\Lambda}\) is the diagonal matrix of scaled eigenvalues.  
Self-attention mechanism: The basic idea is to calculate the correlation between the positions in the sequence, and normalize the correlation into weights [26]. The correlation between positions within a sequence is usually represented by the result of a vector dot product. The computation process of attention weights can be viewed as a query-key-value mechanism, where the query is the current input vector, while the keys and values are the other input vectors. The output is as follows.  
\[Attention(Q,K,V)=softmax(\frac{QK^{T}}{\sqrt{d_{k}}})V \tag{4}\]