## 2 Graph-Based Smoothness Harmonic
### Smoothness Harmonic  
The concept of smoothness harmonics is based on the eigendecomposition analysis of the graph Laplacian. Formally, considering the previously defined graph \(\mathcal{G}\) with \(N\) nodes (RoIs) and \(N\times N\) edges (connectomes), we can compute the symmetric graph Laplacian \(\mathbf{L}(\mathcal{G})\) using \(\mathbf{L}(\mathcal{G})=D-A\), where \(D\) is the degree matrix and \(A\) is the adjacency matrix. In an undirected graph, the adjacency matrix \(A\) can be represented as \(\mathcal{E}\), where \(A_{ij}=\mathcal{E}_{ij}\). This adjacency matrix can be derived from the functional connectome [17]. The degree matrix \(D\) can be computed as \(D=\sum_{j=1}^{N}A(i,j)\).  
According to graph theory [18; 19], smoothness refers to the degree of variation exhibited by a graph signal \(f\) across the nodes of a graph \(\mathcal{G}\). Conventionally, the smoothness of a graph can be quantified using the quadratic form of the graph Laplacian, \(f^{\mathsf{T}}\mathbf{L}(\mathcal{G})f\). In the context of fMRI signals, which are predominantly characterized by ultra-slow oscillations, we propose measuring the smoothness of a graph based on the least-varying component, namely the first eigenvector associated with the smallest eigenvalue in the eigendecomposition of the graph Laplacian.  
Let \(k\) denote the number of eigenvectors. We can obtain the eigenvector matrix \(\mathbf{U}_{k}\) as follows: \(\mathbf{L}(\mathcal{G})=\mathbf{U}_{k}\Lambda\mathbf{U}_{k}^{\mathsf{T}}\), where \(\Lambda\) is a diagonal matrix with the Laplacian eigenvalues \(\lambda_{1},\lambda_{2},...,\lambda_{k}\), and \(\mathbf{U}_{k}\) is the orthonormal matrix consisting of the corresponding eigenvectors (arranged as columns).  
Since the first eigenvector of the eigendecomposed graph Laplacian is associated with the low-frequency and slowly varying component of the graph [20], we extract the first eigenvector, denoted as \(\mathbf{U}_{k=1}\), to represent the slowly varying harmonic of the graph [17]. These slowly varying harmonics in cortical dynamics are believed to reflect changes in the brain's local network architecture, such as the reconfiguration of small-scale functional modules or alterations in the strength of local connections [21]. By utilizing the extracted slowly varying harmonic \(\mathbf{U}_{k=1}\), we derive the resulting smoothness harmonics (\(\mathbb{E}_{\text{temp}}(f)\)), which quantify the extent to which a graph signal \(f\) varies on the slowly varying harmonic, as follows:  
\[\mathbb{E}_{\text{temp}}(f)=(f\otimes\mathbf{U}_{k=1})^{\mathsf{T}}\cdot f, \tag{1}\]  
where \(\otimes\) denotes the broadcastable element-wise multiplication, allowing the resulting matrix \((f\otimes\mathbf{U}_{k=1})^{\mathsf{T}}\) to obtain the size of \([T,N]\).  
Figure 1: Time-varying brain activity as a graph. The sizes and widths of nodes and edges represent the strength of their respective feature values. We characterize the brain activity that evolves through \(T\) time points (representing the duration of recorded brain activity) as a graph. In this graph, the nodal features represent the region-of-interest (RoI) brain activations \(\mathcal{V}\), while the edges represent the connectomes \(\mathcal{E}\) between RoIs. As time progresses, i.e., from \(\mathcal{G}_{t_{1}}\) to \(\mathcal{G}_{t_{n}}\), the nodal features change accordingly, while the edge features remain the same.