Fig. 5

Visual illustration of the multi-scale graph wavelet transform, corresponding to the mathematical formulations in Eqs. (2)–(8). Given the input node features \(X^{(0)}\) and the adjacency matrix, the model first performs multi-hop neighborhood aggregation using a propagation operator \(G(\cdot )\) to obtain features at different scales \(X^{(s_1)}, X^{(s_2)}, \ldots , X^{(s_J)}\). These features are concatenated to form a unified multi-scale representation \(U\). To capture structural dynamics, the model computes first-order differences (\(F_1\)) between adjacent scale features and propagates \(U\) further to derive higher-level features \(U^{(1)}, U^{(2)}, \ldots , U^{(t)}\), from which second-order differences (\(F_2\)) are calculated. Finally, all features are concatenated and passed through a PReLU-activated linear layer to produce the final multi-scale representation \(Z\). This approach enables the model to capture both local interactions and global graph structure.