Pole-Tuned Graph Residual Layer

Integrate the block as a propagation module in a GNN, before a trainable node-wise MLP. For adjacency matrix \(A\), construct a symmetric normalized Laplacian \(\mathcal L=I-D^{-1/2}AD^{-1/2}\), or use the exact Laplacian convention consistently when estimating eigenvalues. Estimate \(\lambda_2\) and \(\lambda_N\) with a few Lanczos iterations; for disconnected graphs, apply the method separately per connected component or add a small graph regularizer. Compute \(\kappa\), \(\gamma^*\), \(\alpha^*\), and \(\varepsilon^*\). Initialize \(X_0=H\), where \(H\) is the input node-feature tensor, and \(Y_0=0\). For each propagation block run X_new = X - eps * L @ X - alpha * Y; Y_new = Y - eps * L @ Y + X_new - X; then assign X,Y = X_new,Y_new. Apply LayerNorm and a trainable MLP to \(X\), with any conventional external skip connection kept identical across baselines. The paper supplies the gain formula and spectral stability interval; the graph eigenvalues, feature Dirichlet energy, and nonlinear-layer behavior are measured numerically. Test on Cora and ogbn-arxiv using 4, 8, and 16 propagation blocks. Compare a standard GCN layer, a fixed-coefficient residual propagation layer, and the pole-tuned two-state block at matched parameter count and message-passing FLOPs. Track validation accuracy, training loss, gradient norm, activation norm, node-feature variance, and Dirichlet energy \(\operatorname{tr}(X^T\mathcal LX)\). Success means higher accuracy at depth, slower collapse of node-feature variance than GCN oversmoothing, and no exploding activations or gradients on graphs with large spectral condition numbers.