Activity-Gated Spectral Message Passing

1. Integration point: implement this as a recurrent graph message-passing block for node features H in a GNN, preferably replacing a residual GCN layer. Let W be the sparse adjacency with nonnegative edge weights, V be a trainable feature projection, and maintain a bounded scalar state x_i for each node or compute x_i=sigmoid((Qh_i)_scalar). Compute a_i=sigmoid(u^T h_i+b) from the current source-node representation. 2. Pseudocode: initialize H and optional x; for each layer, compute gate g=1+delta*a with delta=softplus(raw_delta)-1; compute MvH = W @ (g[:,None] * (H @ V.T)); if spectral normalization is enabled, run 3-5 power iterations on M=W*diag(g) to estimate lambda_hat; choose eta so the target linear growth factor rho_target is 0.9-0.99 using eta=(rho_target+epsilon*zeta-1)/(beta*lambda_hat), clipped to [eta_min,eta_max]; update x=sigmoid(QH), H=H+eta*((1-x)*beta*MvH-zeta*H), then apply a standard output projection and optional LayerNorm. 3. The paper supplies the source-side multiplicative gate, saturation term, and threshold logic. Estimate lambda_hat empirically by sparse power iteration; estimate delta, beta, zeta, and u by gradient descent. For signed hidden features, use the gated propagation term directly but apply the saturation factor to a separate nonnegative scalar channel, or omit it and retain only the mathematically justified gated residual term. 4. First experiment: use a 3-6 layer model on Cora, ogbn-arxiv-subset, or a synthetic stochastic block model, comparing fixed-normalized GCN, GAT, ordinary residual GCN, and this module at equal hidden size and FLOPs. Measure validation accuracy, gradient norm, activation oversmoothing via pairwise cosine similarity across layers, and the estimated dominant eigenvalue. A successful result should show less oversmoothing and fewer unstable runs, or faster loss reduction at equal training steps, especially when node activity is correlated with class or community membership. Ablate the activity gate, spectral normalization, and saturation term separately.