Hodge-Selective Edge Dynamics

Integrate this into an edge-state graph neural network immediately before or after the ordinary edge message/update MLP. Inputs are an incidence matrix B, edge features F in R^{m x d}, and optionally a learned or data-derived cut target F_*; apply the same projection independently to each feature channel. First construct the sparse graph Laplacian L = B B^T. For small graphs, compute P_C explicitly using a pseudoinverse of L; for larger graphs, avoid materializing it and compute P_C X by solving L Y = B X with a pinned node or zero-mean constraint, then returning B^T Y. Compute P_H X = X - P_C X. A layer should execute: (1) compute ordinary edge message Z = MLP([F, edge_attributes, node_messages]); (2) split Z_C=P_C Z and Z_H=P_H Z; (3) update F_new = F_* + (1-alpha) Z_C + (1-kappa) Z_H, or use learned residual branches F_new = F + g_C(Z_C)+g_H(Z_H) followed by scalar gates constrained as alpha=sigmoid(a), kappa=sigmoid(b); (4) optionally feed B F_new to the node update. The paper's exact contribution is the orthogonal decomposition and independent contraction law; the MLP transformations, gates, and target F_* are engineering choices. Estimate projection correctness with ||P_C^2-P_C|| and ||P_C P_H||, and monitor separate norms ||P_C F|| and ||P_H F||. The first experiment should use synthetic ring, grid, and random connected graphs on a graph signal forecasting task, comparing an ordinary edge-GNN residual block, a single projected block, and the proposed two-rate block at matched parameters. Measure long-horizon rollout error, gradient norm, and node prediction accuracy after injecting a large cycle-flow perturbation. Success means lower rollout error and faster recovery from harmonic perturbations without degrading cut-space accuracy.