Phase-Margin Graph Propagation

(1) Integration point: use this construction in a graph-convolution or recurrent message-passing layer. Give each node two auxiliary scalar parameters a_i and phi_i, either learned globally per node or predicted from its current hidden state by a small linear head. Set v_i=softplus(a_i)+1e-4 and theta_i=pi*tanh(phi_i). Preserve the physical graph sparsity pattern and do not densify the adjacency. (2) Pseudocode: compute c_ij=cos(theta_i-theta_j); clamp it with c_safe=max(c_ij,epsilon_c), using epsilon_c=0.1 initially; compute w_ij=B_ij*v_i*v_j/c_safe; compute d_i=sum_{j:(i,j) in E}w_ij; set k_i=1/(2*d_i+softplus(r_i)+tau); set T_i=1/k_i-2*d_i; normalize messages with w_ij/sqrt((d_i+eps)(d_j+eps)); and apply h_next=h+eta*activation(W0*h+message). Add L_margin=lambda*mean(relu(tau-T_i)^2) if k_i is not hard-parameterized. (3) Computed from the paper: the sine/cosine operating-point dependence, edge weight w_ij, and diagonal correction T_i. Estimated empirically: amplitudes and phases if predicted from hidden states, together with eta, tau, epsilon_c, and lambda. For diagnostics, estimate the smallest eigenvalue of the symmetrized propagation Jacobian on small graphs. (4) First experiment: train a 6-layer GCN and a 10-step recurrent GNN on Cora, ogbn-arxiv subgraphs, and synthetic graphs with strong degree heterogeneity. Compare standard symmetric-normalized adjacency, a GCN with learned scalar edge gates, and this phase-margin layer at matched parameter count and FLOPs. Measure validation accuracy, gradient-norm variance, hidden-state norm across recurrent steps, and the largest stable learning rate. Success means fewer exploding or vanishing runs, stable propagation for more steps, or improved accuracy at equal depth; a useful target is at least twice the stable propagation depth without accuracy degradation.