Einstein Edge Normalization

Integrate this into the edge-weight or attention-logit construction of a message-passing GNN, before the first graph convolution and optionally as a differentiable regularizer during training. For a synthetic regular sun with cycle length g and cycle-vertex degree d, use the following procedure: (1) set all pendant-edge conductances to 1; (2) compute q=max(6-g,0); (3) compute a_star=((4-q)+sqrt((4-q)^2+16*(d-2)))/4; (4) assign conductance a_star to every cycle edge; (5) normalize each node's incoming conductances by their sum and use them in place of standard 1/degree weights. For a general graph, classify short local neighborhoods into approximate cycle and branch roles, estimate the local outer-neighbor distance d(x_prime,y_prime) by bounded Dijkstra, and initialize each edge weight w_j with a positive parameter satisfying w_j=softplus(theta_j). A cheaper trainable version uses the loss L_E=lambda_E Var_e[kappa_e], where kappa_e is computed from local shortest-path distances using kappa_e=0.5*(3-d_outer/w_e) whenever the balanced geodesic condition holds; clip or stop-gradient through shortest paths. Pseudocode: compute local distances; compute kappa; total_loss=task_loss+lambda_E*variance(kappa); update theta with AdamW. The closed-form a_star and monotonicity are taken directly from the paper; the general-neighborhood curvature estimate and the value of lambda_E are empirical approximations. First experiment: train 3-layer GCN and GAT models on cycle-plus-leaf graph classification and node-label propagation, comparing degree normalization, learned unconstrained edge gates, and Einstein initialization with or without L_E. Use 1k-node graphs and equal parameter/FLOP budgets. Success means lower validation loss, reduced seed variance, and faster convergence at equal steps; a useful target is a 10% improvement in accuracy or a 20% reduction in steps to baseline accuracy.