Curvature-certified cycle suppression

(1) Integrate the method into a message-passing GNN or differentiable graph-rewiring layer. The exact integration point is the binary adjacency A, or a learned edge logit S before top-k sparsification. During training, compute a curvature penalty on the current graph and add lambda_kappa times R_kappa to the supervised task loss. Apply the penalty only when the application favors locally coherent, nonredundant connectivity; low-curvature cycles may be useful for molecular, maze, or relational tasks.

(2) Pseudocode:

input: adjacency A, node features X, labels Y
for each training step:
    E = current hard edges from A or top_k_edges(sigmoid(S))
    sampled_edges = random subset of E
    for each edge (x,y) in sampled_edges:
        construct m_x_alpha and m_y_alpha from the one-hop neighborhoods
        solve W1 = min_P sum_i,j P[i,j] * shortest_path_distance(i,j)
            subject to P @ ones = m_x_alpha
                       P.T @ ones = m_y_alpha
                       P >= 0
        kappa_hat[x,y] = (1 - W1) / (1 - alpha)
    R_kappa = mean(relu(0.5 - kappa_hat)^2)
    predictions = GNN(X, E)
    loss = task_loss(predictions, Y) + lambda_kappa * R_kappa
    update network parameters and S with AdamW

(3) The exact paper-derived computation is the transport linear program, graph shortest-path cost, and 1/2 threshold. Estimate the limiting LLY curvature using alpha values 0.8, 0.9, and 0.95, then extrapolate kappa_alpha divided by 1-alpha to alpha equal to 1. For a cheap implementation, use alpha=0.9, sample 10 to 25 percent of edges each step, cache neighborhood supports, and recompute all edges every 5 to 10 epochs. Estimate W_1 with an exact LP on small graphs or a Sinkhorn approximation for larger graphs. The edge-sampling rate, alpha, transport approximation, and lambda_kappa are empirical choices.

(4) First experiment: use a 3-layer GCN and GAT on Cora and Citeseer, plus synthetic graphs formed from 5-cycles with random chords. Compare the baseline, random edge dropout at the same expected edge count, and curvature regularization at matched FLOPs. Record validation accuracy, calibration, oversmoothing measured by pairwise node-embedding cosine similarity, gradient norm variance, and the fraction of edges with estimated curvature below 1/2. The method is supported if it improves validation accuracy or calibration at equal compute, reduces oversmoothing and unstable runs, and decreases low-curvature edges. On synthetic C5 graphs, verify the limitation predicted by the paper: strong regularization can destroy task-relevant cycle information, so accuracy should exhibit a measurable tradeoff as lambda_kappa increases.