Cycle-Spectrum Preservation Loss

1. Integration point: add the loss to a graph autoencoder, molecular graph generator, graph diffusion model, or teacher-student graph distillation objective. Compute it after decoding edge logits A_hat and before the optimizer update; do not change the message-passing layers initially. 2. Pseudocode: obtain target adjacency A and decoder probabilities A_hat=sigmoid(logits); for every k in 3..K, enumerate canonical simple cycles C in the target graph and calculate C_k(A)=sum_C product(A[e]); calculate C_k(A_hat) using the same cycle list for reconstruction, or enumerate cycles in the union graph when generated edge support can differ; normalize both vectors using s_k; form L_cycle=sum alpha[k]*(s[k]-s_hat[k])**2; backpropagate L_task+lambda_cycle*L_cycle. For generation, use a random subset of candidate cycles or capped DFS per minibatch. 3. The paper contributes the exact simple-cycle statistic and its squarefree trace/log-determinant derivation. The implementation computes cycle lists empirically; use exact computation for K<=4 on graphs below roughly 100 nodes. Set alpha_k inversely proportional to minibatch variance so rare triangles and abundant 4-cycles contribute comparably. 4. First experiment: train a VGAE or graph diffusion generator on QM9, comparing reconstruction validity, degree-distribution error, and triangle/4-cycle divergence with and without the loss. Also test synthetic graphs with identical degree distributions but different planted cycle counts. Sweep lambda_cycle in {0.01,0.1,1}. Success is lower held-out cycle-statistic error and improved validity or downstream property prediction without a material increase in edge reconstruction loss.