Defect-Localized Cycle Positional Encoding

Integrate the feature into a graph transformer or GNN as an additional node or edge positional channel. For a graph containing a long cycle, identify approximate cycle backbones using a cycle-basis routine; place one coordinate on each cycle edge and append coordinates for pendant or off-cycle edges. Construct a sparse symmetric matrix R: for an unmodified cycle edge index j, set R[j,j]=-1 and R[j,j-1]=R[j,j+1]=1/2; for edges incident to a defect, replace these entries with local weighted-neighborhood couplings and add branch coordinates using the same local coupling rule. Run K=20-50 normalized power iterations, z_next=R z/(||R z||_2+epsilon), starting from random vectors or several localized seeds; retain the eigenvector with the largest Rayleigh quotient lambda=z^T R z/(z^T z). Map edge values to node values by averaging incident absolute values, then concatenate this scalar and optionally its signed version to node embeddings. If the largest Rayleigh quotient is nonpositive, use z=0 and fall back to ordinary positional features. The paper supplies the backbone operator, essential-spectrum argument, and reason a positive defect mode should localize; finite-graph defect entries, cycle detection, and threshold tau must be estimated empirically. First experiment: compare a 4-layer GIN, GraphGPS, and a small graph transformer on synthetic graphs consisting of cycles of lengths 32-512 with one pendant tree whose attachment position determines the label. Compare no positional encoding, Laplacian eigenvectors, random-walk features, and this defect mode at equal hidden size. Measure test accuracy versus cycle length, power-iteration overhead, localization ratio (mass within radius r of the defect), and embedding norm stability. Success is improved extrapolation to cycle lengths unseen during training, especially if the defect mode matches or beats Laplacian features with less than 5% runtime overhead.