Essential-Edge Harmonic Attention

Integrate the method at the edge-logit computation of a graph transformer or graph attention network. First construct a fixed filtration for every input graph, preferably by increasing edge weight, geometric distance, or a supplied node/edge scalar. Compute one-dimensional persistent bars and retain the top K bars or bars whose persistence \(t_b-s_b\) exceeds a threshold. For every retained bar, construct a linear constraint system \(A_bz=d_b\), with one variable per graph edge and constraints encoding the representative cycle and the filtration endpoints. Compute the least-norm representative with the pseudoinverse formula \(z_b=A_b^{\mathsf T}(A_bA_b^{\mathsf T}+\lambda I)^{-1}d_b\), where \(\lambda\) is a small numerical ridge, then normalize z_b by its Euclidean norm. Accumulate \(q_e=\sum_bw_b|z_{b,e}|\), standardize q independently within each graph, and stop gradients through q in the initial experiment. Replace each attention logit \(Q_iK_j^{\mathsf T}/\sqrt d\) by that logit plus \(\alpha\log(q_{ij}+\epsilon)\), with learnable \(\alpha\) initialized at zero. Also test a regularizer \(L_{\mathrm{harm}}=-\sum_{ij}q_{ij}\bar a_{ij}\), where \(\bar a_{ij}\) is mean attention over heads. The paper supplies the least-norm objective, essential-content inequality, and one-dimensional coefficient-gap rationale; filtration choice, K, threshold, ridge, and alpha are empirical. Begin with synthetic graphs containing one predictive cycle and distractor chords, using a small graph transformer. Compare against the unmodified model, Laplacian positional encodings, explicit cycle-count features, and randomized q with the same histogram. Measure accuracy, attention mass on known essential edges, robustness after adding non-essential shortcuts, and performance after essential-edge deletion. Success is higher test accuracy or stronger edge-deletion robustness at equal FLOPs, with attention and learned edge masks concentrating on essential cycle edges.