Reliability-gated Laplacian positional encodings

(1) Integration point: add this module before message passing in a GNN or before graph-token attention in a graph transformer. Given node coordinates or any metric $\rho$, construct the weighted graph at one or several candidate bandwidths $h$; compute the normalized or unnormalized empirical Laplacian, its first $K+1$ nontrivial eigenpairs $(\hat\lambda_k,u_k)$, and concatenate the gated coordinates $z_i$ to node features. Keep the gating differentiability optional: detach $r_k$ initially.

(2) Pseudocode:

for h in candidate_bandwidths:
    A[i,j] = kernel(rho(x_i,x_j)/h)
    L = degree(A) - A
    lam, U = eigsh(L, K+1, smallest_nonzero=True)
    vhat = min_i mean_j[rho(x_i,x_j) <= h]
    for k in 1..K:
        gap = min(lam[k]-lam[k-1], lam[k+1]-lam[k])
        r[k] = min(1, tau*gap*sqrt(n*vhat)/(lam[k]+eps))
    Z_h = U[:,1:K+1] * r
select h with highest median(r) subject to median(lam[:K]) < c/h**2
feed concat(node_features, Z_h) to the network

(3) The theorem supplies the $1/\sqrt{n v_\mu(h)}$ reliability scale, the low-frequency cutoff $h^{-2}$, and the need for an eigengap. The implementation estimates unknown population quantities using empirical local counts, eigenvalues, and gaps; $\tau$, $c$, and $K$ are validation parameters. Use randomized Lanczos for large graphs and cache the eigenpairs when graph structure is fixed.

(4) First experiment: train a 3-layer GCN and a graph transformer on a small molecule or synthetic geometric-graph node-classification benchmark, comparing ordinary Laplacian positional encodings against gated encodings at equal $K$ and FLOPs. Vary training-set size and add isolated or low-density regions. Success means higher validation accuracy and lower run-to-run variance at small $n$, with the largest gain on nodes where $\widehat v(h)$ is low; also test whether discarding modes with $\hat\lambda_k\gtrsim h^{-2}$ improves accuracy without increasing inference cost.