Coherence-to-Diffusion Graph Layer

Integrate the mechanism as a residual graph layer or as the recurrent transition of a graph state-space model. Construct a sparse graph from the input graph, token locality, or a k-nearest-neighbor embedding graph. Parameterize an unconstrained matrix Q and set K equal to Q minus Q transpose; normalize K by its spectral norm if necessary. For the current feature matrix h, compute p equal to sigmoid(h multiplied by w plus b). For every edge, compute r_ij equal to gamma times one plus alpha times p_i plus p_j, then construct the sparse weighted Laplacian L_p. Use the update h_new equal to h plus dt times a K h minus gamma L_p h. Pseudocode is: p = sigmoid(h @ w + b); rates = gamma * (1 + alpha * (p_i + p_j)); L = weighted_laplacian(rates); h = h + dt * (a * K @ h - gamma * L @ h). For improved stability, use operator splitting: first apply a Cayley approximation to the skew channel, h <- solve(I - dt*a*K/2, (I + dt*a*K/2) @ h), then apply an implicit or sufficiently small-step Laplacian update. Compute lambda_max of L_p with five power-iteration steps and enforce dt times gamma times lambda_max below one for explicit Euler. The paper supplies the coherent/dephasing separation and energy mechanism; the neural parameterization and finite-step implementation are adaptations. First test a 20-layer graph MLP on Cora and on a synthetic long-range diffusion task, comparing a standard GCN, a purely skew layer, and this hybrid at matched parameter count. Sweep gamma with a fixed a equal to one. Measure feature influence as a function of graph distance. The quantitative prediction is a crossover near distance ell_c equal to one divided by the square root of gamma: oscillatory or sign-changing influence should dominate below this distance, while influence should become monotone and smoothing above it. A second prediction is that the continuous-time feature energy never increases; under the explicit-step condition, measured norm growth should remain below numerical tolerance, while instability should appear as dt times gamma times lambda_max approaches or exceeds one.