Spectral-Gap Synchronizing Neural Graph Dynamics

Replace independent latent ODE or recurrent updates in a graph neural network with a coupled flow, $z_i^{k+1}=z_i^k+h[f_\theta(z_i^k,x_i)-\kappa\sum_jL_{ij}z_j^k]$. Keep the node model shared across nodes and expose $\kappa$ as a learned or scheduled parameter. Compute the graph Laplacian and estimate its second eigenvalue $\lambda_2$ with a sparse Lanczos iteration. Evaluate the node Jacobian contraction rate $\alpha$ on minibatches, then choose $\kappa$ for a target disagreement rate $r$ using $\kappa=\max(0,(r-\alpha)/\lambda_2)$. Add a penalty when the measured transverse generalized eigenvalue exceeds $-(\alpha+\kappa\lambda_2)$. The Laplacian and spectral gap are computed from the graph; $\alpha$ is obtained from the Hermitian certificate or finite-time perturbation decay; nonlinear deviations are empirical. The first cheap experiment should use a 20-node graph to predict trajectories generated by a stable complex oscillator, comparing path, ring, Erdős–Rényi, and complete graphs at matched node count and edge budget where possible. Sweep $\kappa$ while holding the node model and solver tolerance fixed. The falsifiable prediction is that the semilog slope of disagreement energy grows approximately linearly with $\kappa\lambda_2(L)$ and agrees with $\alpha+\kappa\lambda_2(L)$ within 20%. Increasing $\lambda_2$ by rewiring should accelerate synchronization, while the average-node trajectory should remain largely unchanged because Laplacian coupling vanishes in the consensus mode.