Ordered Diffusion Message Passing

Integrate this at the edge-weight computation of a local graph neural network or point-cloud transformer. The inputs are node coordinates $x_i\in\mathbb{R}^d$ and node features $h_i$. First construct a $k$-nearest-neighbor graph using coordinates or a fixed encoder embedding. Add a small MLP $s_\theta(x_i)$ that outputs one scalar ordering value per node. For each edge $(i,j)$, calculate $d_{ij}=\|x_j-x_i\|^2$ and logits $\ell_{ij}=-d_{ij}/(4\varepsilon)+\beta(s_j-s_i)$. Apply a neighbor-wise softmax, $P_{ij}=\operatorname{softmax}_{j\in N_i}(\ell_{ij})$, then aggregate $m_i=\sum_{j\in N_i}P_{ij}V(h_j)$ and update $h_i'=h_i+\operatorname{MLP}([h_i,m_i])$. Initialize $\beta=0$ so optimization starts from ordinary isotropic diffusion. Set $\varepsilon$ from the median squared distance to the $k$th neighbor divided by four, and optionally learn a positive bandwidth through softplus. The mathematical quantities computed exactly are the Gaussian locality term, order difference, row normalization, and generator scaling $(P-I)/\varepsilon$; $s_\theta$ and $\beta$ are estimated by downstream-loss backpropagation. Clip logits and monitor neighbor entropy $H_i=-\sum_jP_{ij}\log P_{ij}$. First test on 2D and 10D nonlinear SDE trajectories with time labels hidden, training a small next-state predictor. Compare against symmetric Gaussian message passing and standard graph attention at equal neighbors, width, and FLOPs. Success means lower next-state MSE, better 50-step rollout error, and stable non-collapsed entropy.