Reactive Mass-Weighted Message Passing

(1) Replace one graph-attention or graph-message-passing block with this layer. The input is X with shape N by d, and the layer maintains one scalar log-mass r_i per node. Initialize r_i to zero or compute it from a learned linear projection of the node feature. Use a directed adjacency matrix W with entries w_ij; W may be the observed graph adjacency, a learned blockwise matrix, or a low-rank product U V transposed. (2) Pseudocode: compute D_ij=X_i-X_j for every permitted edge; compute transport messages T_ij=K1_theta(D_ij) in R^d and inhibition scores H_ij=softplus(K2_phi(D_ij)); set M_i=exp(clip(r_i,-10,10)) and p_i=M_i/(sum_k M_k+epsilon); compute dx_i=sum_j p_j W_ij T_ij and q_i=sum_j p_j W_ij H_ij; update X_i <- X_i+h dx_i and r_i <- r_i+h(A-q_i); add X to the original representation through a residual connection and apply normalization. (3) The paper contributes the coupled transport/reaction form, directed weights, and positivity-preserving multiplicative dynamics. Estimate or tune A, h, epsilon, the kernel MLP parameters, and the rank of W empirically. For sparse graphs compute only existing edges; for dense interactions use W=U V transposed or a small number of learned node-type blocks. Clip r only for numerical protection, and monitor whether p collapses onto a few nodes. (4) First run a cheap experiment on ZINC or ogbn-arxiv using a 10-layer GraphSAGE or graph-transformer baseline with the same hidden width and training budget. Compare ordinary uniform message passing, fixed learned attention weights, and reactive masses. Record validation accuracy, loss at equal steps and FLOPs, gradient-norm variance, maximum p_i, and runtime. A positive result is higher validation accuracy or faster loss descent without mass collapse; with low-rank W, also require lower interaction cost at equal accuracy.