Kinetic Forest Sparsification

Integration point: apply the mechanism to a sparse graph neural network, locally connected CNN, or attention module whose candidate edges form an explicitly bipartite graph. Maintain a learnable score \(a_e\) for every candidate edge and sample a binary mask. A practical exact forest sampler uses union–find: compute \(s_e=a_e+\mu+g_e\), with independent Gumbel noise \(g_e\); sort edges by decreasing score; activate an edge only when its endpoints currently belong to different connected components. This implements the no-cycle constraint in near-linear time. Use the masked layer \(h'_v=\phi(\sum_{u:(u,v)\in E}c_{uv}W_{uv}h_u)\), and use a straight-through estimator for the discrete mask. The task loss is augmented only by an optional entropy penalty on \(p_e=\sigma(a_e)\). Slowly sweep \(\mu\), or update it by dual control to reach a target edge density. The paper supplies the forest constraint, kinetic rule, partition-function form, and cubic transition estimate; the neural critical point must be measured rather than assumed to equal -3.252. First experiment: train a two-layer GNN on Cora or a synthetic bipartite node-classification task, comparing dense, random-pruned, magnitude-pruned, and annealed-forest models at matched edge counts. Sweep \(\mu\), recording validation loss, gradient norm, active density, component count, and \(m\). The falsifiable prediction is a reproducible kink or continuous rapid increase in \(m\) at a calibrated \(\mu_c\), with \(\mu_c\) varying predictably under graph degree changes; forest masks should also reduce gradient amplification relative to matched random cyclic masks. Reject the mechanism if no order-parameter transition appears across seeds or if stability is not improved at equal sparsity.