{
 "artifacts": null,
 "category": "dynamics",
 "description": "Use a fixed sparse graph for local message passing, but let each edge input be generated recursively from non-adjacent node states or latent states. This represents long-range interactions without densifying the graph, while retaining an explicit separation between local edge physics and learned global feedback.",
 "download_zip": "https://synthcore.org/download/1120",
 "formulas_latex": [
  "$$P_j=f_j\\left(x_{j}^{\\mathrm{tail}},x_{j}^{\\mathrm{head}},u_j\\right)$$",
  "$$u_j^{(k)}=g_{\\phi,j}\\left(x_{t(j)}^{(k)},x_{h(j)}^{(k)},z_j^{(k)},r^{(k)}\\right),\\qquad z_j^{(k+1)}=a_\\phi\\left(z_j^{(k)},\\operatorname{Pool}_{\\ell\\notin\\{t(j),h(j)\\}}x_\\ell^{(k)}\\right)$$",
  "$$P_j^{(k)}=f_{\\theta,j}\\left(x_{t(j)}^{(k)},x_{h(j)}^{(k)},u_j^{(k)}\\right),\\qquad x^{(k+1)}=F_\\theta\\left(x^{(k)},B P^{(k)}\\right)$$",
  "$$\\rho\\left(J_F(x)\\right)\u003c1\\ \\Longrightarrow\\ \\text{local contraction},\\qquad J_F(x)=\\frac{\\partial F_\\theta}{\\partial x}$$"
 ],
 "id": 2869,
 "implementation": "1. Integration point: replace the edge MLP in a standard message-passing GNN or graph neural ODE with two modules. The first is a recursive edge-context module g_phi; the second is the local edge-flow network f_theta. Keep the physical or observed graph sparse and do not add all-pairs edges. Initialize z_j to zero or to an edge-type embedding. 2. Pseudocode: at rollout step k, compute a pooled global summary r from node states; update each edge context with z_j = GRU(z_j, [x_tail, x_head, r]); compute u_j = MLP([x_tail, x_head, z_j, r]); compute P_j = EdgeMLP([x_tail, x_head, u_j]); aggregate q = B P; update x = NodeMLP([x, q]). Train with one-step loss plus a multi-step rollout loss. Estimate the largest Jacobian singular value or spectral radius by power iteration and optionally penalize values above one. 3. Computed from the paper: endpoint-conditioned edge flow and recursive state-to-input feedback. Estimated empirically: the feedback network, the rollout Jacobian, and the useful recursion depth. 4. First cheap experiment: use a sparse ring of coupled oscillators whose ground truth includes a distant coupling. Compare a local GNN, a dense GNN, and this model at equal parameter count. The predicted signature is a sharp rollout-stability boundary near rho(J_F) = 1: models below one should have bounded or geometrically decaying perturbations, while models above one should show exponential long-horizon divergence. At fixed sparse edge count, nonlocal-feedback error should approach dense-GNN error without proportional edge-count growth.",
 "math_summary": "For a directed edge e_j with tail node t(j), head node h(j), endpoint states x_t(j) and x_h(j), and scalar edge input u_j, the paper models its flow as P_j = f_j(x_t(j), x_h(j), u_j). Adapt this to a GNN by setting u_j^(k) = g_phi(x_t(j)^(k), x_h(j)^(k), z_j^(k), r^(k)), where z_j is a recursively propagated nonlocal summary and r is an optional global readout. The edge flow is P_j^(k) = f_theta(x_t(j)^(k), x_h(j)^(k), u_j^(k)), and node states update by x^(k+1) = F_theta(x^(k), B P^(k)), where B is the signed node-edge incidence matrix. The relevant stability signature is the spectral radius of the rollout Jacobian: a local rollout is contractive when rho(J_F) is less than one.",
 "math_tags": [
  "graph-theory",
  "dynamical-systems",
  "control-theory",
  "linear-algebra"
 ],
 "ml_areas": [
  "graph-nn",
  "world-model",
  "training-dynamics"
 ],
 "paper": {
  "arxiv_id": "2608.30157",
  "arxiv_url": "https://arxiv.org/abs/2608.30157",
  "summary_what_math_gives_to_ml": "The paper contributes two constructive graph-modeling mechanisms: recursive state-to-input feedback for representing non-adjacent dependencies without adding graph edges, and parallel decomposition of a composite edge into single-input edges while preserving vertex-level power conservation. These mechanisms transfer naturally to graph neural networks by separating topology from learned long-range feedback and by enforcing conservation through signed incidence aggregation. The strongest neural-network use is in physical or relational forecasting, where the constructions predict measurable reductions in conservation residuals and long-horizon rollout drift rather than relying only on benchmark accuracy.",
  "title": "Multi-Domain Graph-Based Modeling of Energy Systems with Applications to Lithium-Ion Batteries",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 6,
  "usefulness": 7
 },
 "solves": [
  "accuracy",
  "scalability",
  "stability"
 ],
 "title": "Recursive Nonlocal Edge Feedback GNN",
 "url": "https://synthcore.org/idea/2869/recursive-nonlocal-edge-feedback-gnn",
 "verification": {
  "peer_reviewed": false,
  "status": "queued_mechanism",
  "status_label": "Queued — mechanism check",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}