{
 "artifacts": null,
 "category": "dynamics",
 "description": "Use unit-gain directed propagation as a drop-in graph filter whose spectral radius is provably bounded by that of the underlying nonnegative adjacency. Add an inexpensive power-iteration monitor or rescaling rule so learned graph propagation remains no more expansive than the corresponding magnitude-only operator.",
 "formulas_latex": [
  "$$\\left|A(\\Phi)\\right|=A(D),\\qquad \\rho\\!\\left(A(\\Phi)\\right)\\le \\rho\\!\\left(\\left|A(\\Phi)\\right|\\right)\\le\\rho\\!\\left(A(D)\\right).$$",
  "$$\\text{If }|M|\\le N\\text{ and }N\\ge0,\\qquad \\rho(M)\\le\\rho(|M|)\\le\\rho(N).$$",
  "$$P_\\alpha=(1-\\alpha)I+\\alpha D_+^{-1}A(\\Phi),\\qquad h^{(\\ell+1)}=f\\!\\left(P_\\alpha h^{(\\ell)}W_\\ell\\right).$$",
  "$$x_{t+1}=P_\\alpha x_t/\\|P_\\alpha x_t\\|_2,\\qquad \\widehat\\rho_t=\\|P_\\alpha x_t\\|_2/\\|x_t\\|_2.$$"
 ],
 "id": 106,
 "implementation": "1. Integration point: modify the propagation matrix in a directed GNN, graph preprocessing step, or graph diffusion block. For each edge use A_uv=e^{iθ_uv}, multiply by inverse out-degree D_+^{-1}, and retain a real-valued output projection after the complex propagation. 2. Pseudocode: form P=(1−α)I+αD_inv@A_gain; draw a fixed random complex probe x with unit norm; run K=5–10 power iterations x←P@x/norm(P@x), recording rho_hat=norm(P@x); if rho_hat\u003er_target, replace P by (r_target/(rho_hat+ε))*P. Apply h_next=equivariant_activation(P@h@W). Compute the monitor every several hundred optimizer steps, with gradients stopped through rho_hat, or use a soft penalty max(0,rho_hat−r_target)^2. 3. The exact comparison theorem is used for the unnormalized gain adjacency; rho_hat is an empirical estimate for the normalized residual operator, and separately estimate rho(A(D)) using the same power iteration. 4. First cheap experiment: train 2- to 6-layer directed GCNs on Cora, PubMed, and a synthetic long-chain graph, comparing ordinary real adjacency propagation, unconstrained complex gains, and this rescaled version. Record exploding activations, gradient norms, validation accuracy, and feature oversmoothing versus depth. The target signal is fewer exploding runs and stable training at depths where the unconstrained phase model diverges, without reducing accuracy or requiring smaller learning rates.",
 "math_summary": "The gain adjacency A(Φ) has entries of modulus one wherever the underlying digraph has an arc, hence |A(Φ)|=A(D) entrywise for the unweighted underlying adjacency A(D). The paper recalls the Perron-Frobenius comparison result: if N is nonnegative and |M|≤N, then ρ(M)≤ρ(|M|)≤ρ(N), where ρ(M)=max_{λ∈spec(M)}|λ| is spectral radius. Therefore ρ(A(Φ))≤ρ(A(D)). For a propagation operator P_α=(1−α)I+αD_+^{-1}A(Φ), estimate its spectral radius by power iteration and rescale it to a target r≤1. The guarantee applies directly to the raw gain adjacency; the normalized or residual operator should be monitored empirically because degree normalization and self-loops alter the comparison matrix.",
 "math_tags": [
  "spectral-theory",
  "linear-algebra",
  "dynamical-systems"
 ],
 "ml_areas": [
  "graph-nn",
  "training-dynamics",
  "optimizer"
 ],
 "paper": {
  "arxiv_id": "2608.23655",
  "arxiv_url": "https://arxiv.org/abs/2608.23655",
  "summary_what_math_gives_to_ml": "The paper provides a principled way to attach unit-modulus complex phases to directed graph edges while preserving a useful gauge symmetry: node-wise phase reparameterizations change individual edge gains but leave cycle products invariant. This suggests graph-neural layers whose messages are complex and gauge-equivariant, allowing models to represent directional frustration, circulation, and phase consistency rather than only edge magnitudes. The spectral comparison theorem also gives a safety guarantee: phase-weighted propagation cannot have larger spectral radius than the nonnegative underlying adjacency, which can be used to build stable graph filters or constrain learned propagation operators.",
  "title": "On Spectra of $\\mathbb{T}$-Gain Digraphs",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 4,
  "novelty": 6,
  "usefulness": 6
 },
 "solves": [
  "stability",
  "scalability"
 ],
 "title": "Spectrally Safe Phase Propagation",
 "url": "https://synthcore.org/idea/106/spectrally-safe-phase-propagation",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)",
  "verification_axes": {
   "benchmark_mechanism": {
    "confirmed": null,
    "tested": false
   },
   "practical_benchmark": {
    "beats_baseline": null,
    "tested": false
   },
   "toy_mechanism_gate": {
    "confirmed": null,
    "tested": false
   }
  }
 }
}